Orbital stability of smooth solitons for the modified Camassa-Holm equation

Abstract

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved $H^1$ and $L^1$ quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space $H^3$. Furthermore, the global well-posedness of the solution is established for certain initial data in $H^s$ with $s\ge 3$.