Global solutions for the generalized Camassa–Holm equation

In this paper, we study the Cauchy problem of the generalized Camassa–Holm equation. Firstly, we prove the existence of the global strong solutions provide the initial data satisfying a certain sign condition. Then, we obtain the existence and the uniqueness of the global weak solutions under the same sign condition of the initial data.     

Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa–Holm equation

In this paper, we investigate the generalization of the Camassa–Holm equation $u_t+K{\left(u^m\right)}_x?{\left(u^n\right)}_{xxt}={[\frac{{\left({\left(u^n\right)}_x\right)}^2}{2}+u^n{\left(u^n\right)}_{xx}]}_x\text{,}$ where $K$ is a positive constant and $m,n\in \mathbb{N}$. The bifurcation and some explicit expressions of peakons and periodic cusp wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. Further, in the process of obtaining the bifurcation of phase portraits, we show that $K=\frac{m+n}{1+n}{c}^{\frac{n?m+1}{n}}$ is the peakon bifurcation parameter value for the equation. From the bifurcation theory, in general, the peakons can be obtained by taking the limit of the corresponding periodic cusp waves. However, we find that in the cases of $n\ge 2,m=n+1$, when $K$ tends to the corresponding bifurcation parameter value, the periodic cusp waves will no longer converge to the peakons, instead, they will still be the periodic cusp waves. To the best of our knowledge, up until now, this phenomenon has not appeared in any other literature. By further studying the cause of this phenomenon, we show that this planar system has some different characters from the previous Camassa–Holm systems. What is more, we obtain some periodic cusp wave solutions in the form of polynomial functions, which are different from those in the form of exponential functions. Some previous results are extended.     

Riemann–Hilbert approach for the Camassa–Holm equation on the line

We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation $u_t?u_{txx}+2\omega u_x+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}$ on the line (CH). We show that: (i) for all $\omega >0$, the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann–Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for $\omega =0$. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The Camassa–Holm equation on the half-line

We study the initial-boundary-value problem for the Camassa–Holm equation on the half-line by associating to it a matrix Riemann–Hilbert problem in the complex k-plane; the jump matrix is determined in terms of the spectral functions corresponding to the initial and boundary values. We prove that if the boundary values $u(0,t)$ are ?0 for all t then the corresponding initial-boundary-value problem has a unique solution, which can be expressed in terms of the solution of the associated RH problem. In the case $u(0,t)<0$, the compatibility of the initial and boundary data is explicitly expressed in terms of an algebraic relation to be satisfied by the spectral functions. To cite this article: A. Boutet de Monvel, D. Shepelsky, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Global attractor for the viscous modified two-component Camassa–Holm equation

We obtain the existence of global attractor for the Cauchy problem of a viscous modified two-component Camassa–Holm equation. The existence of global strong solutions is obtained using Kato’s theory. The key elements in our analysis are the uniform Gronwall lemma and some estimates of the solutions.     

Blow-up for a weakly dissipative modified two-component Camassa–Holm system

In this paper, we study the Cauchy problem of a weakly dissipative modified two-component Camassa–Holm (MCH2) system. We first derive the precise blow-up scenario and then give several criteria guaranteeing the blow-up of the solutions. We finally discuss the blow-up rate of the blowing-up solutions.     

Orbital stability of floating periodic peakons for the Camassa–Holm equation

Considered herein is the orbital stability of floating periodic peakons for the Camassa–Holm (CH) equation, which describes one-dimensional surface waves at a free surface of shallow water under the influence of gravity. The floating periodic peakons shift up or down according to the change of the parameter. The result shows that the floating periodic peakons are orbitally stable and their stability is independent of the parameter in the CH equation.     

Optimal control of the viscous Camassa–Holm equation

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.     

Algebraic traveling waves for the generalized Newell–Whitehead–Segel equation

In this paper we provide the only possible algebraic traveling wave solutions for the celebrated general Newell–Whitehead–Segel equation.     

Continuity properties of the data-to-solution map for the generalized Camassa–Holm equation

This work studies a generalized Camassa–Holm equation with higher order nonlinearities (g-kbCH). The Camassa–Holm, the Degasperis–Procesi and the Novikov equations are integrable members of this family of equations. g-kb  CH is well-posed in Sobolev spaces H^s$H^s$, s>3/2$s>3/2$, on both the line and the circle and its solution map is continuous but not uniformly continuous. In this work it is shown that the solution map is Hölder continuous in H^s$H^s$ equipped with the H^r$H^r$-topology for 0?r<s$0?r<s$, and the Hölder exponent is expressed in terms of s and r.     

Single peak solitary wave solutions for the generalized Camassa–Holm 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.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.     

Orbital stability of traveling waves for the one-dimensional Gross–Pitaevskii equation

In this paper, we prove the nonlinear orbital stability of the stationary traveling wave of the one-dimensional Gross–Pitaevskii equation by using Zakharov–Shabat's inverse scattering method.