Non-existence of internal mode for small solitary waves of the 1D Zakharov system

TL;DR

This paper proves that small solitary waves of the 1D Zakharov system lack internal modes, a key spectral property that may influence their asymptotic stability analysis.

Contribution

It establishes the non-existence of internal modes for small solitary waves in the 1D Zakharov system, advancing spectral understanding of these solutions.

Findings

No internal modes for small solitary waves

Spectral analysis confirms stability properties

Supports future stability investigations

Abstract

We prove that the linearised operator around any sufficiently small solitary wave of the one-dimensional Zakharov system has no internal mode. This spectral result, along with its proof, is expected to play a role in the study of the asymptotic stability of solitary waves.