Nonexistence of travelling wave solution of the Korteweg-de Vries Benjamin Bona Mahony equation

Abstract

This paper is devoted to the Korteweg-de Vries Benjamin Bona Mahony equation in an infinite domain. The paper discusses weak solutions of the Korteweg-de Vries Benjamin Bona Mahony equation without any conditions at infinity. This particular problem arises from the phenomenon of long breaking wave with small amplitude in fluid. In fluid dynamics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy. For the Korteweg-de Vries Benjamin Bona Mahony equation, we obtain the conditions of blowing-up of travelling wave solutions in finite time. Moreover, there is an explicit upper bound estimate for the wavelength of the corresponding singular traveling wave, depending on the speed of waves. The proof of the results is based on the nonlinear capacity method. In closing, we provide the numerical examples.