System of two-dimensional Monge--Ampere equations: reductions and exact solutions

I. V. Rakhmelevich

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 121-136

Résumé

We study the system of two two-dimensional nonhomogeneous Monge–Ampere equations. Such systems are applied in the problems of hydrodynamics of incompressible two-fluid media. We receive the simplest reductions of this system to the systems of ordinary differential equations (ODE) using the method of additive separation of variables in the case when the right-hand sides of the system are represented as products of multipliers depending on the derivatives of the desired functions for each variable. Also we receive the reduction of the considered system to the systems of ODE by the method of multiplicative separation of variables in the case when the right-hand sides of the system contain power-law nonlinearities with respect to the desired functions and their derivatives. We also constructed reductions and some exact solutions in cases where the solution has a given dependence on one of the variables. In particular, solutions linear in one of the variables are considered in the case when the right parts of the system linearly depend on the desired functions and their derivatives. Solutions exponentially dependent on one of the variables are also considered in the case when the right-hand sides of the system have the form of quadratic polynomials of the desired functions and their derivatives. It is shown that the system has solutions of the classical traveling wave type if the right- \eject hand sides of the system are identically equal to 0 on these solutions. Solutions of the type of generalized traveling waves and the conditions of their existence are obtained in the case when the right parts of the system contain products of power-law nonlinearities with respect to the desired functions and their derivatives. Conditions for the existence of power-law and exponential self-similar solutions are also obtained. Examples of exact solutions of these types for the system under consideration are given.