This talk is focused on recent progress of studies for the Camassa-Holm equation. First, we will give a brief review on the derivations, well-posedness for the strong solution, blow-up phenomenon and existence of the weak solutions. Then, infinite propagation speed for the Camassa-Holm equation will be proved in the following sense: the corresponding solution u(x, t) with compactly supported in...

This study deals with asymptotic models for the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with a flat bottom. We present a new Green-Naghdi type model in the Camassa-Holm (or medium amplitude) regime. This model is fully justified, in the sense that it is consistent, well-pose...

We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa–Holm system, ut − utxx + κux + 3uux − 2uxuxx − uuxxx + ηρρx = 0 and ρt + (uρ)x = 0, for initial data (u, ρ)|t=0 in H1 per ×Lper. It is necessary to augment the system with an associated energy to identify the conservative solution. We study the stability of these periodic solutions by...

In this article we study a new kind of unbounded solutions to the Novikov equation, found via Lie symmetry analysis. These exhibit peakon creation, i.e., these are smooth up u ...

We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation wi...

In this paper, we give a construction of u 0 ∈ B p , ∞ σ such that the corresponding solution to Camassa-Holm equation starting from is discontinuous at t = in metric which implies ill-posedness for . We also apply our method b-equation and Novikov equation.

Extended shallow water wave equations are derived, using the method of asymptotic expansions, from Euler (or wave) equations. These extended models valid one order beyond usual weakly nonlinear, long approximation, incorporating all appropriate dispersive and nonlinear terms. Specifically, first we derive Korteweg–de Vries (KdV) equation, then proceed with Benjamin–Bona–Mahony Camassa–Holm in (...