Related papers: On the isospectral problem of the dispersionless C…
We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the…
A spectral and the inverse spectral problem are studied for the two-component modified Camassa-Holm type for measures associated to interlacing peaks. It is shown that the spectral problem is equivalent to an inhomogenous string problem…
We study a class of (conservative) low regularity solutions to the Camassa-Holm equation on the line by exploiting the classical moment problem (in the framework of generalized indefinite strings) to develop the inverse spectral transform…
The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most…
An Inverse Scattering Method is developed for the Camassa-Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are explicitly constructed in terms of the scattering data. The main…
We solve the inverse spectral problem associated with periodic conservative multi-peakon solutions of the Camassa-Holm equation. The corresponding isospectral sets can be identified with finite dimensional tori.
In this paper we consider a class of isospectral deformations of the inhomogeneous string boundary value problem. The deformations considered are generalizations of the isospectral deformation that has arisen in connection with the…
Motivated by the study of certain nonlinear wave equations (in particular, the Camassa-Holm equation), we introduce a new class of generalized indefinite strings associated with differential equations of the form…
The inverse problem which arises in the Camassa--Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the…
In this Letter the main steps in the inverse spectral construction of a family of non-smooth solitons (peakons) to the modified Camassa-Holm equation are oulined. It is shown that the inverse problem is solvable in terms of multi-point…
In this paper, we show that the peakon (peaked soliton) solutions can be recovered from the smooth soliton solutions, in the sense that there exists a sequence of smooth N-soliton solutions of the dispersion Camassa-Holm equation converging…
Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the…
We consider the interior inverse problem associated with the global conservative {multipeakon} solution of the Camassa-Holm equation. Based on the inverse spectral theory on the half-line and the oscillation property of eigenfunctions, some…
We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be…
We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case when the underlying isospectral problem has purely discrete spectrum. To this end, we introduce a natural coupling problem for…
We use an inverse scattering approach to study multi-peakon solutions of the Degasperis-Procesi (DP) equation, an integrable PDE similar to the Camassa-Holm shallow water equation. The spectral problem associated to the DP equation is…
We prove spectral instability of peakons in the $b$-family of Camassa--Holm equations in $L^2(\mathbb{R})$ that includes the integrable cases of $b = 2$ and $b = 3$. We start with a linearized operator defined on functions in…
Discrete integrable systems are closely related to orthogonal polynomials and isospectral matrix transformations. In this paper, we use these relationships to propose a nonautonomous time-discretization of the Camassa-Holm (CH) peakon…
We consider a one-parameter family of non-evolutionary partial differential equations which includes the integrable Camassa-Holm equation and a new equation first isolated by Degasperis and Procesi. A Lagrangian and Hamiltonian formulation…
In this paper, a generalized nonlinear Camassa-Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define…