Bifurcation of travelling wave solutions for the generalized Camassa–Holm–KP equations

By using the bifurcation theory of planar dynamical systems to the generalized Camassa–Holm–KP equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

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.     

Peakon weak solutions for the rotation-two-component Camassa–Holm system

Recently, Fan, Gao and Liu proposed a kind of rotation-two-component Camassa–Holm system. In this paper, we investigate whether the rotation-two-component Camassa–Holm system admits peakon-delta weak solutions in distribution sense. As special reductions, all peakon solutions for generalized Dullin–Gottwald–Holm system, two-component Camassa–Holm system, Dullin–Gottwald–Holm equation and Camassa–Holm equation are recovered from the corresponding results of rotation-two-component Camassa–Holm system.     

A Note on the analytic solutions of the Camassa–Holm equation

In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to $H^s(\mathbb{R})$ with $s>3/2$, $6u_0{6}_{L^1}<\infty$ and $u_0?u_{0xx}$ does not change sign, we prove that the solution stays analytic globally in time. To cite this article: M.C. Lombardo et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations

In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa–Holm (GCH) equations. A recent, novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon and cuspon solutions. One of the considered GCH equations supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. The second equation does not support singular traveling waves and the last one supports four-segmented, non-smooth M-wave solutions.Moreover, smooth traveling waves of the three GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of their traveling-wave equations, corresponding to pulse (kink or shock) solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding GCH equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. We also show the traveling wave nature of these pulse and front solutions to the GCH NLPDEs.     

Global existence and blow-up phenomena for a periodic 2-component Camassa–Holm equation

We first establish local well-posedness for a periodic 2-component Camassa?CHolm equation. We then present two global existence results for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation.     

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.     

Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa–Holm system

We study the Cauchy problem of a weakly dissipative 2-component Camassa–Holm system. We first establish local well-posedness for a weakly dissipative 2-component Camassa–Holm system. We then present a global existence result for strong solutions to the system. We finally obtain several blow-up results and the blow-up rate of strong solutions to the system.     

On questions of decay and existence for the viscous Camassa–Holm equations

We consider the viscous n  -dimensional Camassa–Holm equations, with n=2,3,4$n=2,3,4$ in the whole space. We establish existence and regularity of the solutions and study the large time behavior of the solutions in several Sobolev spaces. We first show that if the data is only in L²$L^2$ then the solution decays without a rate and that this is the best that can be expected for data in L²$L^2$. For solutions with data in Hm∩L¹$H^m\cap L^1$ we obtain decay at an algebraic rate which is optimal in the sense that it coincides with the rate of the underlying linear part.     

Wave Breaking of the Camassa–Holm Equation

This paper gives a new and direct proof for McKean’s theorem (McKean in Asian J. Math. 2:867–874, 1998) on wave breaking of the Camassa–Holm equation. The blow-up profile is also analyzed.     

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).     

A generalized nonisospectral Camassa–Holm equation and its multipeakon solutions

Motivated by the paper of Beals, Sattinger and Szmigielski (2000) [3], we propose an extension of the Camassa–Holm equation, which also admits the multipeakon solutions. The novel aspect is that our approach is mainly based on classic determinant technique. Furthermore, the proposed equation is shown to possess a nonisospectral Lax pair.     

Low-Regularity Solutions of the Periodic Camassa–Holm Equation

We prove local existence and uniqueness of weak solutions of the Camassa–Holm equation with periodic boundary conditions in various spaces of low-regularity which include the periodic peakons. The proof uses the connection of the Camassa–Holm equation with the geodesic flow on the diffeomorphism group of the circle with respect to the L ² metric.     

Global Solutions for the Two-Component Camassa–Holm System

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa–Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa–Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.     

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.     

Nonuniform continuity of the solution map to the two component Camassa–Holm system

We prove that the solution map of the two-component Camassa–Holm system is not uniformly continuous as a map from a bounded subset of the Sobolev space H^s(T)×H^r(T)$H^s(\mathbb{T})\times H^r(\mathbb{T})$ to C([0,1],H^s(T)×H^r(T))$C([0,1],H^s(\mathbb{T})\times H^r(\mathbb{T}))$ when s?1$s?1$ and r?0$r?0$. We also demonstrate the nonuniform continuous property in the continuous function space C¹(T)×C¹(T)$C^1(\mathbb{T})\times C^1(\mathbb{T})$.     

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.