An Evans-Function Approach to Spectral Stability of Internal Solitary Waves in Stratified Fluids

TL;DR

This paper introduces a novel Evans-function based method for analyzing the spectral stability of internal solitary waves in stratified fluids, providing a rigorous mathematical framework and results for small-amplitude waves.

Contribution

It develops a new approach involving spatial-dynamical systems, finite-dimensional truncations, and Evans functions to study stability of internal solitary waves.

Findings

Demonstrates well-defined Evans functions for all truncation orders

Proves absence of small zeros in Evans function for small-amplitude waves

Establishes spectral stability of small-amplitude internal solitary waves

Abstract

Frequently encountered in nature, internal solitary waves in stratified fluids are well-observed and well-studied from the experimental, the theoretical, and the numerical perspective. From the mathematical point of view, these waves are exact solutions of the 2D Euler equations for incompressible, inviscid fluids. Contrasting with a rich theory for their existence and the development of methods for computing these waves, their stability analysis has hardly received attention at a rigorous mathematical level. This paper proposes a new approach to the investigation of stability of internal solitary waves in a continuously stratified fluidic medium and carries out the following four steps of this approach: (I) to formulate the eigenvalue problem as an infinite-dimensional spatial-dynamical system, (II) to introduce finite-dimensional truncations of the spatial-dynamics description, (III) to demonstrate that each truncation, of any order, permits a well-defined Evans function, (IV) to prove absence of small zeros of the Evans function in the small-amplitude limit. The latter notably implies the low-frequency spectral stability of small-amplitude internal solitary waves to arbitrarily high truncation order.