Abstract
In this paper, first we survey some recent advances in the study of traveling wave solutions to the Burgers–Korteweg–de Vries equation and some comments are given. Then, we study a Burgers–Korteweg–de Vries-type equation with higher-order nonlinearities. A qualitative analysis to a two-dimensional autonomous system which is equivalent to the Burgers–KdV-type equation is presented, and indicates that under certain conditions, the Burgers–Korteweg–de Vries-type equation has neither nontrivial bell-profile solitary waves, nor periodic waves. Finally, a solitary wave solution is obtained by means of the first-integral method which is based on the ring theory of commutative algebra.
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