On traveling wave solutions of the θ-equation of dispersive type

Traveling wave solutions to a class of dispersive models,ut−utxx+uux=θuuxxx+(1−θ)uxuxx, are investigated in terms of the parameter θ, including two integrable equations, the Camassa-Holm equation, θ=1/3, and the Degasperis-Procesi equation, θ=1/4, as special models. It was proved in H. Liu and Z. Yin (2011) [39] that when 1/2<θ≤1 smooth solutions persist for all time, and when 0≤θ≤12, strong solutions of the θ-equation may blow up in finite time, yielding rich traveling wave patterns. This work therefore restricts to only the range θ∈[0,1/2]. It is shown that when θ=0, only periodic travel wave is permissible, and when θ=1/2 traveling waves may be solitary, periodic or kink-like waves. For 0<θ<1/2, traveling waves such as periodic, solitary, peakon, peaked periodic, cusped periodic, or cusped soliton are all permissible.