We consider an integrable generalization of the sine-Gordon (sG) equation that was earlier derived by one of the authors using bi-Hamiltonian methods. This equation is related to the sG equation in the same way that the Camassa-Holm equation is related to the KdV equation. In this paper we: (a) Derive a Lax pair. (b) Use the Lax pair to solve the initial value problem on the line. (c) Analyze s...

In this paper, we provide a blow-up mechanism to the modified Camassa-Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics...

3 Distance defined by optimal transportation problem 33 3.1 A distance functional in the spatially periodic case. Introduction This thesis deals with two strongly nonlinear evolution Partial Differential Equation (in the following named P.D.E.) arising from mathematical physics. The first one was introduced first by Fokas and Fuchssteiner [28] as a bi-Hamiltonian equation, and then was rediscov...

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.