On the linearized log-KdV equation

TL;DR

This paper analyzes the linearized log-KdV equation around Gaussian solitary waves, revealing spectral properties, stability criteria, and dissipative behavior using Hermite functions and convolution methods.

Contribution

It provides a detailed spectral analysis and stability characterization of Gaussian solitary waves in the linearized log-KdV equation, introducing a novel convolution representation.

Findings

Time evolution relates to a Jacobi difference operator.

Spectral and linear orbital stability are characterized.

Solutions exhibit exponential decay in weighted spaces.

Abstract

The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the linearized log-KdV equation at the Gaussian solitary waves. By using the decomposition of solutions in the energy space in terms of Hermite functions, we show that the time evolution is related to a Jacobi difference operator with a limit circle at infinity. This exact reduction allows us to characterize both spectral and linear orbital stability of solitary waves. We also introduce a convolution representation of solutions to the log-KdV equation with the Gaussian weight and show that the time evolution in such a weighted space is dissipative with the exponential rate of decay.