The scattering approach for the Camassa-Holm equation

We present an approach proving the integrability of the Camassa--Holm equation for initial data of small amplitude

On a two-component π\pi-Camassa--Holm system

A novel $\pi$-Camassa--Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa--Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature.Comment: 12 page

Poisson structure and Action-Angle variables for the Camassa-Holm equation

The Poisson brackets for the scattering data of the Camassa-Holm equation are computed. Consequently, the action-angle variables are expressed in terms of the scattering data.Comment: 20 pages, LaTeX. The original publication is available at www.springerlink.co

Stability of smooth solitary waves for the generalized Korteweg - de Vries equation with combined dispersion

The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation.Розглянуто задачу про орбітальну стійкість гладких відокремлених хвиль для узагальненого рівняння Кортевега – де Фріза з комбінованою дисперсією. Отримані результати показують, що гладкі відокремлені хвилі є стійкими при довільній швидкості поширення хвиль

Negative order KdV equation with both solitons and kink wave solutions

In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be transformed to the Camassa-Holm equation through a gauge transform. The Lax pair of the equation is derived to guarantee its integrability, and furthermore the equation is shown to have classical solitons, periodic soliton and kink solutions