Dynamical systems in the theory of nonlinear waves with allowance for nonlocal interactions (the Whitham-Benjamin equation).

Abstract

In this paper we briefly present a general approach to the description of the nonlinear and nonlocal Whitham-Benjamin model, based on the introduction of a system of auxiliary fields that interact locally with the initial nonlinear field. In the case of stationary waves a corresponding dynamical system is defined that admits of a Hamiltonian representation. Some results are presented of a qualitative and numerical analysis of the stationary solitary waves of the Whitham-Benjamin equation with a rapidly decreasing oscillatory kernel. An investigation is made into a phenomenon related to the loss of smoothness of the solution of the original equation and the noncontinuability of these solutions when the structural parameters of the system are changed (this phenomenon is analogous to the formation of limiting Stokes waves).