Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities (Q864656)

The main goal of this paper is to focus on travelling wave solutions of equations \[ u_t+\alpha uu_x+\beta u_{xx}+ su_{xxx}= 0\tag{1} \] and \[ u_t+\alpha u^p u_x+\beta u^{2p} u_x+\gamma u_{xx}+\mu u_{xxx}= 0,\tag{2} \] where \(\alpha\), \(\beta\), \(\gamma\), \(\mu\) and \(s\) are real constants, and \(p\) is a positive number. The authors transform the so-called Burgers-KdV-type equation (2) to a two-dimensional autonomous system and apply the qualitative theory of planar dynamical systems to analyze the resultant system for its solitary waves. A qualitative analysis to the equation (2) is presented, which indicates that under given parametric conditions, the equation (2) has neither nontrivial bell-profile solitary waves, nor periodic waves. The authors show that a solitary wave solution is obtained by using the first-integral method.