Search

In this paper, we construct two families of new explicit exact traveling wave solutions of the Camassa–Holm equation which solve an open problem in the previous paper [Zhang G, Qiao ZJ, Liu F. Cusp...

The dynamical stability of the periodic peaked solitons for a generalized Camassa–Holm equation is studied in this paper. Its generalization version is known to admit a single-peaked soliton soluti...

In this paper, we study global existence and blow-up for a periodic modified Camassa–Holm equation in nonhomogeneous Sobolev spaces. Also, we provide a key blow-up criteria to investigate norm infl...

In this paper, we mainly study the Cauchy problem of a generalized Camassa–Holm equation. We prove the existence of global weak solutions for this generalized Camassa–Holm equation by the viscous a...

In this paper, we mainly devote to investigate a weakly dissipative shallow water wave model. By overcoming the difficulties caused by the complicated higher-order nonlinear terms and dissipative t...

In this paper, we first establish the local well-posedness and the rotational speed limit for the rotation Camassa–Holm equation modelling the equatorial water waves with the weak Coriolis effect i...

In this paper, we mainly devote to study an integrable generalized Camassa–Holm equation with both quadratic and cubic nonlinearity proposed by Xia, Qiao and Li [An integrable system with peakon, c...

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well posed in Sobolev...

In this paper we consider the Cauchy problem for a generalized fifth-order Camassa–Holm equation. The infinite propagation speed is proved in the following sense: the corresponding solution u(x, t) with compactly supported initial data does not have compact support in its lifespan. Moreover, the asymptotic behaviors of the solution at infinity are considered when the initial data decays exponentially and algebraically, respectively.

This paper deals with the Cauchy problem for the generalized Camassa–Holm equation. The existence of weak solutions for the generalized Camassa–Holm equation in lower order Sobolev space with is obtained.

In this paper, we consider a generalized Camassa–Holm equation with the flow generated by the vector field and its gradient. We first establish the local well-posedness of equation in the sense of Hadamard in both critical Besov spaces and supercritical Besov spaces. Then we gain a blow-up criterion. Under a sign condition we reach the sign-preserved property and a precise blow-up criterion. Applying this precise criterion we finally present two blow-up results and the precise blow-up rate for strong solutions to equation.

This Letter presents all possible smooth, peaked and cusped solitary wave solutions for the generalized Camassa–Holm equation under the inhomogeneous boundary condition.The parametric conditions of existence of the smooth, peaked and cusped solitary wave solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, peaked and cusped solitary wave solutions of the generalized Camassa–Holm equation.

In this paper, we study the dependence on initial data of solutions to Camassa–Holm equation with periodic boundary condition in Besov spaces. We show that when and , the solution map is not uniformly continuous from into for or from into for . Moreover, we prove that if a weaker -topology is used, then the solution map becomes Holder continuous in . It seems that the non-uniform dependence on initial data in periodic Besov spaces has not appeared in the previous literature.

In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation. By using the littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the generalized Camassa-Holm equation is locally well posed in the Besov space B-p,r(s) with 1 max {1+1/p, 3/3} which generalizes the local well-posedness result in [1]. We also establish the ill-posedness result of the generalized Camassa-Holm equation and give a blow-up criterion.

In this article, we consider a newly modified two-component Camassa–Holm equation. First, we establish the local well-posedness result, then we present a precise blow-up scenario. Afterwards, we derive a new conservation law, by which and the precise blow-up scenario we prove three blow-up results and a blow-up rate estimate result.