Multidimensional exact solutions of a system of nonlinear Boussinesq type equations

We study the system of two nonlinear partial differential equations of the fourth order. The right parts of the system of equations contain multidimensional analogs of Boussinesq equation, expressed in terms of two-fold Laplace operators and squares of gradients of the required functions, as well as linear functions of the relationship. This kind of equations, similar to Navier-Stokes equations, encountered in problems of hydrodynamics. The paper proposes to search for a solution in the form of anzatz containing quadratic dependence on spatial variables and arbitrary functions on time. The use of the proposed anzatz allows to decompose the process of finding the solution components depending on the spatial variables and time. To find the dependence on spatial variables it is necessary to solve the algebraic system of matrix, vector and scalar equations. The General solution of this system of equations in parametric form is found. To find the time-dependent components of the solution of the initial system, a system of nonlinear ordinary differential equations arises. This system is reduced to one fourth-order equation for which particular solutions are found. A number of examples of the constructed exact solutions of the initial system of Boussinesq equations, including those expressed in terms of Jacobi functions in time and anisotropic in spatial variables, are given.