A generalized Boussinesq-type equation and its exact multidimensional solutions

The article studies a nonlinear fourth-order partial differential equation. The right part of the equation contains multidimensional analogs of Boussinesq equation, expressed in terms of two-fold Laplace operators and squares of gradients of the required function. To find the time-dependent components of the original system solution a system of nonlinear ordinary differential equations has been created. This system is reduced to a single fourth-order equation for which partial solutions are found. We give the examples of the constructed exact solutions of the initial system of Boussinesq-type equations, including those expressed in terms of Jacobi and Weierstrass elliptic functions in time and anisotropic ones in spatial variables. The exact solutions found have not only theoretical, but also applied value, since they can be used for testing and verifying numerical methods and algorithms for constructing approximate solutions of boundary value problems for fourth-order nonlinear partial differential equations modeling hydrodynamic processes and phenomena.