Traveling Wave Solutions to the Burgers-αβ Equations

Abstract

Abstract The Burgers-αβ equation, which was first introduced by Holm and Staley [4], is considered in the special case where ν = 0 ${\nu=0}$ and b = 3 ${b=3}$ . Traveling wave solutions are classified to the Burgers-αβ equation containing four parameters b , α , ν ${b,\alpha,\nu}$ , and β, which is a nonintegrable nonlinear partial differential equation that coincides with the usual Burgers equation and viscous b-family of peakon equation, respectively, for two specific choices of the parameter β = 0 ${\beta=0}$ and β = 1 ${\beta=1}$ . Under the decay condition, it is shown that there are smooth, peaked and cusped traveling wave solutions of the Burgers-αβ equation with ν = 0 ${\nu=0}$ and b = 3 ${b=3}$ depending on the parameter β. Moreover, all traveling wave solutions without the decay condition are parametrized by the integration constant k 1 ∈ ℝ ${k_{1}\in\mathbb{R}}$ . In an appropriate limit β = 1 ${\beta=1}$ , the previously known traveling wave solutions of the Degasperis–Procesi equation are recovered.