New integrable nonlinear integrodifferential equations and related solvable finite-dimensional dynamical systems

Abstract

A new integrable nonlinear integrodifferential equation (NIDE) is proposed. This equation may be interpreted as a model equation for deep-water waves. The N-periodic and N-soliton solutions for the equation are constructed by means of the bilinear transformation method. These solutions have the same structure as that for the Benjamin–Ono equation which describes internal waves in stratified fluids of great depth. Furthermore, it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite-dimensional dynamical systems described by first-order nonlinear ordinary differential equations. The discussion is also made on a more general NIDE that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth.