Bifurcations and Single Peak Solitary Wave Solutions of an Integrable Nonlinear Wave Equation

Abstract

Dynamical system theory is applied to the integrable nonlinear wave equation $u_t±(u^3−u^2)x+(u^3)xxx=0$. We obtain the single peak solitary wave solutions and compacton solutions of the equation. Regular compacton solution of the equation corresponds to the case of wave speed $c$=0. In the case of $c^6$≠0, we find smooth soliton solutions. The influence of parameters of the traveling wave solutions is explored by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for these soliton solutions of the nonlinear wave equation.