On exact multidimensional solutions to a nonlinear system of fourth-order hyperbolic equations

We study the system of two fourth-order nonlinear hyperbolic partial differential equations. The right-hand sides of the equations contain double Laplace operators and the squares of the gradients of the sought functions. Such equations, close to the Boussinesq equation and the Navier–Stokes equations, occur in problems of hydrodynamics. We propose to search for a solution in the form of an ansatz containing quadratic dependence on the spatial variables and arbitrary functions of time. The use of the proposed ansatz allows us to decompose the process of finding the components of the solution depending on the space variables and time. For finding the dependence on the spatial variables, it is necessary to solve an algebraic system of matrix, vector, and scalar equations. We find the general solution to this system in parametric form. In finding the time-dependent components of the solution to the original system, there arises a system of nonlinear ordinary differential equations. In the particular case when the squares of the gradients are not included in the system, we establish the existence of exact solutions of a certain kind to the original system expressed through arbitrary harmonic functions of the spatial variables and exponential functions of time. Some examples are given of the constructed exact solutions including solutions periodic in time and anisotropic in space variables. The exact solutions can be used to verify numerical methods for the approximate construction of the solutions to applied boundary value problems.