Spectral stability of smooth solitary waves for the Degasperis-Procesi Equation

TL;DR

This paper proves the spectral stability of smooth solitary waves in the Degasperis-Procesi equation, an integrable shallow-water wave model, using refined spectral analysis of the associated linear operator.

Contribution

It establishes the existence and spectral stability of smooth solitons for the Degasperis-Procesi equation, advancing understanding of their stability properties.

Findings

Spectral stability of smooth solitary waves is confirmed.

Refined spectral analysis techniques are developed for the linear operator.

Results contribute to the mathematical theory of shallow-water wave models.

Abstract

The Degasperis-Procesi equation is an approximating model of shallow-water wave propagating mainly in one direction to the Euler equations. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations with the same asymptotic accuracy, and is integrable with the bi-Hamiltonian structure. In the present study, we establish existence and spectral stability results of localized smooth solitons to the Degasperis-Procesi equation on the real line. The stability proof relies essentially on refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the Hamiltonian of the Degasperis-Procesi equation.