D’Alembert wave and soliton molecule of the modified Nizhnik–Novikov–Veselov equation

Abstract

The wave motion equation is one of the fundamental equations to describe vibrations of continuous systems. The D’Alembert solution of the wave motion equation is an important basic formula in linear partial differential equations. The study of the D’Alembert wave deserves deep consideration in nonlinear equations. In this paper, the D’Alembert-type wave of the (2 + 1)-dimensional modified Nizhnik–Novikov–Veselov (mNNV) equation is derived by the Ansatze method. The Hirota bilinear form of the mNNV equation is constructed by introducing the dependent variable transformation. The multi-soliton solution is obtained by solving the corresponding bilinear form. By combining the velocity resonance mechanism, a three-soliton molecule, the interaction between a soliton molecule and one soliton, and the interaction between a soliton–solitoff molecule and one soliton of the mNNV equation are obtained. The dynamics of these solutions are shown by selecting the appropriate parameters. These phenomena for the mNNV equation have not yet been given via other methods.