Abstract
This paper deals with the Cauchy problem for a generalized Camassa-Holm equation with high-order nonlinearities, u t − u x x t +k u x +a u m u x =(n+2) u n u x u x x + u n + 1 u x x x , where k,a∈R and m,n∈ Z + . This equation is a generalization of the famous equation of Camassa-Holm and the Novikov equation. The local well-posedness of strong solutions for this equation in Sobolev space H s (R) with s> 3 2 is obtained, and persistence properties of the strong solutions are studied. Furthermore, under appropriate hypotheses, the existence of its weak solutions in low order Sobolev space H s (R) with 1<s≤ 3 2 is established.
Highlights
1 Introduction This work is concerned with the following one-dimensional nonlinear dispersive PDE: ut – uxxt + kux + aumux = (n + )unuxuxx + un+ uxxx, t >, x ∈ R, ( . )
It is worth pointing out that solutions of this type are not mere abstractions: the peakons replicate a feature that is characteristic for the waves of great height - waves of largest amplitude that are exact solutions of the governing equations for irrotational water waves
On the other hand, taking m =, a =, k = in ( . ) we found the Novikov equation [ ]: ut – uxxt + u ux = uuxuxx + u uxxx, t >, x ∈ R, ( . )
Summary
The Cauchy problem for the Camassa-Holm equation Wazwaz [ , ] studied the solitary wave solutions for the generalized Camassa-Holm equation Applying the method of pseudoparabolic regularization, Lai and Wu [ ] investigated the local well-posedness and existence of weak solutions for the following generalized Camassa-Holm equation with dissipative term: ut – uxxt + kux + aumux =
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