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 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.     

Non-uniform dependence on initial data for the periodic modified Camassa–Holm equation

In this paper, we study the periodic Cauchy problem for the modified Camassa–Holm equation $$m_t+um_x+2u_xm=0,\quad m=(1-\partial_x^2)^2u$$ , and show that the solution map is not uniformly continuous in Sobolev spaces ${H^s(\mathbb T)}$ for s > 7/2. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.     

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).     

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.     

Riemann–Hilbert formulation for the KdV equation on a finite interval

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I. Hitzazis, D. Tsoubelis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Blow-up solution near the traveling waves for the second-order Camassa–Holm equation

This paper studies the blow-up solution and its blow-up rate near the traveling waves of the second-order Camassa–Holm equation. The sufficient condition for the existence of blow-up solution is obtained by a rather ingenious method. Applying the extended pseudo-conformal transformation, an equivalent proposition of the solution breaking in finite time near the traveling waves is constructed. The relation is established between the blow-up time and rate of the solution and the residual’s.     

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.     

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.     

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.     

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.