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.