Orbital stability of smooth solitary waves for the Degasperis-Procesi equation

Abstract:

The Degasperis-Procesi (DP) equation is an 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 DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of $L^3$, but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-norm, resulting in $L^3$ higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of $L^2$ norm for DP equation, which cannot be used to control the $L^3$ higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the $L^\infty$ orbital norm of the perturbation is bounded above by a function of its $L^2$ orbital norm, yielding the higher order control and the orbital stability in the $L^2\cap L^\infty$ space.