A 2b 3c D-3 1 Find the Homogeneous Solution

I. INTRODUCTION

Section:

It is a well-known fact that the Einstein equations of the gravitational field do not appear to be generically integrable. In order to find exact solutions, one needs to consider spacetimes with symmetries. In Ref. 1 1. H. Stephani, D. Kramer, E. H. M. MacCallum, and E. Herlt, Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge, 2001). , there is a survey including the exact solutions of Einstein equations. One of such cases of symmetries of spacetime is when the spacetime has two Killing vectors, then, the field equations reduce to the so-called Ernst equation. ² 2. F. J. Ernst, Phys. Rev. 167, 1175 (1968). https://doi.org/10.1103/physrev.167.1175 This equation has been discussed in Ref. 3 3. C. Klein and O. Richter, Ernst Equation and Riemann Surfaces, Lecture Notes in Physics 685 (Springer, Berlin, Heidelberg, 2005). . Another interesting and well-known case of reducing Einstein equations is the case of the so-called, second heavenly equation. It was derived by Plebański in Ref. 4 4. J. F. Plebański, J. Math. Phys. 16, 2395 (1975). https://doi.org/10.1063/1.522505 . The symmetries of this equation were investigated in Ref. 5 5. C. P. Boyer and P. Winternitz, J. Math. Phys. 30, 1081 (1989). https://doi.org/10.1063/1.528379 . In Ref. 6 6. B. Abraham-Shrauner, Phys. Lett. A 369, 299 (2007). https://doi.org/10.1016/j.physleta.2007.04.100 , some hidden symmetry of type II and some exact solutions of the second heavenly equation were obtained by a reduction of this equation to the homogeneous Monge–Ampère equation in similarity variables. One of the very important classes of solutions of elliptic complex Monge–Ampère equation are the solutions, which generate metrics, not possessing the Killing vector because such solutions are non-invariant and, thus, are the candidates for K3 gravitational instantons. ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 Gravitational instantons can be defined analogically to instantons in Yang–Mills theory, as solutions of Einstein equations, non-singular on a section of complexified spacetime, where the curvature decays at large distances. ¹⁵ 15. S. W. Hawking, Phys. Lett. A 60, 81 (1977). https://doi.org/10.1016/0375-9601(77)90386-3 The most wanted gravitational instanton is the so-called Kummer surface K3. ^7,16 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 16. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003). https://doi.org/10.1088/0305-4470/36/39/304 Some gravitational instantons were obtained from a reduction of complex Monge–Ampère equation to a 2-dimensional Monge–Ampère equation, in Ref. 17 17. Y. Nutku, M. B. Sheftel, and A. A. Malykh, Classical Quantum Gravity 14, L59 (1997). https://doi.org/10.1088/0264-9381/14/3/004 . K3 with a Ricci-flat Calabi–Yau metric and the complex torus with the flat metric are the only compact 4-dimensional Riemannian hyper-Kähler manifolds. ¹⁸ 18. M. Dunajski, Solitons, Instantons and Twistors (Oxford University Press, 2010). Finding of the explicit form of the metric corresponding to the Kummer surface K3 is a challenging problem, among others, because of the requirement of non-existence of the Killing vector for such metric, which implicates the requirement of non-invariance of the solution of the homogeneous elliptic complex Monge–Ampère equation. This stated the motivation of our looking for non-invariant solutions of the homogeneous elliptic complex Monge–Ampère equation; however, as it has turned out, it is hard to find simultaneously non-invariant and real solutions in this case and just such solutions can describe the Kummer surface. Some exact solutions of the elliptic complex Monge–Ampère equation, generating metrics not admitting the Killing vectors, were obtained in Ref. 16 16. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003). https://doi.org/10.1088/0305-4470/36/39/304 . In Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , the real solutions of the hyperbolic complex Monge–Ampère equation were found and their non-invariance was also checked. Some non-invariant solutions for the hyperbolic complex Monge–Ampère equation were found in Ref. 8 8. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Theor. 40, 9371 (2007). https://doi.org/10.1088/1751-8113/40/31/014 , by a lift of non-invariant solutions of the Boyer–Finley equation. A lift of invariant solutions of complex Monge–Ampère equations: elliptic and hyperbolic to non-invariant solutions of the latter equations in four dimensions, provided by the partner symmetries of these Monge–Ampère equations, was presented in Ref. 13 13. M. B. Sheftel and A. A. Malykh, J. Nonlinear Math. Phys. 15(sup3), 385 (2008). https://doi.org/10.2991/jnmp.2008.15.s3.37 . A symmetry reduction of second heavenly equation to a two-component (2+1)-dimensional multi-Hamiltonian integrable system was done in Ref. 14 14. D. Yazici and M. B. Sheftel, J. Nonlinear Math. Phys. 15(sup3), 417 (2008). https://doi.org/10.2991/jnmp.2008.15.s3.40 . A formal solution of the Cauchy problem for the second heavenly equation was presented in Ref. 9 9. S. V. Manakov and P. M. Santini, Phys. Lett. A 359, 613 (2006). https://doi.org/10.1016/j.physleta.2006.07.011 . In Ref. 19 19. M. Robaszewska, Commun. Anal. Geom. 16, 437 (2008). https://doi.org/10.4310/cag.2008.v16.n2.a5 , it was shown that one was able to describe locally, non-degenerate complex surfaces by a solution for some Monge–Ampère type equations. Some exact solutions of the multidimensional Monge–Ampère equation were found in Ref. 20 20. V. M. Fedorchuk and O. S. Leibov, Ukr. Math. J. 48, 775 (1996). https://doi.org/10.1007/bf02384226 by using the subgroup of the generalized Poincarè group P(1, 4). In Ref. 21 21. A. G. Kushner, Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki 151, 60 (2009). , a reduction of the Monge–Ampère type equation to the Euler–Poisson equation was studied. Some aspects of the complex Monge–Ampère equation (among others, the existence and stability of its weak solutions on compact Kähler manifolds) were studied in Ref. 23 23. S. Kołodziej, Mem. Am. Math. Soc. 178, 840 (2005). https://doi.org/10.1090/memo/0840 . In Ref. 22 22. B. Doubrov and E. V. Ferapontov, J. Geom. Phys. 60, 1604 (2010); arXiv:0910.3407. https://doi.org/10.1016/j.geomphys.2010.05.009 , the integrability of the symplectic Monge–Ampère equation was studied and some functionally invariant solutions were found. A relation between the notion of complete exceptionality (introduced in Ref. 10 10. G. Boillat and T. Ruggeri, Boll. Un. Mat. Ital. A 15, 197 (1978). ), and the definition of the Monge–Ampère equation, was investigated in Ref. 11 11. G. Moreno, Banach Cent. Publ. 113, 275 (2017). https://doi.org/10.4064/bc113-0-15 . In Ref. 24 24. Z. Kuznetsova, Z. Popowicz, and F. Toppan, J. Phys. A: Math. Gen. 38, 7773 (2005); arXiv:hep-th/0504214. https://doi.org/10.1088/0305-4470/38/35/011 , the sl(2n|2n)¹ super-Toda-lattices and the heavenly equations as the continuum limit were investigated. The N = 2 heavenly equation was studied in Ref. 25 25. F. Toppan, Czech J. Phys. 54, 1387 (2004). https://doi.org/10.1007/s10582-004-9806-y . Symmetries and conservation laws for the second heavenly equation were studied in Ref. 12 12. F. Neyzi, M. B. Sheftel, and D. Yazici, Phys. At. Nucl. 70, 584 (2007). https://doi.org/10.1134/s1063778807030209 . In Ref. 26 26. A.-M. Wazwaz, Proc. Rom. Acad., Ser. A 17, 210 (2016). , the multikink solutions of the equations: second heavenly and asymmetric heavenly were obtained. In Ref. 27 27. M. B. Sheftel and A. A. Malykh, J. Phys. A: Math. Theor. 42, 395202 (2009). https://doi.org/10.1088/1751-8113/42/39/395202 , a classification of scalar partial differential equations of the second order, non-invariant solutions of the mixed heavenly equation, and a connection between this equation and Husain equation were presented. On the other hand, there in Ref. 28 28. M. Jakimowicz and J. Tafel, Classical Quantum Gravity 23, 4907 (2006). https://doi.org/10.1088/0264-9381/23/15/010 , it was shown that every solution of the Husain equation (related to the chiral model of self-dual gravity ²⁹ 29. V. Husain, Phys. Rev. Lett. 72, 800 (1994). https://doi.org/10.1103/physrevlett.72.800 ) defined some solution of the well-known Plebański first heavenly equation.

In addition, in Ref. 27 27. M. B. Sheftel and A. A. Malykh, J. Phys. A: Math. Theor. 42, 395202 (2009). https://doi.org/10.1088/1751-8113/42/39/395202 , the so-called asymmetric heavenly equation was derived. This equation is connected to the so-called evolution form of the second heavenly equation. ^30,31,32 30. J. D. Finley, J. F. Plebański, M. Przanowski, and H. García-Compeán, Phys. Lett. A 181, 435 (1993). https://doi.org/10.1016/0375-9601(93)91145-u 31. J. F. Plebański and M. Przanowski, Phys. Lett. A 212, 22 (1996). https://doi.org/10.1016/0375-9601(96)00025-4 32. C. P. Boyer and J. F. Plebański, J. Math. Phys. 18, 1022 (1977). https://doi.org/10.1063/1.523363 In Ref. 30 30. J. D. Finley, J. F. Plebański, M. Przanowski, and H. García-Compeán, Phys. Lett. A 181, 435 (1993). https://doi.org/10.1016/0375-9601(93)91145-u , the evolution form of the second heavenly equation was derived by some symmetry reductions of Lie algebra of the area preserving group of diffeomorphisms of the two-surface Σ² of self-dual Yang–Mills equations. The same result was obtained in the case of the second heavenly equation in Ref. 33 33. J. F. Plebański, M. Przanowski, B. Rajca, and J. Tosiek, Acta Phys. Pol., B 26, 889 (1995). . In Ref. 34 34. A. A. Malykh and M. B. Sheftel, J. Phys. A: Math. Theor. 44, 155201 (2011); arXiv:1011.2479. https://doi.org/10.1088/1751-8113/44/15/155201 , it was shown that the so-called general heavenly equation governed anti-self-dual (ASD) gravity, and also some exact solutions of these equations were presented.

In this paper, we show that by applying the so-called decomposition method (one should not confuse it with Adomian's decomposition method, presented among others, in Ref. 35 35. G. Adomian, Solving Frontiers Problems of Physics: The Decomposition Method (Springer Science+Business Media, Dordrecht, 1994). ), it is possible to find some new classes of the exact non-invariant solutions (the so-called functionally invariant solutions) of hyperbolic complex Monge–Ampère equation and the second heavenly equation. Additionally, we apply the decomposition method in order to find some new solutions of the elliptic complex Monge–Ampère equation. The main version of the decomposition method, mentioned above, was presented in Ref. 36 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075 .

This paper is organized as follows. In Sec. II, we briefly describe the procedure of the decomposition method (of course, one should not confuse this method with the Bogomolny decomposition, which was obtained for some nonlinear models in field theory, in Refs. 37–39 37. K. Sokalski, Acta Phys. Pol., A 65, 457 (1984). 38. Ł. Stȩpień, K. Sokalski, and D. Sokalska, J. Phys. A: Math. Gen. 35, 6157 (2002). https://doi.org/10.1088/0305-4470/35/29/315 39. Ł. T. Stȩpień, in Current Trends in Analysis and its Applications, edited by V. V. Mityushev and M. V. Ruzhansky (Springer International Publishing, 2015), p. 273. ). Section III includes a short introduction of the heavenly equations mentioned above and to the non-invariance of the solutions of these equations. In Sec. IV, we find by using the mentioned decomposition method, the classes of exact solutions of the mentioned equations. Next, in this same section, we investigate the non-invariance of the solutions belonging to the classes mentioned above.

We formulate also a criterion for the non-invariance of the solutions belonging to the found classes, and we show that the metrics generated by the found solutions for the hyperbolic complex Monge–Ampère equation and the second heavenly equation do not admit Killing vectors. In Sec. V, we give some conclusions. The current paper includes a part of the results included in Ref. 40 40. Ł. T. Stȩpień, arXiv:1208.2905 (2012). .

II. A SHORT DESCRIPTION OF THE DECOMPOSITION METHOD

Section:

Now, we shortly describe the decomposition method, introduced for the first time in Ref. 36 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075 .

Let us assume that we have to solve some nonlinear partial differential equation,

$$F\left(x^{\mu },u_1,\dots ,u_m,u_{1,x^{\mu }},\dots ,u_{m,x^{\mu }},u_{1,x^{\mu },x^{\nu }},\dots \right)=0,$$

(1)

where $u_{n,x^{\alpha }}=\frac{\partial u_n}{\partial x^{\alpha }},\cdots$

According to the assumptions of the decomposition method, which was presented first time in Ref. 36 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075 , first, we check whether it is possible to decompose the equation on the fragments characterized by a homogenity of derivatives. For example, such a decomposed investigated equation may be as follows: ³⁶ 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075

$$G_1\cdot \left[{\left(u_{,x}\right)}^2+{\left(u_{,y}\right)}^2\right]+G_2\cdot \left[u_{,xx}+u_{,xy}\right]=0,$$

(2)

where u = u(x, y) is some function of class ${\mathcal{C}}^2$ and the terms G ₁, G ₂ may depend on $x^{\mu },u,u_{x^{\mu }},\dots$ and $u\in \mathbb{R}$ or $u\in \mathbb{C}$ , depending on an investigated problem.

We see that the result of the checking is positive, and, then, we substitute into (2), the following ansatz: ³⁶ 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075

$$u\left(x^{\sigma }\right)={\beta }_1+f\left(a_{\mu }{x}^{\mu }+{\beta }_2,b_{\nu }{x}^{\nu }+{\beta }_3,c_{\rho }{x}^{\rho }+{\beta }_4\right).$$

(3)

In the above ansatz, we try to keep f as an arbitrary function (of class ${\mathcal{C}}^2$ ), so far as it is possible. The class of the solutions given by the ansatz of such kind is called, in the literature, as functionally invariant solution. ⁴¹ 41. S. L. Sobolev, Travaux Inst. Physico–Math. Stekloff, Acad. Sci. 5, 259 (1934). The function f depends on the appropriate arguments like a _μ x ^μ + β ₂. In this paper, a _μ x ^μ = a ₁ x ¹ + a ₂ x ² + a ₃ x ³ + a ₄ x ⁴. The coefficients a _μ , b _ν , and c _ρ may be in general complex numbers, which are to be determined later, β _j may be in general complex constants, j = 1, …, 4, and μ, ν, ρ = 1, …, 4. In general, the set of values of μ, ν, ρ, depends on the investigated equation. We can decrease or increase the number of the arguments of the function f in (3) and also modify the form of ansatz (3), depending on the situation.

We make such a modification later in this section and in Sec. IV.

After substituting the ansatz (3) in the example Eq. (2), instead of partial derivatives of u, the derivatives of the function f appear: D ₁ f, D ₂ f, …, D _1,1 f, D _1,2 f, …, where the indices denote differentiation with respect to the first and, so far, arguments of the function f (like a _μ x ^μ + β ₂).

For example, if we substitute a two-dimensional version of ansatz (3) in (2) and collect appropriate terms by the derivatives D _j f and D _j,k f, then, we get ³⁶ 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075

$$\begin{array}{ccc}\hfill & \hfill & G_1\cdot \left[\left(a_1^2+a_2^2\right){\left(D_{1}f\right)}^2+\left(b_1^2+b_2^2\right){\left(D_{2}f\right)}^2\right.\hfill \\ \hfill & \hfill & +\left(c_1^2+c_2^2\right){\left(D_{3}f\right)}^2+2\left(a_1{b}_1+a_2{b}_2\right)D_{1}f{D}_{2}f\hfill \\ \hfill & \hfill & \left.+2\left(a_1{c}_1+a_2{c}_2\right)D_{1}f{D}_{3}f+2\left(b_1{c}_1+b_2{c}_2\right)D_{2}f{D}_{3}f\right]\hfill \\ \hfill & \hfill & +G_2\cdot \left[\left(a_1^2+a_1{a}_2\right)D_{\mathrm{1,1}}f+\left(2a_1{b}_1+a_1{b}_2+a_2{b}_1\right)D_{\mathrm{1,2}}f\right.\hfill \\ \hfill & \hfill & +\left(2a_1{c}_1+a_1{c}_2+a_2{c}_1\right)D_{\mathrm{1,3}}f+\left(b_1^2+b_1{b}_2\right)D_{\mathrm{2,2}}f\hfill \\ \hfill & \hfill & \left.+\left(2b_1{c}_1+b_1{c}_2+b_2{c}_1\right)D_{\mathrm{2,3}}f+\left(c_1^2+c_1{c}_2\right)D_{\mathrm{3,3}}f\right]=0,\hfill \end{array}$$

where D _j f, D _j,k f denote, respectively, a derivative of the function f with respect to j − nary argument and the mixed derivative of this function with respect to j − nary and k − nary argument.

Now, we require that all algebraic terms in the parentheses must vanish. As a result, we obtain a system of algebraic equations, whose solutions are the parameters a ₁, a ₂, … We call such a system of algebraic equations as the determining algebraic system. Its solutions establish the relations between a _μ , b _ν , c _ρ and, therefore, they constitute, together with (3), some class of solutions of (2). Depending on the situation, we may need to take into consideration additionally some other conditions, which must be satisfied by our class of solutions. These conditions implicate the requirement of satisfying some algebraic equations, which we attach to the determining algebraic system. In this paper, an example of such additional condition is the condition of non-invariance of the solutions. The ansatz (3) appears as an effect of a generalization of some result, obtained in Refs. 37 37. K. Sokalski, Acta Phys. Pol., A 65, 457 (1984). and 38 38. Ł. Stȩpień, K. Sokalski, and D. Sokalska, J. Phys. A: Math. Gen. 35, 6157 (2002). https://doi.org/10.1088/0305-4470/35/29/315 . Namely, in the mentioned papers, some classes of exact solutions of Bogomolny decomposition (Bogomolny equations) for the Heisenberg model of the ferromagnet have been obtained (but by applying some other methods—the so-called concept of strong necessary conditions),

$$\omega =\omega \left[\left(i\alpha +\beta \gamma \right)x_1+\left(i\gamma -\alpha \beta \right)x_2+\left({\beta }^2-1\right)x_3\right],c.c.,$$

(4)

where ω is the arbitrary holomorphic function of class ${\mathcal{C}}^2$ , depending on its argument and $\omega =\frac{S^1+i{S}^2}{1+S^3}$ , S ⁱ (i = 1, 2, 3) —components of the classical Heisenberg spin, and α ² + β ² + γ ² = 1. Therefore, the ansatz (3) is a generalization of (4). The solution [given by the ansatz (3)] has been obtained for Heisenberg model, nonlinear σ model [or O(3) model], and scalar Born–Infeld-like equation in (3+1)-dimensions in Ref. 36 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075 and for Skyrme–Faddeev model in Ref. 39 39. Ł. T. Stȩpień, in Current Trends in Analysis and its Applications, edited by V. V. Mityushev and M. V. Ruzhansky (Springer International Publishing, 2015), p. 273. .

One can easily show that this method can be extended for the class of the equations of the arbitrary order

$$\begin{array}{ccc}\hfill \begin{array}{c}F\left(x^{\mu },u_1,\dots ,u_m,u_{1,x^{\mu }},\dots ,u_{m,x^{\mu }},u_{1,x^{\mu }{x}^{\nu }},\dots ,\right.\\ \left.u_{1,x^{{\alpha }_1}\dots x^{{\alpha }_n}},\dots ,u_{m,x^{\mu }{x}^{\nu }},\dots ,u_{m,x^{{\alpha }_1}\dots x^{{\alpha }_n}}\right)=0,\end{array}& \hfill & \hfill \end{array}$$

(5)

obviously, if the decomposition on the proper fragments, mentioned above, is possible.

Of course, this above decomposition method may be also applied for solving linear partial differential equations, homogeneous with respect to the derivatives.

Just now, in order to find classes of non-invariant solutions, we made a modification of (3). Therefore, the basic form of (3), which can be applied for all equations considered in this paper is

$$u\left(x^{\mu }\right)={\beta }_1+g_1\left({\mathrm{\Sigma }}_1\right)+g_2\left({\mathrm{\Sigma }}_2\right)+g_3\left({\mathrm{\Sigma }}_3\right)+g_4\left({\mathrm{\Sigma }}_4\right),$$

(6)

where

$$\begin{array}{c}{\mathrm{\Sigma }}_1=a_{\mu }{x}^{\mu }+{\beta }_2,{\mathrm{\Sigma }}_2=b_{\mu }{x}^{\mu }+{\beta }_3,\\ {\mathrm{\Sigma }}_3=c_{\mu }{x}^{\mu }+{\beta }_4,{\mathrm{\Sigma }}_4=d_{\mu }{x}^{\mu }+{\beta }_5,\end{array}$$

(7)

g _k , (k = 1, …, 4) are some functions; depending on the situation, they can be complex or real and we wish they were arbitrary functions, but it can change in some cases, a _μ x ^μ = a ₁ x ¹ + ⋯ + a ₄ x ⁴, where x ^μ are the independent variables. We assume that $g_k\in {\mathcal{C}}^2,\left(k=1,\dots ,4\right)$ (of course, we assume g _k are differentiable functions). In the cases of the second heavenly equation, we will extend the ansatz (6) to the functional series.

Of course, all equations, homogeneous with respect to the derivatives, can be solved by using the decomposition method (if there exists at least one solution of the determining algebraic system). However, it is possible that found solutions of the determining algebraic system will determine the classes of the solutions, which are useless from the physical viewpoint. Therefore, the problem of finding solutions of a given equation is reduced to the problem of solving of the determining algebraic system.

It should be also mentioned here that the first method of finding of functionally invariant solutions, applied to the wave equation, comes from, ⁴¹ 41. S. L. Sobolev, Travaux Inst. Physico–Math. Stekloff, Acad. Sci. 5, 259 (1934). but without the idea of the decomposition method, introduced for the first time in Ref. 36 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075 and applied in this paper. In Refs. 42 42. N. Erouguine, CR (Doklady) Acad. Sci. URSS (NS) 42, 371 (1944). and 43 43. N. P. Erugin and M. M. Smirnov, Differ. Uravn. 17, 853 (1981). , some extension (obtained by using a method, called also, as Erugin's method ⁴⁴ 44. S. V. Meleshko, Methods for Constructing Exact Solutions of Partial Differential Equations (Springer, 2005). ) of the results obtained in Ref. 41 41. S. L. Sobolev, Travaux Inst. Physico–Math. Stekloff, Acad. Sci. 5, 259 (1934). has been presented. In Ref. 45 45. O. F. Men'shikh, Dokl. Akad. Nauk SSSR 205, 30 (1972). , some other approach to finding functionally-invariant solutions was presented and applied for nonlinear partial differential equations of second order, and, in Ref. 46 46. O. F. Men'shikh, Differ. Uravn. 11, 533 (1975). , some analogical results were obtained for some kind of quasilinear partial differential equations of the second order.

However, the method of looking for the solutions of the form (3), introduced, for the first time, in Ref. 36 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075 , "looks" at the investigated nonlinear partial differential equation, by the viewpoint of homogenity of some fragments of this equation, with respect to the derivatives and so it differs from these methods mentioned above. By comparison with the methods mentioned above, it seems to be more simple than they are. Moreover, we have stated above that the decomposition method can be applied for the partial differential equation of arbitrary order, if this equation can be decomposed on proper fragments, mentioned above. In Ref. 47 47. Ł. T. Stȩpień, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics, edited by V. S. Gerdjikov, K. Sekigawa, and S. Dimiev (World Scientific, 2009), p. 210. , some extension of this method (we apply among others, this version in the current paper) was presented.

III. HEAVENLY EQUATIONS

Section:

A. Complex Monge–Ampère equations

The Einstein vacuum equation in the complex four-dimensional Riemann space together with the constraint of (anti-)self-duality can be reduced to the following complex Monge–Ampère equation: ⁴ 4. J. F. Plebański, J. Math. Phys. 16, 2395 (1975). https://doi.org/10.1063/1.522505

The metric, corresponding to this equation, is the following: ⁴ 4. J. F. Plebański, J. Math. Phys. 16, 2395 (1975). https://doi.org/10.1063/1.522505

$$d{s}^2={\mathrm{\Omega }}_{,pr}dpdr+{\mathrm{\Omega }}_{,ps}dpds+{\mathrm{\Omega }}_{,qr}dqdr+{\mathrm{\Omega }}_{,qs}dqds,$$

(9)

where $p,q,r,s\in \mathbb{C}$ and $\mathrm{\Omega }\left(p,q,r,s\right)\in \mathbb{C}$ .

Because of physical requirements, we limit our considerations to the case: Ω(p, q, r, s) = v, $v\in \mathbb{R}$ , ( $p,q,r,s\in \mathbb{C}$ ). If we choose: $p=z^1,q=z^2,r=\theta {\overline{z}}^1,s={\overline{z}}^2$ , then, Eq. (8) becomes: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$v_{,z^1{\overline{z}}^1}{v}_{,z^2{\overline{z}}^2}-v_{,z^1{\overline{z}}^2}{v}_{,z^2{\overline{z}}^1}=\theta ,$$

(10)

where θ = ±1, ${\overline{z}}^1$ is the complex conjugation of z ¹, $v_{,z^1}=\frac{\partial v}{\partial z^1}$ , …. The metric (9) has the following form: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$d{s}^2=v_{,z^1{\overline{z}}^1}d{z}^{1}d{\overline{z}}^1+v_{,z^1{\overline{z}}^2}d{z}^{1}d{\overline{z}}^2+v_{,z^2{\overline{z}}^1}d{z}^{2}d{\overline{z}}^1+v_{,z^2{\overline{z}}^2}d{z}^{2}d{\overline{z}}^2.$$

(11)

If θ = 1, then Eq. (10) is called the elliptic complex Monge–Ampère equation and if θ = −1, then Eq. (10) is called the hyperbolic complex Monge–Ampère equation.

1. Elliptic complex Monge–Ampère equation

As we have stated above, the elliptic complex Monge–Ampère equation has the following form:

$$v_{z^1{\overline{z}}^1}{v}_{z^2{\overline{z}}^2}-v_{z^1{\overline{z}}^2}{v}_{{\overline{z}}^1{z}^2}=1.$$

(12)

This equation has many applications in mathematics and physics. Among others, as we stated in Sec. I, Eq. (12) is strictly related to instanton solutions of the Einstein equations of the gravitational field. These solutions are described by 4-dimensional Kähler metrics, ¹⁶ 16. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003). https://doi.org/10.1088/0305-4470/36/39/304

where we sum the two values of both unbarred and barred indices and $v_{z^i{\overline{z}}^k}=\frac{{\partial }^{2}v}{\partial z^i\partial {\overline{z}}^k}$ .

The metric satisfies the vacuum Einstein equations of the gravitational field with Euclidean signature, provided that the Kähler potential is some solution of (12). We will look for non-invariant, real solutions of (12), which can be used for the construction of hyper-Kähler metrics, not possessing any Killing vectors. One of them is the K3 surface (Kummer surface), being the most important gravitational instanton. ¹⁶ 16. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003). https://doi.org/10.1088/0305-4470/36/39/304 In Ref. 16 16. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003). https://doi.org/10.1088/0305-4470/36/39/304 , some exact non-invariant and real solution of (12) was found by some reduction of the problem of solving (12) to solving some linear system of equations. Namely, this solution has the form

$$\begin{array}{ccc}\hfill w& =\hfill & \sum _{k=-\infty }^{\infty }\exp \left\{2\mathfrak{I}\left(\left[A_k^2\left(B_k^2+1\right)+1\right]z^2\right)\right\}\left\{\exp \left[2B_k\mathfrak{R}\left[A_k\left(p+\gamma z^2\right)\right]\right]\right.\hfill \\ \hfill & \hfill & \times \mathfrak{R}\left\{C_k\exp \left[2i\left[\mathfrak{I}\left(A_k\left(p+\gamma z^2\right)\right)-2B_k\mathfrak{R}\left(A_k^2{z}^2\right)\right]\right]\right\}\hfill \\ \hfill & \hfill & +\exp \left[-2B_k\mathfrak{R}\left[A_k\left(p+\gamma z^2\right)\right]\right]\hfill \\ \hfill & \hfill & \left.\times \mathfrak{R}\left\{H_k\exp \left[2i\left[\mathfrak{I}\left[A_k\left(p+\gamma z^2\right)\right]+2B_k\mathfrak{R}\left(A_k^2{z}^2\right)\right]\right]\right\}\right\},\hfill \end{array}$$

(14)

where A _k , C _k , and H _k are arbitrary complex constants, $B_k=\sqrt{1-1/\mid A_k{\mid }^2}$ , γ is the arbitrary real constant, w = e ^−ψ , and ψ is a solution of the Legendre transform of elliptic complex Monge–Ampère equation, ¹⁶ 16. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003). https://doi.org/10.1088/0305-4470/36/39/304

$${\psi }_{p\overline{p}}{\psi }_{z^2{\overline{z}}^2}-{\psi }_{p{\overline{z}}^2}{\psi }_{\overline{p}{z}^2}={\psi }_{pp}{\psi }_{\overline{p}\overline{p}}-{\psi }_{p\overline{p}}^2,$$

(15)

and

$$v=\psi -p{\psi }_p-\overline{p}{\psi }_{,\overline{p}},v_{z^1}=p,v_{{\overline{z}}^1}=\overline{p}.$$

(16)

Some other solutions, functionally invariant ones, have been found in Ref. 48 48. A. Malykh and M. B. Sheftel, Symmetry, Integrability Geom.: Methods Appl. 7, 043 (2011). https://doi.org/10.3842/SIGMA.2011.043 ,

$$w={\int }_{a_0}^{a_1}F\left(a,{\beta }_a+i{\delta }_a\right)da+\sum _k{F}_k\left({\beta }_{a_k}+i{\delta }_{a_k}\right)+c.c.,$$

(17)

where $a\in \mathbb{R}$ and

$$\begin{array}{c}v=w-p{w}_{,p}-\overline{p}{w}_{,\overline{p}}-r{w}_{,r},z^1=-w_{,p},{\overline{z}}^1=-w_{,\overline{p}},\\ \rho =\xi +\overline{\xi }=-w_{,r},{\beta }_{a_k}=p+\overline{p}+ia\left(\overline{p}-p\right),\\ {\delta }_{a_k}=i\sqrt{\gamma }\left(r+\frac{a+i}{a-i}z+\frac{a-i}{a+i}\overline{z}\right),\gamma =a^2+1,\end{array}$$

(18)

and $\xi ,\overline{\xi }$ are parameters of the symmetry group of (12).

2. Hyperbolic complex Monge–Ampère equation transformed by Legendre transformation

After applying Legendre transformation, ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$\begin{array}{l}\hfill w=v-z^1{v}_{,z^1}-{\overline{z}}^1{v}_{,{\overline{z}}^1},p=v_{,z^1},\overline{p}=v_{,{\overline{z}}^1},z^1=-w_{,p},{\overline{z}}^1=-w_{,\overline{p}},\\ \hfill \end{array}$$

(19)

the hyperbolic complex Monge–Ampère equation (10) becomes (if θ = −1) ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$w_{,p\overline{p}}{w}_{,z^2{\overline{z}}^2}-w_{,p{\overline{z}}^2}{w}_{,\overline{p}{z}^2}-w_{,p\overline{p}}^2+w_{,pp}{w}_{,\overline{p}\overline{p}}=0.$$

(20)

The metric (11), governed by (10), after applying the transformation (19), has the following form: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$\begin{array}{ccc}\hfill d{s}^2& =\hfill & \frac{1}{\left(w_{,pp}{w}_{,\overline{p}\overline{p}}-w_{,p\overline{p}}^2\right)}\left[w_{,pp}{\left(w_{,p\overline{p}}dp+w_{,\overline{p}{z}^2}d{z}^2\right)}^2\right.\hfill \\ \hfill & \hfill & +w_{,\overline{p}\overline{p}}{\left(w_{,p\overline{p}}d\overline{p}+w_{,p{\overline{z}}^2}d{\overline{z}}^2\right)}^2+\frac{w_{,pp}{w}_{,\overline{p}\overline{p}}+w_{,p\overline{p}}^2}{w_{,p\overline{p}}}\hfill \\ \hfill & \hfill & \left.\times \mid w_{,p\overline{p}}dp+w_{,\overline{p}{z}^2}d{z}^2{\mid }^2\right]-\frac{w_{,pp}{w}_{,\overline{p}\overline{p}}-w_{,p\overline{p}}^2}{w_{,p\overline{p}}}d{z}^{2}d{\overline{z}}^2.\hfill \end{array}$$

(21)

The condition of existence of Legendre transformation (19) has the following form:

and it must be satisfied for the given solution or class of solutions of (20).

3. Non-invariance of the solutions of the hyperbolic complex Monge–Ampère equation

As it was shown in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , the conditions of non-invariance of the solutions of the hyperbolic complex Monge–Ampère equation are strictly determined by Killing equations.

The condition equivalent to the Killing equation has the following form [after applying the invertible point transformation, generated by Legendre transformation (19)]: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$\begin{array}{ccc}\hfill & \hfill & p{\xi }^1\left(-w_{,p},z^2\right)+w_{,z^2}{\xi }^2\left(-w_{,p},z^2\right)+\overline{p}{\xi }^{\overline{1}}\left(-w_{,\overline{p}},{\overline{z}}^2\right)+w_{,{\overline{z}}^2}{\xi }^{\overline{2}}\left(-w_{,\overline{p}},{\overline{z}}^2\right)\hfill \\ \hfill & \hfill & =h\left(-w_{,p},z^2\right)+\overline{h}\left(-w_{,\overline{p}},{\overline{z}}^2\right).\hfill \end{array}$$

(23)

The Killing vector exists for a given solution of the hyperbolic complex Monge–Ampère equation, only if this solution satisfies (23), and, then, such solution is invariant.

In Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , the search for solutions of the hyperbolic Monge–Ampère equation has been confined to solving some two systems of linear partial differential equations by applying the method of partner symmetries. The solutions of these mentioned systems of the linear equation, found in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , have the following form:

where ${\mathrm{\Sigma }}_j={\gamma }_{j}p+{\overline{\gamma }}_j\overline{p}+{\delta }_j{z}^2+{\overline{\delta }}_j{\overline{z}}^2$ . The coefficients ${\alpha }_j\in \mathbb{R}$ are arbitrary, but γ _j , δ _j must satisfy the following relations: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

1.	for the first system

$$\mid {\gamma }_j{\mid }^2=a{\gamma }_j+ā\overline{{\gamma }_j},{\delta }_j=i\frac{{\gamma }_j^2-\left(ā+i\overline{b}\right){\gamma }_j}{ā},$$

(25)

2.	for the second system

$$\begin{array}{c}{\delta }_j=\left(\nu +i-i\frac{{\gamma }_j}{{\overline{\gamma }}_j}\right){\gamma }_j,\\ c.c.,\end{array}$$

(26)

where ν = const and a, b—arbitrary complex constants.

Therefore, there are two solutions of the hyperbolic complex Monge–Ampère equation and they are non-invariant, if n ≥ 4, because they do not satisfy Killing Eq. (23). ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 Namely, it is provided by the fact that the matrix of coefficients, ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$M=\left(\begin{array}{cccc}1\hfill & e^{-2i{\phi }_1}\hfill & e^{2i{\phi }_1}\hfill & e^{-4i{\phi }_1}\hfill \\ 1\hfill & e^{-2i{\phi }_2}\hfill & e^{2i{\phi }_2}\hfill & e^{-4i{\phi }_2}\hfill \\ 1\hfill & e^{-2i{\phi }_3}\hfill & e^{2i{\phi }_3}\hfill & e^{-4i{\phi }_3}\hfill \\ 1\hfill & e^{-2i{\phi }_4}\hfill & e^{2i{\phi }_4}\hfill & e^{-4i{\phi }_4}\hfill \end{array}\right),$$

(27)

where φ _j = arg(γ _j ) and (j = 1, …, 4) is non-singular. Hence,

$${\mathrm{\Sigma }}_j={\gamma }_{j}p+{\overline{\gamma }}_j\overline{p}+{\delta }_j{z}^2+{\overline{\delta }}_j{\overline{z}}^2$$

(28)

are linearly independent and the transformations from $p,\overline{p},z^2,{\overline{z}}^2$ to Σ _j are invertible. ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 Therefore, as it has been proved in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , after substituting each of these above solutions in Killing equation (23), this equation becomes

Hence, as it has been proved in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , Eq. (23) cannot be satisfied identically for the solution of Legendre-transformed hyperbolic complex Monge–Ampère equation (20), found in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 .

As it has been pointed out in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , the solutions given by (24) and (25) can be generalized, thanks to the functional invariance established in a theorem proved in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 . According to this theorem, we have

where the coefficients in Σ _j satisfy (25) and f is the arbitrary function ( $f\in {\mathcal{C}}^2$ ) and hence, by using these solutions, we can obtain some new classes of ultra-hyperbolic heavenly metrics in four dimensions, which do not admit Killing vectors. ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

B. Second heavenly equation of Plebański

The second heavenly equation of Plebański has the following form: ^4,5,6,7 4. J. F. Plebański, J. Math. Phys. 16, 2395 (1975). https://doi.org/10.1063/1.522505 5. C. P. Boyer and P. Winternitz, J. Math. Phys. 30, 1081 (1989). https://doi.org/10.1063/1.528379 6. B. Abraham-Shrauner, Phys. Lett. A 369, 299 (2007). https://doi.org/10.1016/j.physleta.2007.04.100 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$v_{,xx}{v}_{,yy}-v_{,xy}^2+v_{,xw}+v_{,yz}=0,$$

(31)

where $v\left(x,y,w,z\right)\in \mathbb{C}$ is a holomorphic function, $v_{xx}=\frac{{\partial }^{2}v}{\partial x^2}$ , …. and $x,y,w,z\in \mathbb{C}$ . The heavenly metric has the form ^6,7 6. B. Abraham-Shrauner, Phys. Lett. A 369, 299 (2007). https://doi.org/10.1016/j.physleta.2007.04.100 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$d{s}^2=dwdx+dzdy-v_{,xx}d{z}^2-v_{,yy}d{w}^2+2v_{,xy}dwdz.$$

(32)

In Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , the second heavenly equation (31) has been transformed by partial Legendre transformation

$$𝜗=v-w{v}_{,w}-y{v}_{,y},v_{,w}=t,$$

(33)

$$v_{,y}=r,w=-{𝜗}_{,t},y=-{𝜗}_{,r}.$$

(34)

We need also to remember that this above transformation exists if the following condition is satisfied: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$${𝜗}_{,tt}{𝜗}_{,rr}-{\left({𝜗}_{,rt}\right)}^2\ne 0.$$

(35)

Second heavenly equation, transformed by Legendre transformation (34), has the form: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$${𝜗}_{,tt}\left({𝜗}_{,xx}+{𝜗}_{,rz}\right)+{𝜗}_{,xt}\left({𝜗}_{,rr}-{𝜗}_{,xt}\right)-{𝜗}_{,rt}\left({𝜗}_{,rx}+{𝜗}_{,tz}\right)=0.$$

(36)

The metric (32) transformed by Legendre transformation has the following form: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

$$\begin{array}{ccc}\hfill d{s}^2& =\hfill & \frac{1}{{𝜗}_{,tt}\left({𝜗}_{,tt}{𝜗}_{,rr}-{𝜗}_{,tr}^2\right)}\left({𝜗}_{,tt}\left({𝜗}_{,tt}dt+{𝜗}_{,tr}dr+{𝜗}_{,tx}dx+{𝜗}_{,tz}dz\right)\right.\hfill \\ \hfill & \hfill & +{\left.\left({𝜗}_{,tt}{𝜗}_{,rx}-{𝜗}_{,tr}{𝜗}_{,tx}\right)dz\right)}^2-\frac{{𝜗}_{,tt}{𝜗}_{,xx}-{𝜗}_{,tx}^2}{{𝜗}_{,tt}}d{z}^2\hfill \\ \hfill & \hfill & -\left({𝜗}_{,tt}dt+{𝜗}_{,tr}dr+{𝜗}_{,tx}dx+{𝜗}_{,tz}dz\right)dx\hfill \\ \hfill & \hfill & -\left({𝜗}_{,rt}dt+{𝜗}_{,rr}dr+{𝜗}_{,rx}dx+{𝜗}_{,rz}dz\right)dz,\hfill \end{array}$$

(37)

where ϑ(x, r, t, z) is the potential, which satisfies Legendre transformed second heavenly equation (36).

In the aim of linearization of the above equation, there in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , translational symmetries have been applied. In the case of the so-called equal symmetries, instead of Eq. (36), the following system of equations has been investigated: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

The appropriate form of the Killing equation for second-heavenly equation, transformed by Legendre transformation, was derived in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 .

In Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , solutions of Eqs. (38)–(43) have been obtained as

$$𝜗=\sum _{j=1}^n{m}_j\exp \left({\alpha }_{j}t+{\gamma }_{j}r+{\zeta }_{j}x+{\lambda }_{j}z\right),$$

(44)

where the coefficients must satisfy the following relations: ⁷ 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010

1.	for system (38)–(40),

$${\alpha }_j=\frac{{\gamma }_j^2}{{\zeta }_j-{\gamma }_j},{\lambda }_j=-\frac{{\zeta }_j^2}{{\gamma }_j},$$

(45)

2.	for system (41)–(43),

$${\alpha }_j=\frac{{\gamma }_j^2}{{\zeta }_j},{\lambda }_j=-\frac{{\zeta }_j^2}{{\gamma }_j}.$$

(46)

Hence, these above solutions are non-invariant, if n ≥ 4, then, they generate metrics without the Killing vector. This is provided by the fact that as in the case of the solutions of the hyperbolic complex Monge–Ampere equation, the matrices of the coefficients for solutions given either by (44) and (45) or by (44) and (46) are non-singular (a detailed discussion is in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ).

IV. SOME NEW CLASSES OF EXACT SOLUTIONS AND THEIR NON-INVARIANCE

Section:

In this section, we find the classes of exact solutions of Eqs. (12), (20), and (36), and we check the non-invariance of these classes.

A. Class of exact solutions of elliptic complex Monge–Ampère equation

Now we want to find the class of exact solutions of elliptic complex Monge–Ampère equation (12). In contrast to Subsections IV B and IV C, we do not investigate the Legendre transform of the origin equation. Actually, at the first sight, one can think that the simplest way of finding the wanted class of exact solutions is applying the decomposition method to the Legendre transform of (12).

However, in this case, after obtaining the corresponding determining algebraic system, it turns out that finding of the class of exact solutions, which satisfies three conditions of existence of Legendre transformation, non-invariance, and reality, simultaneously, is very hard and it seems that it is possible that there are no appropriate solutions of the determining algebraic system. Thus, we apply the decomposition method directly to the elliptic complex Monge–Ampère equation (12). Obviously, in this case, we cannot use directly the ansatz (6) to this equation because the main obstacle in applying the original ansatz is the presence of the free term "1" in (12). Therefore, we apply some modifications of the ansatz in such a way that after substituting it in (12), some possibility of balancing of the free term will appear (the necessity of balancing of the free term was taken into consideration in the decomposition method in Ref. 47 47. Ł. T. Stȩpień, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics, edited by V. S. Gerdjikov, K. Sekigawa, and S. Dimiev (World Scientific, 2009), p. 210. ). We do it by choosing two functions in (6); as square functions, we choose the functions g _k , (k = 1, 2) to be square functions and the functions g ₃ is the arbitrary function of class ${\mathcal{C}}^2$ , but g ₄ = $\overline{g}$ ₃, in order to satisfy the condition of reality of the solution. Hence, we apply the following ansatz:

$$v\left(x^{\mu }\right)={\beta }_1+{\left({\mathrm{\Sigma }}_1\right)}^2+{\left({\mathrm{\Sigma }}_2\right)}^2+g_3\left({\mathrm{\Sigma }}_3\right)+g_4\left({\mathrm{\Sigma }}_4\right),$$

(47)

where

$$\begin{array}{c}{\mathrm{\Sigma }}_1=a_{\mu }{x}^{\mu }+{\beta }_2,{\mathrm{\Sigma }}_2=b_{\mu }{x}^{\mu }+{\beta }_3,\\ {\mathrm{\Sigma }}_3=c_{\mu }{x}^{\mu }+{\beta }_4,{\mathrm{\Sigma }}_4=d_{\mu }{x}^{\mu }+{\beta }_5,\end{array}$$

(48)

${\mathrm{\Sigma }}_k\in \mathbb{R},\left(k=\mathrm{1,2}\right)$ , $g_3\in \mathbb{C}$ is the arbitrary function of class ${\mathcal{C}}^2$ , g ₄ = $\overline{g}$ ₃, a _μ x ^μ = a ₁ x ¹ + ⋯ + a ₄ x ⁴ and $x^1=z^1,x^2={\overline{z}}^1,x^3=z^2,x^4={\overline{z}}^2$ are the independent variables. Owing to applying the decomposition method directly to (12), not to its Legendre transform (16), we avoid the necessity of satisfying the condition of existence of Legendre transformation. After substituting the ansatz (47) in (12) and collecting proper terms, we derive the determining algebraic system,

$$4\left(a_3{b}_1-a_1{b}_3\right)\left(a_4{b}_2-a_2{b}_4\right)-1=0,$$

(49)

$$2\left(a_3{d}_1\left(a_4{d}_2-a_2{d}_4\right)+b_3{d}_1\left(b_4{d}_2-b_2{d}_4\right)\right.+,$$

(50)

$$d_3\left.\left(-a_1{a}_4{d}_2-b_1{b}_4{d}_2+a_1{a}_2{d}_4+b_1{b}_2{d}_4\right)\right)=0,$$

(51)

$$2c_2\left(a_3{a}_4{c}_1+b_3{b}_4{c}_1-\left(a_1{a}_4+b_1{b}_4\right)c_3+\right.,$$

(52)

$$2\left(\left.-a_2{a}_3{c}_1-b_2{b}_3{c}_1+a_1{a}_2{c}_3+b_1{b}_2{c}_3\right)c_4\right)=0,$$

(53)

$$\left(c_3{d}_1-c_1{d}_3\right)\left(c_4{d}_2-c_2{d}_4\right)=0.$$

(54)

Its solution such that (47) is a real function, the condition of its non-invariance is satisfied, and mixed derivatives of (47) depend on $z^1,{\overline{z}}^1,z^2,{\overline{z}}^2$ gives a set of conditions, which have to be satisfied by the coefficients in (47). In such a case, (47) determines the metric, which does not admit any Killing vector.

We found the following solutions of this system:

$$a_1={ā}_2,a_3=0,a_4=0,b_1=1,b_2=1,b_3=\frac{1}{2{ā}_2},b_4=\frac{1}{2a_2},$$

(55)

$$c_1=0,c_2=\frac{1}{2{ā}_2},c_3=0,c_4=\overline{d_3},d_1=\frac{1}{2a_2},d_2=0,d_4=0,$$

(56)

$${\beta }_1,{\beta }_2,{\beta }_3\in \mathbb{R},{\beta }_5={\overline{\beta }}_4.$$

(57)

Hence, the class of solutions of (12) has the form

$$\begin{array}{ccc}\hfill v\left(z^1,{\overline{z}}^1,z^2,{\overline{z}}^2\right)& =\hfill & {\beta }_1+{\left({ā}_2{z}^1+a_2{\overline{z}}^1+{\beta }_2\right)}^2\hfill \\ \hfill & \hfill & +{\left(z^1+{\overline{z}}^1+\frac{1}{2{ā}_2}{z}^2+\frac{1}{2a_2}{\overline{z}}^2+{\beta }_3\right)}^2\hfill \\ \hfill & \hfill & +g_3\left(\frac{1}{2{ā}_2}{\overline{z}}^1+{\overline{d}}_3{\overline{z}}^2+{\beta }_4\right)+{\overline{g}}_3\left(\frac{1}{2a_2}{z}^1+d_3{z}^2+{\overline{\beta }}_4\right).\hfill \\ \hfill & \hfill & \hfill \end{array}$$

(58)

Now, we need to check whether the solutions belonging to the found class are non-invariant. We construct the following matrix:

$$M=\left(\begin{array}{cccc}\hfill {\mathrm{\Gamma }}_1{ā}_2\hfill & \hfill {\mathrm{\Gamma }}_1{a}_2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\mathrm{\Gamma }}_2\hfill & \hfill {\mathrm{\Gamma }}_2\hfill & \hfill \frac{{\mathrm{\Gamma }}_2}{2{ā}_2}\hfill & \hfill \frac{{\mathrm{\Gamma }}_2}{2a_2}\hfill \\ \hfill 0\hfill & \hfill \frac{g_3^{\prime }}{2{ā}_2}\hfill & \hfill 0\hfill & \hfill g_3^{\prime }{\overline{d}}_3\hfill \\ \hfill \frac{{\overline{g}}_3^{\prime }}{2a_2}\hfill & \hfill 0\hfill & \hfill {\overline{g}}_3^{\prime }{d}_3\hfill & \hfill 0\hfill \end{array}\right),$$

(59)

where ${\mathrm{\Gamma }}_1=2\left({ā}_2{z}^1+a_2{\overline{z}}^1+{\beta }_2\right),{\mathrm{\Gamma }}_2=2z^1+2{\overline{z}}^1+\frac{z^2}{{ā}_2}+\frac{{\overline{z}}^2}{a_2}+2{\beta }_3$ . In order to provide non-invariance of the solutions belonging to the class (58), the determinant of this above matrix must not vanish,

$$\begin{array}{ccc}\hfill det\left(M\right)& =\hfill & -\frac{1}{2{ā}_2^2{a}_2^2}\left({ā}_2{z}^1+a_2{\overline{z}}^1+{\beta }_2\right)\hfill \\ \hfill & \hfill & \times \left(2z^1{ā}_2{a}_2+2{\overline{z}}^1{ā}_2{a}_2+z^2{a}_2+{\overline{z}}^2{ā}_2+2{\beta }_3{ā}_2{a}_2\right)\hfill \\ \hfill & \hfill & \times \left(4d_3{\overline{d}}_3{ā}_2^2{a}_2-d_3{ā}_2-4{ā}_2{a}_2^2{d}_3{\overline{d}}_3+{\overline{d}}_3{a}_2\right)g_3^{\prime }{\overline{g}}_3^{\prime }\ne 0.\hfill \end{array}$$

(60)

Let us assume that a ₂ ≠ 0. Hence, the solutions belonging to the class (58) depend on four variables and they are non-invariant, in the regions, where simultaneously the conditions, ${ā}_2{z}^1+a_2{\overline{z}}^1+{\beta }_2\ne 0$ and $2z^1{ā}_2{a}_2+2{\overline{z}}^1{ā}_2{a}_2+z^2{a}_2+{\overline{z}}^2{ā}_2+2{\beta }_3{ā}_2{a}_2\ne 0$ and g′₃ ≠ 0, hold.

Of course, this above found class of solutions of the elliptic complex Monge–Ampère equation (12) is the class of real solutions. Obviously, these solutions are not differentiable in the $\mathbb{C}$ sense. They are differentiable in the $\mathbb{R}$ sense. If we repeat these above computations, but in the variables $x^1=\mathfrak{R}\left(p\right),x^2=\mathfrak{I}\left(p\right),x^3=\mathfrak{R}\left(z^2\right),x^4=\mathfrak{I}\left(z^2\right)$ , $x^k\in \mathbb{R},\left(k=\mathrm{1,2,3,4}\right)$ , then it turns out that the function, given by (58), is still the class of exact solutions of elliptic complex Monge–Ampère equation (12), and these solutions are still non-invariant. On other hand, as we see from the form of the solutions presented above, the metric (13) generated by them admits the Killing vector.

B. Classes of exact solutions of hyperbolic complex Monge–Ampère equation

In this subsection, we look for the class of exact solutions of hyperbolic complex Monge–Ampère equation (20). To this effect, we apply directly the ansatz (6) for (20), in contrast with Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , where this equation was linearized and the obtained systems of linear equations were solved. Here, u ≡ w and the independent variables $x^1=p,x^2=\overline{p},x^3=z^2,x^4={\overline{z}}^2$ , and g _j (j = 1, 2, 3, 4) are such functions of their arguments that $w\left(p,\overline{p},z^2,{\overline{z}}^2\right)\in \mathbb{R}$ . We search the class of non-invariant solutions of (20), given by (6) and (7), which satisfy (22).

After making the procedure described in Sec. II, we obtain some systems of nonlinear algebraic equations, the so-called determining algebraic system. Apart from satisfying it, we require also satisfying of the following conditions: condition (22), the condition of non-singularity of Jacobian matrix, and the condition that the solution must be real.

We found three classes of non-invariant, exact solutions of (20), satisfying the mentioned conditions. These classes are given by (6) and by the following sets of relations between the coefficients:

1.	for class I

$$\begin{array}{cc}\hfill a_1& =a_2,a_3=-\frac{a_2\left(d_1{d}_2-d_2^2-d_1{d}_4\right)}{d_1{d}_2},\hfill \\ \hfill a_4& =-\frac{a_2\left(d_1{d}_2-d_2^2-d_1{d}_4\right)}{d_1{d}_2},\hfill \\ \hfill b_1& =d_2,b_2=d_1,b_3=d_4,b_4=-\frac{d_1^2-d_2^2-d_1{d}_4}{d_2},\hfill \\ \hfill c_1& =-iA,c_2=iA,c_3=-\frac{iA\left(d_1{d}_2+d_2^2+d_1{d}_4\right)}{d_1{d}_2},\hfill \\ \hfill c_4& =\frac{iA\left(d_1{d}_2+d_2^2+d_1{d}_4\right)}{d_1{d}_2},d_3=-\frac{d_1^2-d_2^2-d_1{d}_4}{d_2},\hfill \end{array}$$

(61)

in this case: $g_1,g_3\in \mathbb{R}$ , $g_2\in \mathbb{C},g_4={\overline{g}}_2$ (of course ${\beta }_5={\overline{\beta }}_3$ );

2.	for class II

$$\begin{array}{c}{a}_1=0,a_2=0,a_3={ā}_4,b_1=0,b_2=0,b_3={\overline{b}}_4,\\ c_1=0,c_2=d_4,c_3=d_4,c_4=d_3,d_1=d_4,d_2=0,\end{array}$$

(62)

in this case: $g_1,g_2\in \mathbb{R}$ , $g_3\in \mathbb{C},g_4={\overline{g}}_3$ (of course ${\beta }_5={\overline{\beta }}_4$ );

3.	for class III

$$\begin{array}{cc}\hfill a_1& =A_2\left(1+i\right),a_2={ā}_1,a_3=2i{A}_2,a_4={ā}_3,\hfill \\ \hfill b_1& =\frac{\sqrt{B_3^2+B_4^2}}{2}\left(1+i\frac{B_3+B_4-\sqrt{B_3^2+B_4^2}}{B_3-B_4+\sqrt{B_3^2+B_4^2}}\right),b_2={\overline{b}}_1,\hfill \\ \hfill b_3& =B_3+i{B}_4,b_4={\overline{b}}_3,c_1=i{C}_2,c_2={\overline{c}}_1,c_3=0,c_4=0,\hfill \\ \hfill d_1& =H_2\left(-1+i\right),d_2={\overline{d}}_1,d_3=2i{H}_2,d_4={\overline{d}}_3,\hfill \end{array}$$

(63)

in this case: $g_j,\left(j=\mathrm{1,2,3,4}\right)\in \mathbb{R}$ .

In all these above cases, g _j , (j = 1, 2, 3, 4) are the functions of class ${\mathcal{C}}^2$ .

Hence, the found classes of the exact solutions of (20) have the form,

1.

class I

$$\begin{array}{ccc}\hfill w\left(p,\overline{p},z^2,{\overline{z}}^2\right)& =\hfill & {\beta }_1+g_1\left(a_{2}p+a_2\overline{p}-\frac{a_2\left(d_1{d}_2-d_2^2-d_1{d}_4\right)}{d_1{d}_2}{z}^2-\frac{a_2\left(d_1{d}_2-d_2^2-d_1{d}_4\right)}{d_1{d}_2}{\overline{z}}^2+{\beta }_2\right)\hfill \\ \hfill & \hfill & +g_2\left(d_{2}p+d_1\overline{p}+d_4{z}^2-\frac{d_1^2-d_2^2-d_1{d}_4}{d_2}{\overline{z}}^2+{\beta }_3\right)\hfill \\ \hfill & \hfill & +g_3\left(-iAp+iA\overline{p}-\frac{iA\left(d_1{d}_2+d_2^2+d_1{d}_4\right)}{d_1{d}_2}{z}^2+\frac{iA\left(d_1{d}_2+d_2^2+d_1{d}_4\right)}{d_1{d}_2}{\overline{z}}^2+{\beta }_4\right)\hfill \\ \hfill & \hfill & +{\overline{g}}_2\left(d_{2}p+d_1\overline{p}+d_4{z}^2-\frac{d_1^2-d_2^2-d_1{d}_4}{d_2}{\overline{z}}^2+{\beta }_3\right),\hfill \end{array}$$

(64)

2.

class II

$$w\left(p,\overline{p},z^2,{\overline{z}}^2\right)={\beta }_1+g_1\left({ā}_4{z}^2+a_4{\overline{z}}^2+{\beta }_2\right)+g_2\left({\overline{b}}_4{z}^2+b_4{\overline{z}}^2+{\beta }_3\right)+g_3\left(d_4\overline{p}+d_4{z}^2+d_3{\overline{z}}^2+{\beta }_4\right)+{\overline{g}}_3\left(d_4\overline{p}+d_4{z}^2+d_3{\overline{z}}^2+{\beta }_4\right),$$

(65)

3.

class III

$$\begin{array}{ccc}\hfill w\left(p,\overline{p},z^2,{\overline{z}}^2\right)& =\hfill & {\beta }_1+g_1\left(A_2\left(1+i\right)p+A_2\left(1-i\right)\overline{p}+2i{A}_2{z}^2-2i{A}_2{\overline{z}}^2+{\beta }_2\right)+g_2\left(\frac{\sqrt{B_3^2+B_4^2}}{2}\left[1+i\frac{B_3+B_4-\sqrt{B_3^2+B_4^2}}{B_3-B_4+\sqrt{B_3^2+B_4^2}}\right]p\right.\hfill \\ \hfill & \hfill & \left.+\frac{\sqrt{B_3^2+B_4^2}}{2}\left[1-i\frac{B_3+B_4-\sqrt{B_3^2+B_4^2}}{B_3-B_4+\sqrt{B_3^2+B_4^2}}\right]\overline{p}+\left(B_3+i{B}_4\right)z^2+\left(B_3-i{B}_4\right){\overline{z}}^2+{\beta }_3\right)+g_3\left(i{C}_{2}p-i{C}_2\overline{p}+{\beta }_4\right)\hfill \\ \hfill & \hfill & +g_4\left(H_2\left(-1+i\right)p+H_2\left(-1-i\right)\overline{p}+2i{H}_2{z}^2-2i{H}_2{\overline{z}}^2+{\beta }_5\right),\hfill \end{array}$$

(66)

where

•	for class I: g 1, g 3 are arbitrary real functions (of class ${\mathcal{C}}^2$ ) of their arguments, g 2 is some arbitrary complex function of its argument and $g_2\in {\mathcal{C}}^2$ , $\overline{g}$ 2 is the complex conjugation of g 2, i.e, $g_2=f\left(\mathrm{\Phi }\right),{\overline{g}}_2=f\left(\overline{\mathrm{\Phi }}\right)$ , where $\mathrm{\Phi }\in \mathbb{C}$ is the argument of g 2 given in (64) and $A,a_2,d_1,d_2,d_4,{\beta }_1,{\beta }_2,{\beta }_4\in \mathbb{R}$ and ${\beta }_3\in \mathbb{C}$ .
•	for class II: g 1, g 2 are arbitrary real functions (of class ${\mathcal{C}}^2$ ) of their arguments, g 3 is some arbitrary complex function of its argument and $g_3\in {\mathcal{C}}^2$ , $\overline{g}$ 3 is the complex conjugation of g 3, i.e., $g_3=f\left(\mathrm{\Phi }\right),{\overline{g}}_3=f\left(\overline{\mathrm{\Phi }}\right)$ , where $\mathrm{\Phi }\in \mathbb{C}$ is the argument of g 3 given in (65) and $a_4,b_4\in \mathbb{C}$ , $d_3,d_4,{\beta }_k\in \mathbb{R},\left(k=\mathrm{1,2,3}\right)$ and ${\beta }_4\in \mathbb{C}$ .
•	for class III: g j , (j = 1, 2, 3, 4) are arbitrary real functions (of class ${\mathcal{C}}^2$ ) of their arguments and $A_2,B_3,B_4,C_2,H_2,{\beta }_k\in \mathbb{R},\left(k=1,\dots ,5\right)$ .

Now, we check, whether condition (22) is satisfied by the solutions belonging to the classes I, II, and III. It turns out that it is satisfied, when

1.	for class I

$$\begin{array}{ccc}\hfill & \hfill & a_2^2{\left(d_1-d_2\right)}^2{g}_1^{″}\left({\overline{g}}_2^{″}+g_2^{″}\right)-A^2{\left(d_1+d_2\right)}^2{g}_3^{″}\left({\overline{g}}_2^{″}+g_2^{″}\right)\hfill \\ \hfill & \hfill & +{\left(d_1^2-d_2^2\right)}^2{g}_2^{″}{\overline{g}}_2^{″}-4a_2^2{A}^2{g}_1^{″}{g}_3^{″}\ne 0,\hfill \end{array}$$

(67)

2.	for class II

3.	for class III (some algebraic inequality, which we skip in this paper, due to its complicated structure),

where $g_j^{\prime \prime },\left(j=\mathrm{1,2,3,4}\right)$ denotes second derivative of the function g _j with respect to its argument.

Next, based on the considerations included in Subsection III A 3, we make the analysis of non-invariance of the solutions, belonging to the found classes. The Jacobian matrices have the following forms:

1.	for class I

$$\begin{array}{l}\hfill M=\left(\begin{array}{cccc}\hfill a_2{g}_1^{\prime }\hfill & \hfill a_2{g}_1^{\prime }\hfill & \hfill -a_2\frac{d_1{d}_2-d_2^2-d_1{d}_4}{d_1{d}_2}{g}_1^{\prime }\hfill & \hfill -a_2\frac{d_1{d}_2-d_2^2-d_1{d}_4}{d_1{d}_2}{g}_1^{\prime }\hfill \\ \hfill d_2{g}_2^{\prime }\hfill & \hfill d_1{g}_2^{\prime }\hfill & \hfill d_4{g}_2^{\prime }\hfill & \hfill -\frac{d_1^2-d_2^2-d_1{d}_4}{d_2}{g}_2^{\prime }\hfill \\ \hfill -iA{g}_3^{\prime }\hfill & \hfill iA{g}_3^{\prime }\hfill & \hfill -\frac{iA\left(d_1{d}_2+d_2^2+d_1{d}_4\right)}{d_1{d}_2}{g}_3^{\prime }\hfill & \hfill \frac{iA\left(d_1{d}_2+d_2^2+d_1{d}_4\right)}{d_1{d}_2}{g}_3^{\prime }\hfill \\ \hfill d_1{\overline{g}}_2^{\prime }\hfill & \hfill d_2{\overline{g}}_2^{\prime }\hfill & \hfill -\frac{d_1^2-d_2^2-d_1{d}_4}{d_2}{\overline{g}}_2^{\prime }\hfill & \hfill d_4{\overline{g}}_2^{\prime }\hfill \end{array}\right),\\ \hfill \end{array}$$

(69)

2.	for class II

$$M=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill {ā}_4{g}_1^{\prime }\hfill & \hfill a_4{g}_1^{\prime }\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\overline{b}}_4{g}_2^{\prime }\hfill & \hfill b_4{g}_2^{\prime }\hfill \\ \hfill 0\hfill & \hfill d_4{g}_3^{\prime }\hfill & \hfill d_4{g}_3^{\prime }\hfill & \hfill d_3{g}_3^{\prime }\hfill \\ \hfill d_4{\overline{g}}_3^{\prime }\hfill & \hfill 0\hfill & \hfill d_3{\overline{g}}_3^{\prime }\hfill & \hfill d_4{\overline{g}}_3^{\prime }\hfill \end{array}\right),$$

(70)

3.	for class III

$$\begin{array}{l}\hfill M=\left(\begin{array}{cccc}\hfill A_2\left(1+i\right)g_1^{\prime }\hfill & \hfill A_2\left(1-i\right)g_1^{\prime }\hfill & \hfill 2i{A}_2{g}_1^{\prime }\hfill & \hfill -2i{A}_2{g}_1^{\prime }\hfill \\ \hfill N_1{g}_2^{\prime }\hfill & \hfill {\overline{N}}_1{g}_2^{\prime }\hfill & \hfill \left(B_3+i{B}_4\right)g_2^{\prime }\hfill & \hfill \left(B_3-i{B}_4\right)g_2^{\prime }\hfill \\ \hfill i{C}_2{g}_3^{\prime }\hfill & \hfill -i{C}_2{g}_3^{\prime }\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill H_2\left(-1+i\right)g_4^{\prime }\hfill & \hfill H_2\left(-1-i\right)g_4^{\prime }\hfill & \hfill 2i{H}_2{g}_4^{\prime }\hfill & \hfill -2i{H}_2{g}_4^{\prime }\hfill \end{array}\right),\\ \hfill \end{array}$$

(71)

where g′ _j (j = 1, 2, 3, 4) denotes the first derivative of the function g _j with respect to its argument and $N_1=\frac{\sqrt{B_3^2+B_4^2}}{2}\left(1+i\frac{B_3+B_4-\sqrt{B_3^2+B_4^2}}{B_3-B_4+\sqrt{B_3^2+B_4^2}}\right)$ . From the requirement of non-vanishing of the determinants of these above matrices, we have

1.	for class I

$$detM=\frac{2i{a}_{2}A}{\underset{1}{\overset{2}{d}}{d}_2^2}\left(\underset{1}{\overset{6}{d}}-d_2^6-3d_1^4{d}_2^2+3d_1^2{d}_2^4\right)g_1^{\prime }{g}_2^{\prime }{\overline{g}}_2^{\prime }{g}_3^{\prime }\ne 0,$$

(72)

2.	for the class II

$$detM=-\left({ā}_4{b}_4-a_4{\overline{b}}_4\right)d_4^2{g}_1^{\prime }{g}_2^{\prime }{g}_3^{\prime }{\overline{g}}_3^{\prime }\ne 0,$$

(73)

3.	for the class III

$$detM=16A_2{B}_3{C}_2{H}_2{g}_1^{\prime }{g}_2^{\prime }{g}_3^{\prime }{g}_4^{\prime }\ne 0.$$

(74)

Let us assume additionally for class I

Let us notice that the mixed derivatives of the solutions given by (64) and (66) depend on all four independent variables: $p,\overline{p},z^2,{\overline{z}}^2$ . Therefore, let us fix now the functions g _j , (j = 1, …, 4) in (64) and (66), such that conditions (72) [together with (67) and (75)], (73) [together with (68)] and (74) [together with $B_3-B_4+\sqrt{B_3^2+B_4^2}\ne 0$ and (A.1)] will be still satisfied, respectively.

We can now repeat from the Subsection III A 3, based on Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , that Eq. (23) cannot be satisfied identically for any solution of Legendre-transformed hyperbolic complex Monge–Ampère equation (20) just by a proper choice of the functions ${\xi }^1,{\xi }^{\overline{1}},{\xi }^2,{\xi }^{\overline{2}},h,\overline{h}$ because the variables $p,\overline{p},w_{,z^2},w_{,{\overline{z}}^2}$ explicitly enter into the coefficients of this equation.

Σ _j , (j = 1, 2, 3, 4) are linearly independent for the three above classes of solutions. Therefore, the transformations from $p,\overline{p},z^2,{\overline{z}}^2$ to Σ _j are invertible and we can express $p,\overline{p},z^2,{\overline{z}}^2$ through Σ _j , (j = 1, 2, 3, 4), so that Σ _j , (j = 1, 2, 3, 4), can be chosen as new independent variables in (23), and after substituting each of the above classes of solutions in Eq. (23), this equation becomes

For example, if we choose g _j = exp(Σ _j ) (solutions of such forms were found in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ) in the above found classes of solutions, especially in (66), we obtain a form similar to the form of the ansatz (24), if n = 4. Therefore, the solutions belonging to the above classes, (64) or (66), with such fixed functions g _k , (k = 1, …, 4), do not have functional arbitrariness. Hence, after taking into account these above arguments, we see that these solutions cannot satisfy the first-order Killing equation (23).

Thus, we see that the Killing equation (23) cannot be satisfied identically for any solution, belonging to the classes of the form (64) or (66).

Of course, all these above found classes of solutions of the hyperbolic complex Monge–Ampère equation (20) are the classes of real solutions. Obviously, these solutions are not differentiable in $\mathbb{C}$ sense. They are differentiable in the $\mathbb{R}$ sense. If we repeat these above computations, but in the variables $x^1=\mathfrak{R}\left(p\right),x^2=\mathfrak{I}\left(p\right),x^3=\mathfrak{R}\left(z^2\right),x^4=\mathfrak{I}\left(z^2\right)$ , $x^k\in \mathbb{R},\left(k=\mathrm{1,2,3,4}\right)$ , then it turns out that the functions, given either by (64) or by (65) or by (66), are still the classes of exact solutions of hyperbolic complex Monge–Ampère equation (20), these solutions are still non-invariant and condition (22) is still satisfied.

Hence, we have proved the following theorem:

Theorem 1.

The metric (21) with w, being some solution of hyperbolic complex Monge–Ampère equation (20), belonging to any class, defined by:

1.	(64) —class I, [when the relations (67) , (72) , and (75) hold]
2.	(66) —class III, [when the relations (74) are satisfied, $B_3-B_4+\sqrt{B_3^2+B_4^2}\ne 0$ and some algebraic inequality mentioned above],

where the functions g _j , (j = 1, …, 4), are fixed, does not admit the Killing vector.

C. Classes of exact solutions of the second heavenly equation

Now, in order to find new classes of non-invariant solutions of the heavenly equation, we use the decomposition method, in the cases of equal symmetries and higher symmetries. We apply the ansatz (6) for systems (38)–(40) and (41)–(43), but now u ≡ ϑ, the independent variables are x ¹ = x, x ² = r, x ³ = t, x ⁴ = z, and g _k , (k = 1, 2, 3, 4), are arbitrary holomorphic, complex-valued functions of their arguments. The ansatz (6) presents the class of solutions of systems (38)–(40), (41)–(43), when the relations, which must be satisfied by the coefficients, are as follows:

1.	for Eqs. (38)–(40)—the case of equal symmetries

$$\begin{array}{c}{a}_1=\frac{a_2\left(a_2+a_3\right)}{a_3},a_4=-\frac{a_2{\left(a_2+a_3\right)}^2}{a_3^2},\\ b_1=\frac{b_2\left(b_2+b_3\right)}{b_3},b_4=-\frac{b_2{\left(b_2+b_3\right)}^2}{b_3^2},\\ c_1=\frac{c_2\left(c_2+c_3\right)}{c_3},c_4=-\frac{c_2{\left(c_2+c_3\right)}^2}{c_3^2},\\ d_1=\frac{d_2\left(d_2+d_3\right)}{d_3},d_4=-\frac{d_2{\left(d_2+d_3\right)}^2}{d_3^2},\end{array}$$

(77)

2.	for Eqs. (41)–(43) —the case of higher symmetry • subclass I

$$\begin{array}{c}{a}_3=\frac{a_2^2}{a_1},a_4=-\frac{a_1^2}{a_2},\\ b_3=\frac{b_2^2}{b_1},b_4=-\frac{b_1^2}{b_2},\\ c_3=\frac{c_2^2}{c_1},c_4=-\frac{c_1^2}{c_2},\\ d_3=\frac{d_2^2}{d_1},d_4=-\frac{d_1^2}{d_2},\end{array}$$

(78)

$$\begin{array}{c}{a}_1=\frac{a_2^2}{a_3},a_4=-\frac{a_2^3}{a_3^2},\\ b_1=\frac{b_2^2}{b_3},b_4=-\frac{b_2^3}{b_3^2},\\ c_1=\frac{c_2^2}{c_3},c_4=-\frac{c_2^3}{c_3^2},\\ d_1=\frac{d_2^2}{d_3},d_4=-\frac{d_2^3}{d_3^2}.\end{array}$$

(79)

•	subclass II

The parameters β _k , (k = 1, …, 5), occuring in (6) and (7), are, in this case, arbitrary constants.

We check now, whether condition (35) of the existence of Legendre transformation (34) is satisfied for the ansatz (6) and for the three sets of the relations of the coefficients (77)–(79). It turns out that this condition is satisfied for this ansatz and for these above three sets of relations of coefficients, if these following conditions are satisfied:

$$\begin{array}{l}\hfill {\left(b_2{d}_3-b_3{d}_2\right)}^2{g}_2^{″}{g}_4^{″}+{\left(b_2{c}_3-b_3{c}_2\right)}^2{g}_2^{″}{g}_3^{″}+{\left(a_2{d}_3-a_3{d}_2\right)}^2{g}_1^{″}{g}_4^{″}+{\left(c_2{a}_3-c_3{a}_2\right)}^2{g}_1^{″}{g}_3^{″}+{\left(a_2{b}_3-a_3{b}_2\right)}^2{g}_1^{″}{g}_2^{″}+{\left(c_2{d}_3-c_3{d}_2\right)}^2{g}_3^{″}{g}_4^{″}\ne 0,\\ \hfill \end{array}$$

(80)

$$\begin{array}{ccc}\hfill & \hfill & \frac{\left(b_2^4{a}_1^2{c}_1^2{d}_1^2{a}_2^2-2a_2^3{b}_1{c}_1^2{d}_1^2{b}_2^3{a}_1+a_2^4{b}_1^2{c}_1^2{d}_1^2{b}_2^2\right)g_1^{″}{g}_2^{″}}{a_1^2{b}_1^2{c}_1^2{d}_1^2}+\frac{\left(-2a_2^3{b}_1^2{c}_1{d}_1^2{c}_2^3{a}_1+c_2^4{a}_1^2{b}_1^2{d}_1^2{a}_2^2+a_2^4{b}_1^2{c}_1^2{d}_1^2{c}_2^2\right)g_1^{″}{g}_3^{″}}{a_1^2{b}_1^2{c}_1^2{d}_1^2}\hfill \\ \hfill & \hfill & +\frac{\left(-2a_2^3{b}_1^2{c}_1^2{d}_1{d}_2^3{a}_1+a_2^4{b}_1^2{c}_1^2{d}_1^2{d}_2^2+d_2^4{a}_1^2{b}_1^2{c}_1^2{a}_2^2\right)g_1^{″}{g}_4^{″}}{a_1^2{b}_1^2{c}_1^2{d}_1^2}+\frac{\left(c_2^4{a}_1^2{b}_1^2{d}_1^2{b}_2^2+b_2^4{a}_1^2{c}_1^2{d}_1^2{c}_2^2-2b_2^3{a}_1^2{c}_1{d}_1^2{c}_2^3{b}_1\right)g_2^{″}{g}_3^{″}}{a_1^2{b}_1^2{c}_1^2{d}_1^2}\hfill \\ \hfill & \hfill & +\frac{\left(-2b_2^3{a}_1^2{c}_1^2{d}_1{d}_2^3{b}_1+b_2^4{a}_1^2{c}_1^2{d}_1^2{d}_2^2+d_2^4{a}_1^2{b}_1^2{c}_1^2{b}_2^2\right)g_2^{″}{g}_4^{″}}{a_1^2{b}_1^2{c}_1^2{d}_1^2}+\frac{\left(d_2^4{a}_1^2{b}_1^2{c}_1^2{c}_2^2+c_2^4{a}_1^2{b}_1^2{d}_1^2{d}_2^2-2c_2^3{a}_1^2{b}_1^2{d}_1{d}_2^3{c}_1\right)g_3^{″}{g}_4^{″}}{a_1^2{b}_1^2{c}_1^2{d}_1^2}\ne 0,\hfill \end{array}$$

(81)

where $g_j^{\prime \prime },\left(j=\mathrm{1,2,3,4}\right)$ denotes the second derivative of the function g _j with respect to its argument.

•	for (77) and for (79), [g j , (j = 1, …, 4), are the functions of the arguments, including the coefficients, which satisfy, respectively, (77) and (79)]
•	for (78)

Now, based on the considerations included in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , we make the analysis of non-invariance of these above classes of exact solutions of the second heavenly equation. Namely, we check now, whether Σ₁, Σ₂, Σ₃, Σ₄ are linearly independent, i.e., the transformations from x, r, t, z to Σ _j , (j = 1, 2, 3, 4) are invertible. The Jacobian matrices are the following:

1.	for the case of equal symmetries [when the relations (77) hold]

$$M=\left(\begin{array}{cccc}\hfill \frac{a_2\left(a_2+a_3\right)}{a_3}{g}_1^{\prime }\hfill & \hfill a_2{g}_1^{\prime }\hfill & \hfill a_3{g}_1^{\prime }\hfill & \hfill -\frac{a_2{\left(a_2+a_3\right)}^2}{a_3^2}{g}_1^{\prime }\hfill \\ \hfill \frac{b_2\left(b_2+b_3\right)}{b_3}{g}_2^{\prime }\hfill & \hfill b_2{g}_2^{\prime }\hfill & \hfill b_3{g}_2^{\prime }\hfill & \hfill -\frac{b_2{\left(b_2+b_3\right)}^2}{b_3^2}{g}_2^{\prime }\hfill \\ \hfill \frac{c_2\left(c_2+c_3\right)}{c_3}{g}_3^{\prime }\hfill & \hfill c_2{g}_3^{\prime }\hfill & \hfill c_3{g}_3^{\prime }\hfill & \hfill -\frac{c_2{\left(c_2+c_3\right)}^2}{c_3^2}{g}_3^{\prime }\hfill \\ \hfill \frac{d_2\left(d_2+d_3\right)}{d_3}{g}_4^{\prime }\hfill & \hfill d_2{g}_4^{\prime }\hfill & \hfill d_3{g}_4^{\prime }\hfill & \hfill -\frac{d_2{\left(d_2+d_3\right)}^2}{d_3^2}{g}_4^{\prime }\hfill \end{array}\right),$$

(82)

2.	for the case of higher symmetry [when the relations (78) hold]

$$M=\left(\begin{array}{cccc}\hfill a_1{g}_1^{\prime }\hfill & \hfill a_2{g}_1^{\prime }\hfill & \hfill \frac{a_2^2}{a_1}{g}_1^{\prime }\hfill & \hfill -\frac{a_1^2}{a_2}{g}_1^{\prime }\hfill \\ \hfill b_1{g}_2^{\prime }\hfill & \hfill b_2{g}_2^{\prime }\hfill & \hfill \frac{b_2^2}{b_1}{g}_2^{\prime }\hfill & \hfill -\frac{b_1^2}{b_2}{g}_2^{\prime }\hfill \\ \hfill c_1{g}_3^{\prime }\hfill & \hfill c_2{g}_3^{\prime }\hfill & \hfill \frac{c_2^2}{c_1}{g}_3^{\prime }\hfill & \hfill -\frac{c_1^2}{c_2}{g}_3^{\prime }\hfill \\ \hfill d_1{g}_4^{\prime }\hfill & \hfill d_2{g}_4^{\prime }\hfill & \hfill \frac{d_2^2}{d_1}{g}_4^{\prime }\hfill & \hfill -\frac{d_1^2}{d_2}{g}_4^{\prime }\hfill \end{array}\right),$$

(83)

3.	for the case of higher symmetry [when the relations (79) hold]

$$M=\left(\begin{array}{cccc}\hfill \frac{a_2^2}{a_3}{g}_1^{\prime }\hfill & \hfill a_2{g}_1^{\prime }\hfill & \hfill a_3{g}_1^{\prime }\hfill & \hfill -\frac{a_2^3}{a_3^2}{g}_1^{\prime }\hfill \\ \hfill \frac{b_2^2}{b_3}{g}_2^{\prime }\hfill & \hfill b_2{g}_2^{\prime }\hfill & \hfill b_3{g}_2^{\prime }\hfill & \hfill -\frac{b_2^3}{b_3^2}{g}_2^{\prime }\hfill \\ \hfill \frac{c_2^2}{c_3}{g}_3^{\prime }\hfill & \hfill c_2{g}_3^{\prime }\hfill & \hfill c_3{g}_3^{\prime }\hfill & \hfill -\frac{c_2^3}{c_3^2}{g}_3^{\prime }\hfill \\ \hfill \frac{d_2^2}{d_3}{g}_4^{\prime }\hfill & \hfill d_2{g}_4^{\prime }\hfill & \hfill d_3{g}_4^{\prime }\hfill & \hfill -\frac{d_2^3}{d_3^2}{g}_4^{\prime }\hfill \end{array}\right),$$

(84)

where g′ _j (j = 1, 2, 3, 4) is the first derivative of the function g _j with respect to its argument. We require non-vanishing of the following Jacobians:

$$\begin{array}{ccc}\hfill detM& =\hfill & \frac{g_1^{\prime }{g}_2^{\prime }{g}_3^{\prime }{g}_4^{\prime }}{a_3^2{b}_3^2{d}_3^2{c}_3^2}\left(-a_2^2{a}_3{b}_2{b}_3^2{c}_3^3{d}_2^3+a_2^2{a}_3{b}_2{b}_3^2{c}_2^3{d}_3^3-a_2^2{a}_3{c}_2{c}_3^2{d}_3^3{b}_2^3+a_2^2{a}_3{c}_2{c}_3^2{b}_3^3{d}_2^3\right.\hfill \\ \hfill & \hfill & -a_2^2{a}_3{d}_2{d}_3^2{b}_3^3{c}_2^3+a_2^2{a}_3{d}_2{d}_3^2{c}_3^3{b}_2^3+b_2^2{b}_3{a}_2{a}_3^2{c}_3^3{d}_2^3-b_2^2{b}_3{a}_2{a}_3^2{c}_2^3{d}_3^3+b_2^2{b}_3{c}_2{c}_3^2{d}_3^3{a}_2^3\hfill \\ \hfill & \hfill & -b_2^2{b}_3{c}_2{c}_3^2{a}_3^3{d}_2^3+b_2^2{b}_3{d}_2{d}_3^2{a}_3^3{c}_2^3-b_2^2{b}_3{d}_2{d}_3^2{c}_3^3{a}_2^3+c_2^2{c}_3{a}_2{a}_3^2{d}_3^3{b}_2^3-c_2^2{c}_3{a}_2{a}_3^2{b}_3^3{d}_2^3\hfill \\ \hfill & \hfill & -c_2^2{c}_3{b}_2{b}_3^2{d}_3^3{a}_2^3+c_2^2{c}_3{b}_2{b}_3^2{a}_3^3{d}_2^3-c_2^2{c}_3{d}_2{d}_3^2{a}_3^3{b}_2^3+c_2^2{c}_3{d}_2{d}_3^2{b}_3^3{a}_2^3+d_2^2{d}_3{a}_2{a}_3^2{b}_3^3{c}_2^3\hfill \\ \hfill & \hfill & \left.-d_2^2{d}_3{a}_2{a}_3^2{c}_3^3{b}_2^3-d_2^2{d}_3{b}_2{b}_3^2{a}_3^3{c}_2^3+d_2^2{d}_3{b}_2{b}_3^2{c}_3^3{a}_2^3+d_2^2{d}_3{c}_2{c}_3^2{a}_3^3{b}_2^3-d_2^2{d}_3{c}_2{c}_3^2{b}_3^3{a}_2^3\right)\ne 0,\hfill \end{array}$$

(85)

$$\begin{array}{ccc}\hfill detM& =\hfill & \frac{g_1^{\prime }{g}_2^{\prime }{g}_3^{\prime }{g}_4^{\prime }}{c_1{d}_2{c}_2{d}_1{b}_2{b}_1{a}_2{a}_1}\left(-a_1^2{a}_2{b}_2^2{b}_1{c}_2^3{d}_1^3+a_1^2{a}_2{b}_2^2{b}_1{c}_1^3{d}_2^3-a_1^2{a}_2{c}_2^2{c}_1{d}_2^3{b}_1^3+a_1^2{a}_2{c}_2^2{c}_1{b}_2^3{d}_1^3\right.\hfill \\ \hfill & \hfill & -a_1^2{a}_2{d}_2^2{d}_1{b}_2^3{c}_1^3+a_1^2{a}_2{d}_2^2{d}_1{c}_2^3{b}_1^3+b_1^2{b}_2{a}_2^2{a}_1{c}_2^3{d}_1^3-b_1^2{b}_2{a}_2^2{a}_1{c}_1^3{d}_2^3+b_1^2{b}_2{c}_2^2{c}_1{d}_2^3{a_1}^3\hfill \\ \hfill & \hfill & -b_1^2{b}_2{c}_2^2{c}_1{a}_2^3{d}_1^3+b_1^2{b}_2{d}_2^2{d}_1{a}_2^3{c}_1^3-b_1^2{b}_2{d}_2^2{d}_1{c}_2^3{a}_1^3+c_1^2{c}_2{a}_2^2{a}_1{d}_2^3{b}_1^3-c_1^2{c}_2{a}_2^2{a}_1{b}_2^3{d}_1^3\hfill \\ \hfill & \hfill & -c_1^2{c}_2{b}_2^2{b}_1{d}_2^3{a}_1^3+c_1^2{c}_2{b}_2^2{b}_1{a}_2^3{d}_1^3-c_1^2{c}_2{d}_2^2{d}_1{a}_2^3{b}_1^3+c_1^2{c}_2{d}_2^2{d}_1{b}_2^3{a}_1^3+d_1^2{d}_2{a}_2^2{a}_1{b}_2^3{c}_1^3\hfill \\ \hfill & \hfill & \left.-d_1^2{d}_2{a}_2^2{a}_1{c}_2^3{b}_1^3-d_1^2{d}_2{b}_2^2{b}_1{a}_2^3{c}_1^3+d_1^2{d}_2{b}_2^2{b}_1{c}_2^3{a}_1^3+d_1^2{d}_2{c}_2^2{c}_1{a}_2^3{b}_1^3-d_1^2{d}_2{c}_2^2{c}_1{b}_2^3{a}_1^3\right)\ne 0,\hfill \end{array}$$

(86)

$$\begin{array}{ccc}\hfill detM& =\hfill & \frac{g_1^{\prime }{g}_2^{\prime }{g}_3^{\prime }{g}_4^{\prime }}{a_3^2{b}_3^2{d}_3^2{c}_3^2}\left(-a_2^2{a}_3{b}_2{b}_3^2{c}_3^3{d}_2^3+a_2^2{a}_3{b}_2{b}_3^2{c}_2^3{d}_3^3-a_2^2{a}_3{c}_2{c}_3^2{d}_3^3{b}_2^3+a_2^2{a}_3{c}_2{c}_3^2{b}_3^3{d}_2^3\right.\hfill \\ \hfill & \hfill & -a_2^2{a}_3{d}_2{d}_3^2{b}_3^3{c}_2^3+a_2^2{a}_3{d}_2{d}_3^2{c}_3^3{b}_2^3+b_2^2{b}_3{a}_2{a}_3^2{c}_3^3{d}_2^3-b_2^2{b}_3{a}_2{a}_3^2{c}_2^3{d}_3^3+b_2^2{b}_3{c}_2{c}_3^2{d}_3^3{a}_2^3\hfill \\ \hfill & \hfill & -b_2^2{b}_3{c}_2{c}_3^2{a}_3^3{d}_2^3+b_2^2{b}_3{d}_2{d}_3^2{a}_3^3{c}_2^3-b_2^2{b}_3{d}_2{d}_3^2{c}_3^3{a}_2^3+c_2^2{c}_3{a}_2{a}_3^2{d}_3^3{b}_2^3-c_2^2{c}_3{a}_2{a}_3^2{b}_3^3{d}_2^3\hfill \\ \hfill & \hfill & -c_2^2{c}_3{b}_2{b}_3^2{d}_3^3{a}_2^3+c_2^2{c}_3{b}_2{b}_3^2{a}_3^3{d}_2^3-c_2^2{c}_3{d}_2{d}_3^2{a}_3^3{b}_2^3+c_2^2{c}_3{d}_2{d}_3^2{b}_3^3{a}_2^3+d_2^2{d}_3{a}_2{a}_3^2{b}_3^3{c}_2^3\hfill \\ \hfill & \hfill & \left.-d_2^2{d}_3{a}_2{a}_3^2{c}_3^3{b}_2^3-d_2^2{d}_3{b}_2{b}_3^2{a}_3^3{c}_2^3+d_2^2{d}_3{b}_2{b}_3^2{c}_3^3{a}_2^3+d_2^2{d}_3{c}_2{c}_3^2{a}_3^3{b}_2^3-d_2^2{d}_3{c}_2{c}_3^2{b}_3^3{a}_2^3\right)\ne 0.\hfill \end{array}$$

(87)

•	the case of equal symmetries
•	the case of higher symmetry—subclass I
•	the case of higher symmetry—subclass II

Let us notice analogically the case of the hyperbolic Monge–Ampère equation that the mixed derivatives of these solutions depend on all four independent variables x, r, t, and z. We can now repeat the reasoning, from Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 , that the Killing equation cannot be satisfied identically for any solution of the Legendre transformed second heavenly equation of Plebański (36).

In these above three cases: det(M) ≠ 0, when the corresponding polynomials, included in (85)–(87), do not possess zeroes and g′₁ g′₂ g′₃ g′₄ ≠ 0. Therefore, after the assumption that a ₃ b ₃ c ₃ d ₃ ≠ 0 (the case of equal symmetries) and c ₁ d ₂ c ₂ d ₁ b ₂ b ₁ a ₂ a ₁ ≠ 0, a ₃ b ₃ d ₃ c ₃ ≠ 0 (the case of higher symmetries), we may say that Σ₁ = a ₁ x + a ₂ r + a ₃ t + a ₄ z + β ₂, …, Σ₄ = d ₁ x + d ₂ r + d ₃ t + d ₄ z + β ₅ [where the coefficients satisfy (77)–(79), respectively], are linearly independent and the transformations from x, r, t, z to Σ _j , (j = 1, 2, 3, 4) are invertible. Then, we can express x, r, t, z by Σ _j , j = 1, 2, 3, 4; therefore, we can choose Σ _j , j = 1, 2, 3, 4, as new independent variables in the Killing equation (derived in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ). After substituting each of the above classes of solutions in this equation, we obtain a relation of the form

These above classes of exact solutions have been obtained by solving the systems of second-order linear equations together with Legendre-transformed second heavenly equation (36), they are determined up to arbitrary constants because we may choose each of the function g _j = f(Σ _j ), as f(δ + εΣ _j ), where δ, ε are arbitrary constants. Let us fix now the functions g _k , (k = 1, …, 4) in found classes of solutions, such that conditions (85) [together with (80)], (86) [together with (81)], and (87) [together with (80)] are satisfied, respectively. For example, if we choose g _j = exp(Σ _j ) in the above found classes of solutions, we obtain solutions similar to the solutions given either by (44) and (45) or by (44) and (46), if n = 4. Therefore, having no functional arbitrariness, the solutions belonging to these above classes with fixed functions g _k , (k = 1, …, 4) cannot satisfy in addition the first-order equation (Killing equation derived in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ). Hence, this equation cannot be tautology for the solutions belonging to these above classes for the Legendre-transformed potential Π(x, r, t, z), satisfying the corresponding equation (derived in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ) and for suitable choice of the functions q, c, e, ρ, σ, ψ and the constants a, k.

Thus, we have showed that the metric (37) with ϑ, being a solution, belonging to the classes, given by (6) and Σ₁ = a ₁ x + a ₂ r + a ₃ t + a ₄ z + β ₂, …, Σ₄ = d ₁ x + d ₂ r + d ₃ t + d ₄ z + β ₅ [where in the case of system (38)–(40), the coefficients satisfy the relations (77) and in the case of system (41)–(43), the coefficients satisfy the relations (78) or (79)] and the functions g _j , (j = 1, …, 4) are fixed, does not possess the Killing vector.

In order to obtain some extensions of classes of the solutions, given by (6) and (77)–(79)—the series of n (we assume as yet that n is a finite number) functions g _i , it is convenient to write the ansatz down in the convention applied in formula (44) from Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 . Namely, the argument in each function g _j is now α _j x + γ _j r + ζ _j t + λ _j z + β _j . Hence, our ansatz has the following form:

$$𝜗\left(x,r,t,z\right)=\sum _{j=1}^n{g}_j\left({\mathrm{\Sigma }}_j\right),$$

(89)

where g _j are arbitrary holomorphic functions of

$${\mathrm{\Sigma }}_j={\alpha }_{j}x+{\gamma }_{j}r+{\zeta }_{j}t+{\lambda }_{j}z+{\beta }_j.$$

(90)

Obviously, now the notation of the coefficients changes. We give here this change for j = 1, …, 4,

$$\begin{array}{c}{a}_1={\alpha }_1,a_2={\gamma }_1,a_3={\zeta }_1,a_4={\lambda }_1,\\ b_1={\alpha }_2,b_2={\gamma }_2,b_3={\zeta }_2,b_4={\lambda }_2,\\ c_1={\alpha }_3,c_2={\gamma }_3,c_3={\zeta }_3,c_4={\lambda }_3,\\ d_1={\alpha }_4,d_2={\gamma }_4,d_3={\zeta }_4,d_4={\lambda }_4.\end{array}$$

(91)

Then, (89) is some class of solutions of systems (38)–(40) and (41)–(43) (respectively) and, in consequence, of second heavenly equation (36), if the coefficients satisfy the following relations:

1.	in the case of equal symmetries

$${\alpha }_j=\frac{{\gamma }_j\left({\gamma }_j+{\zeta }_j\right)}{{\zeta }_j},{\lambda }_j=-\frac{{\gamma }_j{\left({\gamma }_j+{\zeta }_j\right)}^2}{{\zeta }_j^2},$$

(92)

2.	in the case of higher symmetry we found two subclasses as follows: • I subclass

•	II subclass

In both cases of equal symmetries and higher symmetry, β _j are arbitrary constants. It turns out that in the case of higher symmetry, the class of the solutions of the second heavenly equation transformed by Legendre transformation is given by functional series, which appear in (89), and this series can be infinite.

However, we have here the functional series (89). Therefore, now we need to apply some properties of the functional series. ⁴⁹ 49. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (In Polish) (PWN, Warszawa, 1983), Vol. 1. Namely, from the requirement of differentiability of (89), we have the requirement that it needs to be uniformly convergent. In addition, from the requirement of differentiability of (89), we see that the corresponding series [for the coefficients satisfying the relations (92)],

$$\begin{array}{c}\sum _{j=1}^n\frac{\partial }{\partial x}{g}_j,\dots ,\sum _{j=1}^n\frac{\partial }{\partial z}{g}_j,\end{array}$$

(95)

need to be uniformly convergent.

Furthermore, from the requirement of differentiability of series (95), the consecutive series, including the terms obtained by computing the derivatives in (95) [for the coefficients satisfying the relations (92)],

$$\sum _{j=1}^n\frac{\partial }{\partial x}\left(g_j^{\prime }\frac{{\gamma }_j\left({\gamma }_j+{\zeta }_j\right)}{{\zeta }_j}\right),\sum _{j=1}^n\frac{\partial }{\partial t}\left(g_j^{\prime }\frac{{\gamma }_j\left({\gamma }_j+{\zeta }_j\right)}{{\zeta }_j}\right),\dots$$

(96)

need to be uniformly convergent too, that system (38)–(40), for (89), when the relations (92) are satisfied.

One can also check that the ansatz (89) with the parameters satisfying (92) satisfies Eq. (36) too.

Analogically, in the case of higher symmetry or of system (41)–(43), series (89) needs also to be uniformly convergent. In addition, for the case of the relations (93) and (94), from the requirement of differentiability of (89), we see that the corresponding series,

$$\begin{array}{c}\sum _{j=1}^n\frac{\partial }{\partial x}{g}_j,\dots ,\sum _{j=1}^n\frac{\partial }{\partial z}{g}_j,\end{array}$$

(97)

need to be uniformly convergent.

Furthermore, from the requirement of differentiability of series (97), the consecutive series, including the terms obtained by computing the derivatives in (97),

$$\sum _{j=1}^n\frac{\partial }{\partial x}\left(g_j^{\prime }{\alpha }_j\right),\sum _{j=1}^n\frac{\partial }{\partial x}\left(g_j^{\prime }{\gamma }_j\right),\dots$$

(98)

$$\sum _{j=1}^n\frac{\partial }{\partial x}\left(g_j^{\prime }\frac{{\gamma }_j^2}{{\zeta }_j}\right),\sum _{j=1}^n\frac{\partial }{\partial x}\left(g_j^{\prime }{\gamma }_j\right),\dots$$

(99)

need to be uniformly convergent too. One can check that system (41)–(43) is satisfied by (89), when the coefficients satisfy the relations (93), and a similar situation is in the case when the coefficients satisfy the relations (94).

•	when the relations (93) hold
•	when the relations (94) hold

As we see, ansatz (89), with the relations (90) and (93), is some direct generalization of the solution given by (44) and (46), found in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 . We check now, whether condition (35) is satisfied for these three classes of solutions given by ansatz (89) and (92)–(94), respectively. It turns out that it is satisfied for these three classes, if the following relations hold:

1.

for (92)

$$\left(\sum _{j=1}^n{g}_j^{″}{\zeta }_j^2\right)\left(\sum _{j=1}^n{g}_j^{″}{\gamma }_j^2\right)-{\left(\sum _{j=1}^n{g}_j^{″}{\gamma }_j{\zeta }_j\right)}^2\ne 0,$$

(100)

2.

for (93)

$$\left(\sum _{j=1}^n\frac{g_j^{″}{\gamma }_j^4}{{\alpha }_j^2}\right)\left(\sum _{j=1}^n{g}_j^{″}{\gamma }_j^2\right)-{\left(\sum _{j=1}^n\frac{g_j^{″}{\gamma }_j^3}{{\alpha }_j}\right)}^2\ne 0,$$

(101)

3.

for (94)

$$\left(\sum _{j=1}^n{g}_j^{″}{\zeta }_j^2\right)\left(\sum _{j=1}^n{g}_j^{″}{\gamma }_j^2\right)-{\left(\sum _{j=1}^n{g}_j^{″}{\gamma }_j{\zeta }_j\right)}^2\ne 0.$$

(102)

Conditions (85)–(87) are satisfied also for (89), (90), and (92)–(94), respectively, but, of course, the notation for the coefficients a _j , b _j , c _j , d _j , (j = 1, …, 4) changes according to (91).

Hence, now we may repeat the similar reasonings (included in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ), as previously, and we may say that if n ≥ 4, then Σ _j = α _j x + γ _j r + ζ _j t + λ _j z + β _j , (j = 1, 2, 3, 4), are linearly independent and the transformations from x, r, t, z to Σ _j , (j = 1, 2, 3, 4) are invertible, when the conditions (85)—for the equal symmetries and (86) and (87)—for the higher symmetry are satisfied [after taking into consideration the relations (91)]. Hence, we can express x, r, t, z by Σ _i and the same for Σ₅, …, Σ _n , so we can choose Σ _j , j = 1, 2, 3, 4, as new independent variables in the Killing equation (derived in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ). Hence, after substituting any solution belonging to each of the classes, given either by (89), (90), (92) or by (89), (90), (93) or by (89), (90), (94), respectively, in the Killing equation mentioned above, this equation will possess the form (88).

The solutions belonging to such classes, obtained by solving the systems of linear equations together with Legendre-transformed second heavenly Eq. (36), are determined up to arbitrary constants, because we may choose each of the function g _j = f(Σ _j ), as f(δ + εΣ _j ), where δ, ε are arbitrary constants. Now, let us fix the functions g _j , (j = 1, …, n) in (89) for each of the obtained classes, such that conditions (85) [together with (100)], (86) [together with (101)], and (87) [together with (102)] are satisfied, respectively, [of course, the notation for the coefficients a _j , b _j , c _j , d _j , (j = 1, …, 4) changes according to (91), and, therefore, the relations ζ ₁ ζ ₂ ζ ₃ ζ ₄ ≠ 0, α ₁ γ ₁ α ₂ γ ₂ α ₃ γ ₃ α ₄ γ ₄ ≠ 0, and ζ ₁ ζ ₂ ζ ₃ ζ ₄ ≠ 0 need to hold, respectively]. For example, if we choose g _j = exp(Σ _j ) in (89), then we obtain a form similar to the form of ansatz (44). Therefore, after fixing functions g _j , (j = 1, …, n), these solutions having no functional arbitrariness cannot solve, in addition, the first-order Killing equation (derived in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ).

Hence, this equation cannot be tautology for the solutions belonging to these above three classes, for the Legendre-transformed potential Π(x, r, t, z), satisfying the appropriate equation (derived in Ref. 7 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010 ), for suitable choice of the functions q, c, e, ρ, σ, ψ and the constants a, k. Then, the metrics generated by these solutions do not admit the Killing vector.

Thus, we have proved the following theorem:

Theorem 2.

The metric (37) with ϑ, being the exact solution of (36) , belonging to any class, defined by (89) [where g _i are the functions of (90) ], when n is an arbitrary natural number, and by the relations:

1.	(92) —class I (the case of equal symmetries),
2.	(93) —subclass I (the case I of higher symmetry),
3.	(94) —subclass II (the case II of higher symmetry),

where the functions g _j , (j = 1, …, n) are fixed, does not possess the Killing vector, when n ≥ 4 and the conditions, (85) , ζ ₁ ζ ₂ ζ ₃ ζ ₄ ≠ 0, (100) , n is finite—for class I (the case of equal symmetries), (86) , α ₁ γ ₁ α ₂ γ ₂ α ₃ γ ₃ α ₄ γ ₄ ≠ 0, (101) —for subclass I (the case of higher symmetry) and (87) , ζ ₁ ζ ₂ ζ ₃ ζ ₄ ≠ 0, (102) —for subclass II (the case of higher symmetry), after taking into consideration the relations (91) , are satisfied. In the case of higher symmetry, the class of the solutions is given by an infinite functional series.

In all these cases, series (89) and some other corresponding series mentioned above need to be uniformly convergent.

In the case of the higher symmetry, one can check by using Maple Waterloo Software that the ansatz $𝜗\left(x,r,t,z\right)={\sum }_{j=1}^{\infty }{g}_j\left({\mathrm{\Sigma }}_j\right)$ (where g _j are arbitrary holomorphic functions of Σ _j = α _j x + γ _j r + ζ _j t + λ _j z + β _j ) also gives the class of the solutions of system (41)–(43) and also of (36), when the relations (93) and (94) hold. One can also check that condition (35) is satisfied in the case of this above ansatz and the relations (93) and (94). We can formulate the following theorem:

Theorem 3.

The second heavenly equation (36) [obtained by a Legendre transformation of (31) ] and original version of the second heavenly equation (31), not transformed by the Legendre transformation, possess the class of solutions of the form of the functional series

$$u\left(x^{\mu }\right)={\beta }_1+\sum _{j=1}^n{g}_j\left({\mathrm{\Sigma }}_j\right),$$

(103)

where u = v in the case of (31) and u = ϑ in the case of (36) , g _j are some arbitrary holomorphic functions of the arguments ${\mathrm{\Sigma }}_m=a_{\nu }^{\left(m\right)}{x}^{\nu }+{\beta }_{m+1}$ , [n can be any natural number, so the series ${\sum }_{j=1}^n{g}_j\left({\mathrm{\Sigma }}_j\right)$ can be finite or infinite], x ^ν are proper independent variables, the series ${\sum }_{j=1}^n{g}_j\left({\mathrm{\Sigma }}_j\right)$ and its corresponding derivatives, are uniformly convergent, $a_{\nu }^{\left(m\right)}$ are some constants satisfying some relations following from satisfying of the system of algebraic equations, following from applying of the decomposition method to the versions of second heavenly equation: (36) and (31) , and β _m+1 are arbitrary constants.

Proof 1.

This is sufficient to prove that for any n > 0 (n ∈ 1, 2, 3, …), of course, after proper changing of the independent variables in ansatz (103) , one can decompose the equation obtained after substituting this ansatz in (36) and (31) , according to the idea of the decomposition method, ³⁶ 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075 and next, that according to this method, one can obtain a system of algebraic equations. The solutions of this system establish the relations between the coefficients, which occur in (103) . If these relations are satisfied, then the ansatz (103) gives some class of the solutions of Eqs. (36) and (31) . We prove this theorem by mathematical induction. ⁵⁰ 50. H. Rasiowa, Introduction to Modern Mathematics, edited by O. Wojtasiewicz (North-Holland Publishing Company Amsterdam, PWN , Warszawa, 1973).

1.	We check whether this Theorem holds for n = 1, after substituting the ansatz (103) into (36) and (31) , we obtain the equation

$$g_1^{″}{a}_1^{\left(1\right)}{a}_3^{\left(1\right)}+g_1^{″}{a}_2^{\left(1\right)}{a}_4^{\left(1\right)}=0,$$

(104)

therefore, $g_1^{\prime \prime }\left(a_1^{\left(1\right)}{a}_3^{\left(1\right)}+a_2^{\left(1\right)}{a}_4^{\left(1\right)}\right)=0$ and it suffices to find the solution of the algebraic equation $a_1^{\left(1\right)}{a}_3^{\left(1\right)}+a_2^{\left(1\right)}{a}_4^{\left(1\right)}=0$ .

2.	We prove that if this Theorem holds for n = k, then this holds also for n = k + 1

After substituting the ansatz $u\left(x^{\mu }\right)={\beta }_1+{\sum }_{j=1}^{n=k}{g}_j\left({\mathrm{\Sigma }}_j\right)$ into (36) and (31) , and collecting the algebraic terms by $g_i^{\prime \prime }g{\prime \prime }_j$ and $g_i^{\prime \prime }$ , one obtains the following differential equation [it follows from the fact that Eqs. (36) and (31) satisfy the assumptions of the decomposition method]:

$$\begin{array}{ll}\hfill & g_1^{\prime \prime 2}{X}_1+g_1^{″}{g}_2^{″}{X}_2+g_1^{″}{g}_3^{″}{X}_3+\cdots +g_1^{″}{g}_m^{″}{X}_m+g_2^{\prime \prime 2}{X}_2+\cdots \hfill \\ \hfill & +g_{k-1}^{″}{g}_k^{″}{X}_{k-1}+{\left(g_k^{″}\right)}^2{X}_k=0,\hfill \end{array}$$

(105)

where X _i , (i = 1, …, k) are some polynomials including the constants $a_{\nu }^{\left(k\right)}$ . If we demand vanishing of these polynomials, we obtain some system of algebraic equations. Its solutions establish the relations between $a_{\nu }^{\left(n\right)}$ . Next, after substituting the ansatz $u\left(x^{\mu }\right)={\beta }_1+{\sum }_{n=1}^{n=k+1}{g}_j\left({\mathrm{\Sigma }}_j\right)$ into (36) and (31) , and collecting the algebraic terms by $g_i^{\prime \prime }{g}_j^{\prime \prime }$ and $g_i^{\prime \prime }$ , one obtains the following differential equation [it also follows from the fact that Eqs. (36) and (31) satisfy the assumptions of the decomposition method]:

$$\begin{array}{ll}\hfill & g_1^{\prime \prime 2}{X}_1+g_1^{″}{g}_2^{″}{X}_2+g_1^{″}{g}_3^{″}{X}_3+\dots +g_1^{″}{g}_m^{″}{X}_m+g_2^{\prime \prime 2}{X}_2+\dots \hfill \\ \hfill & +g_k^{″}{g}_{k+1}^{″}{X}_k+{\left(g_{k+1}^{″}\right)}^2{X}_{k+1}=0,\hfill \end{array}$$

(106)

where X _i , i = 1, …, k + 1 are some polynomials including the constants $a_{\nu }^{\left(k+1\right)}$ . If we demand vanishing of these polynomials, we obtain again some system of algebraic equations. Similarly, as previously, its solutions establish the relations between $a_{\nu }^{\left(n\right)}$ .

q.e.d.

In all these above considerations, the condition $g_j^{\prime \prime }\ne 0$ need to be satisfied.

D. The criterion for non-invariance of found solutions and non-existence of Killing vectors

From the results obtained in Subsections IV A–IV C, especially from the forms of the Jacobians for the found classes of exact solutions, the following criterion follows immediately:

Corollary 1.

Let the ansatz

$$u\left(x^k\right)=\sum _{j=1}^n{g}_j\left({\mathrm{\Sigma }}_j\right),k=\mathrm{1,2,3,4},$$

(107)

where Σ _j = α _j x ¹ + γ _j x ² + ζ _j x ³ + λ _j x ⁴ + β _j , the coefficients α _j , γ _j , ζ _j , λ _j , β _j satisfy some relations, and n = 4 (for the equations: elliptic and hyperbolic complex Monge–Ampère) or n ≥ 4 (for second heavenly equation), $g_j\in {\mathcal{C}}^2$ are arbitrary functions [however, in the case of the elliptic complex Monge–Ampère equation, g ₁, g ₂ are the square functions of their arguments, and in the cases of equal symmetries and higher symmetry for the second heavenly equation, the ansatz (107) needs to be convergent series, some series including terms obtained after computing derivatives of (107) need to be uniformly convergent too; in the case of the higher symmetry, the series appearing in (107) can be infinite; moreover, in the case of elliptic and hyperbolic complex Monge–Ampère equation, the solutions belonging to the corresponding classes, given by (107) , need to be real], gives the class of exact solutions of elliptic and hyperbolic complex Monge–Ampère equations and second heavenly equation, when corresponding conditions of the existence of Legendre transformation are satisfied (in the case of the equations hyperbolic complex Monge–Ampère, second heavenly).

These solutions are non-invariant; if first derivatives of the all functions g _n , (n = 1, …, 4) are non-zero, the polynomials included in Jacobians, corresponding to each of these above mentioned classes of solutions, do not possess zeroes and the coefficients α _j , γ _j , ζ _j , λ _j satisfy some additional relations (in the case of the second heavenly equation, uniform convergence of the series and their proper derivatives is also required). We assume additionally that in the case of the equations, hyperbolic complex Monge–Ampère and second heavenly, the functions g _j in (107) are some fixed functions. Then, one can show that in the case of these equations, the metrics generated by their solutions given by (107) and the coefficients α _j , γ _j , ζ _j , λ _j , β _j satisfying some relations (derived in this paper) do not admit the Killing vectors (in the case of the hyperbolic complex Monge–Ampère equation, these solutions are given by classes I and III).

V. CONCLUSIONS

Section:

We applied the decomposition method for the finding of classes of exact solutions (functionally invariant solutions) of the complex Monge–Ampere equations, the elliptic and hyperbolic one, and of the second heavenly equation. For each of these equations, we have obtained the algebraic determining system, following from substituting the ansatz into the investigated equation. Apart from satisfying such an algebraic determining system and the condition of non-invariance of wanted solutions, belonging to our classes (which implicated non-vanishing of the Jacobians), some additional conditions must be satisfied by the functions included in ansatz and by the coefficients. It depends on the following investigated equations:

1.	elliptic complex Monge–Ampère equation—the condition of reality of the solutions,
2.	hyperbolic complex Monge–Ampère equation—the condition of existence of Legendre transformation and the condition of reality of the solutions,
3.	second heavenly equation—the condition of existence of Legendre transformation; the class of exact solutions is given by general functional series (for any n); obviously, this series needs to be uniformly convergent, and some series obtained by twice differentiating of this series need to be also uniformly convergent.

Obviously, in the case of the hyperbolic complex Monge–Ampère equation and the second heavenly equation, the corresponding conditions of existence of Legendre transformation should also be satisfied.

We have also shown that the metrics generated by the found solutions of the hyperbolic complex Monge–Ampère equation (classes of solutions: I and III) and second heavenly equation do not admit the Killing vector. The subclasses I and II (in the case of higher symmetry), of the second heavenly equation, were applied to construct in Ref. 51 51. Ł. T. Stȩpień, in New Trends in Analysis and Interdisciplinary Applications, edited by T. Q. Pei Dang, M. Ku, and L. G. Rodino (Springer International Publishing, 2017), p. 327. , some exact solutions of self-dual Yang–Mills (SDYM) equations, in the non-R gauge case, owing to the reduction of SDYM equations to the second heavenly equation, done by Plebański et al. in Ref. 52 52. J. F. Plebański, M. Przanowski, and H. García-Compeán, Acta Phys. Pol., B 25, 1079 (1994). . As we have established earlier, in the case of the higher symmetry of the second heavenly equation, the series appearing in (107) can be infinite.

We tried to keep the generality of the functions included in the ansatz as it was possible too.

To sum up, we have found some new classes of exact, non-invariant solutions of each of all the above mentioned equations, and we have established also the criterion for the non-invariance of the solutions belonging to these classes.

Non-invariant solutions of heavenly equations, obtained in this paper, cannot generate metrics on K3; however, the decomposition method gives some opportunity for finding solutions for the elliptic Monge–Ampère equation, which can generate such metrics. Finding such solutions can be possible, provided one finds proper solutions of the algebraic system (49)–(54), which setup the relations between the coefficients appearing in (47), such that all conditions necessary to generate the solution of the elliptic Monge–Ampère equation, some metrics not admitting Killing vectors, will be satisfied—this is one of the directions of research. ⁶⁶ 66. Ł. T. Stȩpień, "On some exact solutions of elliptic Monge-Ampère equation," (unpublished).

Moreover, although there are several methods of solving the nonlinear partial differential equations (presented for e.g., in Ref. 35 35. G. Adomian, Solving Frontiers Problems of Physics: The Decomposition Method (Springer Science+Business Media, Dordrecht, 1994). , 44 44. S. V. Meleshko, Methods for Constructing Exact Solutions of Partial Differential Equations (Springer, 2005). , and 53–64 53. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 13, 166 (1979). https://doi.org/10.1007/bf01077483 54. W. I. Fushchich, W. M. Shtelen, and N. Serov, Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics (Kluwer Academic Publishers, Dordrecht, 1993). 55. J. Cieslinski, in Proceedings of First Non-Orthodox School , edited by D. Wójcik and J. Cieślinski (PWN, 1998), p. 81. 56. Zhaosheng Feng, J. Phys. A: Math. Gen. 35, 343 (2002). https://doi.org/10.1088/0305-4470/35/2/312 57. N. A. Kudryashov, Analytic Theory of Nonlinear Differential Equations (Institute of Computer Studies, Moscow-Izhevsk, 2004) (in Russian). 58. P. J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, New York, 1986). 59. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers (Springer Science+BusinessMedia, Birkhäuser, 2012). 60. A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (CRC Press Taylor & Francis Group, 2012). 61. Z.-Y. Zhang, J. Zhong, S. S. Dou, J. Liu, D. Peng, and T. Gao, Rom. Rep. Phys. 65, 1155 (2013). 62. T. M. Elzaki and J. Biazar, World Appl. Sci. J. 24, 944 (2013). https://doi.org/10.5829/idosi.wasj.2013.24.07.1041 63. N. K. Vitanov, Z. I. Dimitrova, and K. N. Vitanov, Appl. Math. Comput. 269, 363 (2015). https://doi.org/10.1016/j.amc.2015.07.060 64. L. I. Rubina and O. N. Ul'yanov, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki 27, 355 (2017). https://doi.org/10.20537/vm170306 ), the decomposition method applied in this paper is not a general method; one can say that this method can offer sometimes a more easy way of finding exact solutions of some nonlinear partial differential equations, like heavenly equations, in comparison with other exact methods.

In some cases, this method can give a possibility of finding classes of exact solutions of a given nonlinear PDE, by applying this method directly to the equation, without the necessity of linearization of this PDE. A good example is the case of the hyperbolic complex Monge–Ampère equation.

Of course, the mentioned above solutions of the equations investigated in this paper are not first found functionally invariant solutions of these equations. Actually, some functionally invariant solutions were found for the second heavenly equation, for example, in Ref. 65 65. M. Dunajski and L. J. Mason, J. Math. Phys. 44, 3430 (2003). https://doi.org/10.1063/1.1588466 . However, they possess a different form from that of the solutions presented in Sec. IV of the current paper.

It is easy to check that the functionally invariant solutions found in the current paper for the second heavenly equation have a more general functional form than the multikink solutions found in Ref. 26 26. A.-M. Wazwaz, Proc. Rom. Acad., Ser. A 17, 210 (2016). .

ACKNOWLEDGMENTS

The author is indebted to the referee for valuable remarks, which allowed us to improve the quality of this paper. The author thanks Professor M. Sheftel for a very interesting discussion. The author is also indebted to Dr Z. Lisowski, Professor V. Mityushev, Professor K. Sokalski, and Professor J. Szczȩsny for their valuable remarks.

The computations were carried out by using Waterloo MAPLE Software on the computers, "mars" and "saturn" (Grant Nos. MNiI/IBM BC HS21/AP/057/2008 and MNiI/Sun6800/WSP/008/2005, respectively) in ACK-Cyfronet AGH in Kraków (Poland). Some part of the computations was done by using Waterloo Maple Software, owing to the financial support provided by The Pedagogical University of Cracow for the research project "Application of some analytical and numerical methods for solving problems in quantum mechanics and field theory" (the leader of this project: Dr. K. Rajchel). This research was supported also by the PL-Grid Infrastructure. The computations were carried out also in the Interdisciplinary Center for Mathematical and Computer Modeling (ICM) under Grant No. G31-6.

The author thanks The Pedagogical University of Cracow for financial support of the publication of this paper.

REFERENCES

Section:

1. 1. H. Stephani, D. Kramer, E. H. M. MacCallum, and E. Herlt, Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge, 2001). Google Scholar
2. 2. F. J. Ernst, Phys. Rev. 167, 1175 (1968). https://doi.org/10.1103/physrev.167.1175, Google Scholar Crossref
3. 3. C. Klein and O. Richter, Ernst Equation and Riemann Surfaces, Lecture Notes in Physics 685 (Springer, Berlin, Heidelberg, 2005). Google Scholar Crossref
4. 4. J. F. Plebański, J. Math. Phys. 16, 2395 (1975). https://doi.org/10.1063/1.522505, Google Scholar Scitation , ISI
5. 5. C. P. Boyer and P. Winternitz, J. Math. Phys. 30, 1081 (1989). https://doi.org/10.1063/1.528379, Google Scholar Scitation , ISI
6. 6. B. Abraham-Shrauner, Phys. Lett. A 369, 299 (2007). https://doi.org/10.1016/j.physleta.2007.04.100, Google Scholar Crossref
7. 7. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 37, 7527 (2004). https://doi.org/10.1088/0305-4470/37/30/010, Google Scholar Crossref
8. 8. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Theor. 40, 9371 (2007). https://doi.org/10.1088/1751-8113/40/31/014, Google Scholar Crossref
9. 9. S. V. Manakov and P. M. Santini, Phys. Lett. A 359, 613 (2006). https://doi.org/10.1016/j.physleta.2006.07.011, Google Scholar Crossref
10. 10. G. Boillat and T. Ruggeri, Boll. Un. Mat. Ital. A 15, 197 (1978). Google Scholar
11. 11. G. Moreno, Banach Cent. Publ. 113, 275 (2017). https://doi.org/10.4064/bc113-0-15, Google Scholar Crossref
12. 12. F. Neyzi, M. B. Sheftel, and D. Yazici, Phys. At. Nucl. 70, 584 (2007). https://doi.org/10.1134/s1063778807030209, Google Scholar Crossref
13. 13. M. B. Sheftel and A. A. Malykh, J. Nonlinear Math. Phys. 15(sup3), 385 (2008). https://doi.org/10.2991/jnmp.2008.15.s3.37, Google Scholar Crossref
14. 14. D. Yazici and M. B. Sheftel, J. Nonlinear Math. Phys. 15(sup3), 417 (2008). https://doi.org/10.2991/jnmp.2008.15.s3.40, Google Scholar Crossref
15. 15. S. W. Hawking, Phys. Lett. A 60, 81 (1977). https://doi.org/10.1016/0375-9601(77)90386-3, Google Scholar Crossref
16. 16. A. A. Malykh, Y. Nutku, and M. B. Sheftel, J. Phys. A: Math. Gen. 36, 10023 (2003). https://doi.org/10.1088/0305-4470/36/39/304, Google Scholar Crossref
17. 17. Y. Nutku, M. B. Sheftel, and A. A. Malykh, Classical Quantum Gravity 14, L59 (1997). https://doi.org/10.1088/0264-9381/14/3/004, Google Scholar Crossref
18. 18. M. Dunajski, Solitons, Instantons and Twistors (Oxford University Press, 2010). Google Scholar
19. 19. M. Robaszewska, Commun. Anal. Geom. 16, 437 (2008). https://doi.org/10.4310/cag.2008.v16.n2.a5, Google Scholar Crossref
20. 20. V. M. Fedorchuk and O. S. Leibov, Ukr. Math. J. 48, 775 (1996). https://doi.org/10.1007/bf02384226, Google Scholar Crossref
21. 21. A. G. Kushner, Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki 151, 60 (2009). Google Scholar
22. 22. B. Doubrov and E. V. Ferapontov, J. Geom. Phys. 60, 1604 (2010); arXiv:0910.3407. https://doi.org/10.1016/j.geomphys.2010.05.009, Google Scholar Crossref
23. 23. S. Kołodziej, Mem. Am. Math. Soc. 178, 840 (2005). https://doi.org/10.1090/memo/0840, Google Scholar Crossref
24. 24. Z. Kuznetsova, Z. Popowicz, and F. Toppan, J. Phys. A: Math. Gen. 38, 7773 (2005); arXiv:hep-th/0504214. https://doi.org/10.1088/0305-4470/38/35/011, Google Scholar
25. 25. F. Toppan, Czech J. Phys. 54, 1387 (2004). https://doi.org/10.1007/s10582-004-9806-y, Google Scholar Crossref
26. 26. A.-M. Wazwaz, Proc. Rom. Acad., Ser. A 17, 210 (2016). Google Scholar
27. 27. M. B. Sheftel and A. A. Malykh, J. Phys. A: Math. Theor. 42, 395202 (2009). https://doi.org/10.1088/1751-8113/42/39/395202, Google Scholar Crossref
28. 28. M. Jakimowicz and J. Tafel, Classical Quantum Gravity 23, 4907 (2006). https://doi.org/10.1088/0264-9381/23/15/010, Google Scholar Crossref
29. 29. V. Husain, Phys. Rev. Lett. 72, 800 (1994). https://doi.org/10.1103/physrevlett.72.800, Google Scholar Crossref
30. 30. J. D. Finley, J. F. Plebański, M. Przanowski, and H. García-Compeán, Phys. Lett. A 181, 435 (1993). https://doi.org/10.1016/0375-9601(93)91145-u, Google Scholar Crossref
31. 31. J. F. Plebański and M. Przanowski, Phys. Lett. A 212, 22 (1996). https://doi.org/10.1016/0375-9601(96)00025-4, Google Scholar Crossref
32. 32. C. P. Boyer and J. F. Plebański, J. Math. Phys. 18, 1022 (1977). https://doi.org/10.1063/1.523363, Google Scholar Scitation , ISI
33. 33. J. F. Plebański, M. Przanowski, B. Rajca, and J. Tosiek, Acta Phys. Pol., B 26, 889 (1995). Google Scholar
34. 34. A. A. Malykh and M. B. Sheftel, J. Phys. A: Math. Theor. 44, 155201 (2011); arXiv:1011.2479. https://doi.org/10.1088/1751-8113/44/15/155201, Google Scholar
35. 35. G. Adomian, Solving Frontiers Problems of Physics: The Decomposition Method (Springer Science+Business Media, Dordrecht, 1994). Google Scholar Crossref
36. 36. Ł. T. Stȩpień, J. Comput. Appl. Math. 233, 1607 (2010). https://doi.org/10.1016/j.cam.2009.02.075, Google Scholar Crossref
37. 37. K. Sokalski, Acta Phys. Pol., A 65, 457 (1984). Google Scholar
38. 38. Ł. Stȩpień, K. Sokalski, and D. Sokalska, J. Phys. A: Math. Gen. 35, 6157 (2002). https://doi.org/10.1088/0305-4470/35/29/315, Google Scholar Crossref
39. 39. Ł. T. Stȩpień, in Current Trends in Analysis and its Applications, edited by V. V. Mityushev and M. V. Ruzhansky (Springer International Publishing, 2015), p. 273. Google Scholar Crossref
40. 40. Ł. T. Stȩpień, arXiv:1208.2905 (2012). Google Scholar
41. 41. S. L. Sobolev, Travaux Inst. Physico–Math. Stekloff, Acad. Sci. 5, 259 (1934). Google Scholar
42. 42. N. Erouguine, CR (Doklady) Acad. Sci. URSS (NS) 42, 371 (1944). Google Scholar
43. 43. N. P. Erugin and M. M. Smirnov, Differ. Uravn. 17, 853 (1981). Google Scholar
44. 44. S. V. Meleshko, Methods for Constructing Exact Solutions of Partial Differential Equations (Springer, 2005). Google Scholar
45. 45. O. F. Men'shikh, Dokl. Akad. Nauk SSSR 205, 30 (1972). Google Scholar
46. 46. O. F. Men'shikh, Differ. Uravn. 11, 533 (1975). Google Scholar
47. 47. Ł. T. Stȩpień, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics, edited by V. S. Gerdjikov, K. Sekigawa, and S. Dimiev (World Scientific, 2009), p. 210. Google Scholar Crossref
48. 48. A. Malykh and M. B. Sheftel, Symmetry, Integrability Geom.: Methods Appl. 7, 043 (2011). https://doi.org/10.3842/SIGMA.2011.043, Google Scholar Crossref
49. 49. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (In Polish) (PWN, Warszawa, 1983), Vol. 1. Google Scholar
50. 50. H. Rasiowa, Introduction to Modern Mathematics, edited by O. Wojtasiewicz (North-Holland Publishing Company Amsterdam, PWN , Warszawa, 1973). Google Scholar
51. 51. Ł. T. Stȩpień, in New Trends in Analysis and Interdisciplinary Applications, edited by T. Q. Pei Dang, M. Ku, and L. G. Rodino (Springer International Publishing, 2017), p. 327. Google Scholar Crossref
52. 52. J. F. Plebański, M. Przanowski, and H. García-Compeán, Acta Phys. Pol., B 25, 1079 (1994). Google Scholar
53. 53. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl. 13, 166 (1979). https://doi.org/10.1007/bf01077483, Google Scholar Crossref
54. 54. W. I. Fushchich, W. M. Shtelen, and N. Serov, Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics (Kluwer Academic Publishers, Dordrecht, 1993). Google Scholar Crossref
55. 55. J. Cieslinski, in Proceedings of First Non-Orthodox School , edited by D. Wójcik and J. Cieślinski (PWN, 1998), p. 81. Google Scholar
56. 56. Zhaosheng Feng, J. Phys. A: Math. Gen. 35, 343 (2002). https://doi.org/10.1088/0305-4470/35/2/312, Google Scholar Crossref
57. 57. N. A. Kudryashov, Analytic Theory of Nonlinear Differential Equations (Institute of Computer Studies, Moscow-Izhevsk, 2004) (in Russian). Google Scholar
58. 58. P. J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, New York, 1986). Google Scholar Crossref
59. 59. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers (Springer Science+BusinessMedia, Birkhäuser, 2012). Google Scholar Crossref
60. 60. A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations (CRC Press Taylor & Francis Group, 2012). Google Scholar
61. 61. Z.-Y. Zhang, J. Zhong, S. S. Dou, J. Liu, D. Peng, and T. Gao, Rom. Rep. Phys. 65, 1155 (2013). Google Scholar
62. 62. T. M. Elzaki and J. Biazar, World Appl. Sci. J. 24, 944 (2013). https://doi.org/10.5829/idosi.wasj.2013.24.07.1041, Google Scholar Crossref
63. 63. N. K. Vitanov, Z. I. Dimitrova, and K. N. Vitanov, Appl. Math. Comput. 269, 363 (2015). https://doi.org/10.1016/j.amc.2015.07.060, Google Scholar Crossref
64. 64. L. I. Rubina and O. N. Ul'yanov, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki 27, 355 (2017). https://doi.org/10.20537/vm170306, Google Scholar Crossref
65. 65. M. Dunajski and L. J. Mason, J. Math. Phys. 44, 3430 (2003). https://doi.org/10.1063/1.1588466, Google Scholar Scitation , ISI
66. 66. Ł. T. Stȩpień, "On some exact solutions of elliptic Monge-Ampère equation," (unpublished). Google Scholar

1. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

A 2b 3c D-3 1 Find the Homogeneous Solution

Source: https://aip.scitation.org/doi/10.1063/1.5144327

Share :