Orbital Stability of the sum of Smooth solitons in the Degasperis-Procesi Equation

TL;DR

This paper proves the orbital stability of multi-soliton wave trains in the Degasperis-Procesi equation, addressing challenges from its nonlocal structure and establishing stability in the energy space.

Contribution

It establishes the $L^2\cap L^\infty$ orbital stability of well-separated smooth solitons in the DP equation, overcoming nonlocality and energy space limitations.

Findings

Proved orbital stability of multi-soliton solutions.

Developed a priori estimates for stability analysis.

Addressed nonlocal structure challenges in the DP equation.

Abstract

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the $L^2\cap L^\infty$ orbital stability of a wave train containing $N$ smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce \textit{a priori } estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation.