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Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities. (English) Zbl 1119.35075

The main goal of this paper is to focus on travelling wave solutions of equations

u t +αuu x +βu xx +su xxx =0(1)


u t +αu p u x +βu 2p u x +γu xx +μu xxx =0,(2)

where α, β, γ, μ 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.

35Q53KdV-like (Korteweg-de Vries) equations
37K40Soliton theory, asymptotic behavior of solutions