Traveling Wave Solutions to the Burgers-αβ Equations

The Burgers-αβ equation, which was first introduced by Holm and Staley [4], is considered in the special case where ν = 0 ν=0 and b = 3 b=3 . Traveling wave solutions are classified to the Burgers-αβ equation containing four parameters b , α , ν b,α,ν , 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 β=0 and β = 1 β=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 ν=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 k₁ . In an appropriate limit β = 1 β=1 , the previously known traveling wave solutions of the Degasperis–Procesi equation are recovered.