A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts. Camassa and Holm [2],[3] introduced the strongly nonlinear equation (1 − 1 4 D 2)u t = 3 2 (u 2) x − 1 8 (u 2 x) x − 1 4 (uu xx) x , (1) as a possible model for dispersive waves in shallow water and show...

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kote...

Peakons (peaked solitons) are particular solutions admitted by certain nonlinear PDEs, most famously the Camassa–Holm shallow water wave equation. These take form of a train peak-shaped waves, interacting in particle-like fashion. In this article we give an overview mathematics peakons, with emphasis on connections to classical problems analysis, such as Padé approximation, mixed Hermite–Padé m...

In this paper, we develop, analyze and test a local discontinuous Galerkin (LDG) method for solving the Camassa-Holm equation which contains nonlinear high order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions and give a detailed error estimate for smooth solutions, and provide numerical simulation results for dif...

This paper develops a new approach in the analysis of the Camassa-Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we ob...

An analytic study of two nonlinear variants of the (2+1)-dimensional Camassa-Holm-KP equation is presented in this paper. The (G'/G)-expansion method is employed to obtain exact travelling wave solutions of the equations. The solutions gained from the proposed method have been verified with those obtained by the sine-cosine method and the tanh method. More importantly, other new and more genera...

This paper is devoted to the stability of smooth solitary waves for b-family Camassa–Holm equations. We verify criterion analytically general case b>1 by idea monotonicity period function planar Hamiltonian systems and show that are orbitally stable, which gives a positive answer open problem proposed Lafortune Pelinovsky (2022).

We show that the smooth traveling waves of the Camassa-Holm equation naturally correspond to traveling waves of the Korteweg-de Vries equation.

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 −u′′ = z uω + zuυ on an interval [0, L), where ω is a real-valued distribution in H−1 loc [0, L), υ is a non-negative Borel measure on [0, L) and z is a complex spectral parameter. Apa...