In this paper we study numerically the KdV-top equation and compare it with the Boussinesq equations over uneven bottom. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa-Holm equat...

We consider a stochastic Camassa-Holm equation driven by one-dimensional Wiener process with first order differential operator as diffusion coefficient. prove the existence and uniqueness of local strong solutions this equation. In to do so, we transform it into random quasi-linear partial apply Kato's theory methods. Some results have potential find applications other nonlinear equations.

Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa– Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon. The strongly nonlinear equation ( 1 − 1 4 D ) ut = 3 2 ( u ) x − 1 8 ( u2x ) x − 1 4 (uuxx)x, (1) introduced by Camassa and Holm [5] as a possible model for...

The Camassa–Holm equation ut−uxxt+3uux−2uxuxx−uuxxx = 0 enjoys special solutions of the form u(x, t) = Pn i=1 pi(t)e −|x−qi(t)|, denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data u|t=0 = u0 in H1(R) such that u − uxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in Lloc(R, H1 loc(R)) to the unique...

We consider a family of non-local evolution equations including the $0-$Holm-Staley equation. show that considered does not posses compactly supported solutions as long initial data is non-trivial. Also, we prove different unique continuation results for studied. In addition, some special solutions, such peakons and kinks, are studied their dynamics analyzed. Persistence properties also investi...

In the paper, we gave a strengthening of our previous work in \cite{Li1} (J. Differ. Equ. 269 (2020)) and proved that data-to-solution map for Camassa-Holm equation is nowhere uniformly continuous $B^s_{p,r}(\R)$ with $s>\max\{1+1/{p},3/2\}$ $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to b-family equations which contain Degasperis-Procesi equations.