Long-time Asymptotics for the Camassa–Holm Equation

Abstract

Abstract. We apply the method of nonlinear steepest descent to compute the longtime asymptotics of the Camassa-Holm equation for decaying initial data, completing previous results by A. Boutet de Monvel and D. Shepelsky.

Wave-Breaking and Peakons for a Modified Camassa–Holm Equation

“…where m is as above, (1.2), can itself be derived from the Korteweg-deVries equation by tri-Hamiltonian duality. The Camassa-Holm (CH) equation [6,24] (see also [17,36]) was originally proposed as a model for surface waves, and has been studied extensively in the last twenty years because of its many remarkable properties: infinity of conservation laws and complete integrability [6,19,24], with action angle variables constructed using inverse scattering [3,4,14,18], existence of peaked solitons and multi-peakons [1,6,7], geometric formulations [8,37,44], well-posedness and breaking waves, meaning solutions that remain bounded while their slope becomes unbounded in finite time [10,11,12,13,40]. Note that the nonlinearity in the CH equation is quadratic.…”

On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

“…Alternative derivations of Camassa-Holm equation as an equation for geodesic flow on the diffeomorphism group of the circle were presented by Constantin and Kolev [6]. The equation has bi-Hamiltonian structure [14] and is completely integrable [1,2,5,8,9].…”

Long-time asymptotic behavior for an extended modified Korteweg-de Vries equation

“…Before processing the next deformation, we first introduce analytic approximations of {r j (k)} 4 1 following the idea of [22]. We define the open subsets {Ω j } 4 1 , as displayed in Fig. 3 such that…”