Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

Abstract

By using Frobenius’ idea together with integral bifurcation method, we study a third order nonlinear equation of generalization form of the modified KdV equation, which is an important water wave model. Some exact traveling wave solutions such as smooth solitary wave solutions, nonsmooth peakon solutions, kink and antikink wave solutions, periodic wave solutions of Jacobian elliptic function type, and rational function solution are obtained. And we show their profiles and discuss their dynamic properties aim at some typical solutions. Though the types of these solutions obtained in this work are not new and they are familiar types, they did not appear in any existing literatures because the equationut+ux+νuxxt+βuxxx+αuux+1/3να(uuxxx+2uxuxx)+3μα2u2ux+νμα2(u2uxxx+ux3+4uuxuxx)+ν2μα2(ux2uxxx+2uxuxx2)=0is very complex. Particularly, compared with the cited references, all results obtained in this paper are new.