Orbital Stability of smooth solitary waves for the Degasperis-Procesi Equation

Ji Li, Yue Liu, Qiliang Wu

Published 2020-02-03Version 1

We study localized smooth solitary waves in the Desgasperis-Procesi (DP) equation on the real line. This work extends our previous work on spectral stability of these solitons \cite{LLW} by establishing their orbital stability. The main difficulty stems from the fact that the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. The remedy is to observe that, given a sufficiently smooth initial condition satisfying a measurable constraint, the $L^\infty$ orbital norm of the perturbation is bounded above by a function of its $L^2$ orbital norm, yielding the orbital stability in the $L^2\cap L^\infty$ space.