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, Contemporary Mathematics, 2011, 526, pp273–294] that when 1/2 < θ ≤ 1 smooth solutions persist for all time, and when 0 ≤ θ ≤ 2 , 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.