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

The Degasperis-Procesi equation is the integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the Desgasperis-Procesi (DP) equation on the real line. %extending our previous work on their spectral stability \cite{LLW}. The main difficulty stems from the fact that the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-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 $L^{\mathrm{\infty }}$ orbital norm of the perturbation is bounded above by a function of its $L^2$ orbital norm, yielding the orbital stability in the $L^2\cap L^{\mathrm{\infty }}$ space.