On Exact Multidimensional Solutions of a Nonlinear System of First Order Partial Differential Equations

Abstract

This study is concerned with a system of two nonlinear first order partial differential equations. The right-hand sides of the system contain the squares of the gradients of the unknown functions. Such type of Hamilton-Jacobi like equations are considered in mechanics and control theory. In the paper, we propose to search a solution in the form of an ansatz, the latter containing a quadratic dependence on the spatial variables and arbitrary functions of time. The use of this ansatz allows us to decompose the search of the solution’s components depending on the spatial variables and time. In order to find the dependence on the spatial variables one needs to solve an algebraic system of some matrix and vector equations and of a scalar equation. A general solution of this system of equations is found in a parametric form. To find the time-dependent components of the solution of the original system, we are faced with a system of nonlinear differential equations. The existence of exact solutions of a certain kind for the original system is established. A number of examples of the constructed exact solutions, including periodic in time and anisotropic in the spatial variables ones, are given. The spatial structure of the solutions is analyzed revealing that it depends on the rank of the matrix of the quadratic form entering the solution.

Keywords

• nonlinear system
• Hamilton-Jacobi type equations
• exact solutions