We present an approach proving the integrability of the Camassa–Holm equation for initial data of small amplitude.

In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in critical Besov space B 2 , 1 / ( T ) . Our results show that comparison with well-known work for classic Camassa-Holm-type equations, existence solution only relies on L -integrability variable coefficients α t and γ but nothing do shape initial data. Th...

The infinite-time asymptotic behavior of a modified version of Camassa–Holm equations is studied. In order to describe the peakon-type solutions of the problem, a family of self-similar solutions of the second kind is constructed by an asymptotic analysis. Then the asymptotic profiles are compared to some numerical computations which indicate a curious property of the evolution of the solution ...

In this paper, we study the following initial boundary value problem for a generalized 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 fa...

The Camassa–Holm equation is rich in geometric structures, it is completely integrable, bi-Hamiltonian, and it represents geodesics for a certain metric in the group of diffeomorphism. Here two new multi-symplectic formulations for the Camassa–Holm equation are presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-...

We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm andBenjamin-Ono equations. The stability these is leveraged prove rigorous bounds resulting error. present several numerical experiments demonstrate thatPINNs can very accurately