Orbital Stability of Periodic Waves for the Log-KdV Equation

TL;DR

This paper proves the orbital stability of periodic waves in the logarithmic Korteweg-de Vries equation using spectral analysis and abstract stability theories, extending understanding of wave stability in nonlinear dispersive equations.

Contribution

It introduces a method to establish orbital stability of periodic waves for the log-KdV equation, building on recent spectral and well-posedness results.

Findings

Proved orbital stability of periodic waves in the energy space.

Constructed a smooth branch of periodic solutions.

Analyzed spectral properties of the linearized operator.

Abstract

In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work \cite{carles}, in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in \cite{natali1} to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in \cite{grillakis1} and \cite{weinstein1} to deduce the orbital stability of the periodic traveling waves in the energy space.