Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities.

*(English)*Zbl 1119.35075The main goal of this paper is to focus on travelling wave solutions of equations

$${u}_{t}+\alpha u{u}_{x}+\beta {u}_{xx}+s{u}_{xxx}=0\phantom{\rule{2.em}{0ex}}\left(1\right)$$

and

$${u}_{t}+\alpha {u}^{p}{u}_{x}+\beta {u}^{2p}{u}_{x}+\gamma {u}_{xx}+\mu {u}_{xxx}=0,\phantom{\rule{2.em}{0ex}}\left(2\right)$$

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.

Reviewer: Messoud A. Efendiev (Berlin)