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<p>We study a system of two two-dimensional inhomogeneous Monge–Ampère equations.Such systems arise in problems of hydrodynamics of incompressible two-fluid media.The simplest reductions of this system to systems of ordinary differential equations are obtained using additive separation of variables, when the right-hand sides are written as products of factors depending on derivatives of the unknown functions with respect to each variable, and multiplicative separation of variables, when the right-hand sides contain power nonlinearities in the unknown functions and their derivatives.Reductions and some exact solutions are also constructed in cases where a solution exhibits a prescribed dependence on one of the variables.In particular, solutions linear in one of the unknown variables are considered when the right-hand sides depend linearly on the unknown functions and their derivatives.Solutions with exponential dependence on one of the variables are also considered when the right-hand sides are quadratic polynomials in the unknown functions and their derivatives.It is shown that the system has classical traveling wave solutions if the right-hand sides vanish identically on these solutions.Solutions of generalized traveling wave type and conditions for their existence are derived for the case where the right-hand sides contain products of power nonlinearities in the unknown functions and their derivatives.Conditions for the existence of power and exponential self-similar solutions are also established.Examples of exact solutions of the specified types for the system under&#xa0;consideration&#xa0;are&#xa0;provided.</p>

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System of Two-Dimensional Monge–Ampère Equations: Reductions and Exact Solutions

  • 期刊论文

  • 发表日期：

    2025年7月23日

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  • I. V. Rakhmelevich

摘要

We study a system of two two-dimensional inhomogeneous Monge–Ampère equations.Such systems arise in problems of hydrodynamics of incompressible two-fluid media.The simplest reductions of this system to systems of ordinary differential equations are obtained using additive separation of variables, when the right-hand sides are written as products of factors depending on derivatives of the unknown functions with respect to each variable, and multiplicative separation of variables, when the right-hand sides contain power nonlinearities in the unknown functions and their derivatives.Reductions and some exact solutions are also constructed in cases where a solution exhibits a prescribed dependence on one of the variables.In particular, solutions linear in one of the unknown variables are considered when the right-hand sides depend linearly on the unknown functions and their derivatives.Solutions with exponential dependence on one of the variables are also considered when the right-hand sides are quadratic polynomials in the unknown functions and their derivatives.It is shown that the system has classical traveling wave solutions if the right-hand sides vanish identically on these solutions.Solutions of generalized traveling wave type and conditions for their existence are derived for the case where the right-hand sides contain products of power nonlinearities in the unknown functions and their derivatives.Conditions for the existence of power and exponential self-similar solutions are also established.Examples of exact solutions of the specified types for the system under consideration are provided.