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We suggest a finite dfference scheme for the Camassa-Holm equation that can handle general H1 initial data. The form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality. We prove that the difference scheme converges strongly in $H^1$ towards an exact dissipative weak solution of Camassa-Holm equation.

We consider higher-order Camassa--Holm equations describing exponential curves of the manifold of smooth orientation preserving diffeomorphisms of the unit circle in the plane. We establish the existence of a strongly continuous semigroup of global weak solutions. We also present some invariant spaces under the action of that semigroup. Moreover, we prove a "weak equals strong" uniqueness result.

Recent work [4] has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper we construct numerical schemes and prove that they converge to entropy solutions. Additionally, we provide several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data. Our focus on discontinuous solutions contrasts notably with the existing literature on the Degasperis-Procesi equation, which seems to emphasize similarities with the Camassa-Holm equation (bi-Hamiltonian structure, integrabillity, peakon solutions, H1 as the relevant functional space).

We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the Lp setting.

We consider a weakly dissipative hyperelastic-rod wave equation (or weakly dissipative Camassa-Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. By fixed a smooth solution, we establish the existence of a strongly continuous semigroup of global weak solutions for any initial perturbation from $H^1({\mathbb R})$. In particular, the supersonic solitary shock waves [8] are included in the analysis.

We consider a generalized hyperelastic-rod wave equation (or generalized Camassa--Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global weak solutions for any initial data from $H^1(\R)$. We also prove a ``weak equals strong'' uniqueness result.

We show existence of a unique, regular global solution of the parabolic-elliptic system $u_t +f(t,x,u)_x+g(t,x,u)+P_x=(a(t,x) u_x)_x$ and $-P_{xx}+P=h(t,x,u,u_x)+k(t,x,u)$ with initial data $u|_{t=0} = u_0$. Here $\inf_{(t,x)}a(t,x)>0$. Furthermore, we show that the solution is stable with respect to variation in the initial data $u_0$ and the functions $f$, $g$ etc. Explicit stability estimates are provided. The regularized generalized Camassa--Holm equation is a special case of the model we discuss.

The existence of a positive solution in a weighted Sobolev space for an homogeneous semilinear elliptic integro-differential Dirichlet problem is proved. The integral operator of the equation depends on a nonlinear function with a singularity at the origin.

In this paper we study the solitary waves for the coupled system of Schrödinger-Maxwell equations in three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed L2 norm. We study the asymptotic behavior and the smoothness of these solutions. We show also the fact that the eigenvalues are negative and the first one is isolated.