On the Spectral Stability of Internal Solitary Waves in Fluids with Density Stratification

Abstract

Frequently encountered in nature, internal solitary waves in stratified fluids are well-observed and well-studied objects from the experimental, the theoretical and the numerical perspective. Contrasting with a rich theory for their existence and the development of methods for computing these waves, their stability analysis has received less attention at a rigorous mathematical level.In this thesis, we propose a method for investigating the spectral stability of internal solitary waves which consists of five steps:(i) reformulation of the eigenvalue problem as an infinite-dimensional spatial-dynamical system,(ii) procedure to obtain finite-dimensional truncations,(iii) definition of an Evans function for the finite-dimensional problems,(iv) investigation of the Evans functions for zeros with positive real part,(v) identification or preclusion of eigenvalues with positive real part of the infinite-dimensional system.While steps (i)-(iv) are carried out within this thesis, step (v) is left to future work.