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

In this thesis we investigate the spectral stability of internal solitary waves occurring frequently in large natural water bodies, such as lakes and oceans, where they play an important role in mechanisms of mixing and energy transport. A commonly used mathematical model in this context is given by the Euler equations for a heterogeneous incompressible fluid in a two-dimensional channel. The rest state supports localized, rapidly decaying disturbances called internal solitary waves travelling horizontally with constant speed and unaltered in shape. While mathematical results on the existence of internal solitary waves for quite general stratifications are well known, their stability analysis has received only little attention on a rigorous level. In order to study the spectral stability of such waves we consider the associated eigenvalue problem, which is obtained from the Euler equations by linearizing about the internal solitary wave, specializing the time-dependence to an exponential factor eκt and eliminating this factor subsequently. If the eigenvalue problem has a bounded solution for some κ ∈ C+ := {z ∈ C : Re z > 0} then the wave under consideration is called spectrally unstable; otherwise, the wave is called spectrally stable. The central subject of this thesis is a method for the investigation of spectral stability of internal solitary waves. Our approach consists in five steps: (i) reformulation of the eigenvalue problem as an infinite-dimensional spatial-dynamical system (EVP), (ii) procedure to obtain finitedimensional truncations (EVPN ) of (EVP), (iii) definition of an Evans function DN (κ) for the finite-dimensional problems (EVPN ), (iv) investigation of DN for zeros κ ∈ C+, (v) identification or preclusion of eigenvalues κ ∈ C+ of the infinite-dimensional system (EVP). While steps (i)-(iv) are carried out within this thesis, step (v) is left to future work. In the following we outline the results of this thesis. We prove in Theorem I that the eigenvalue problem permits a spatial-dynamics formulation, which is a dynamical system (EVP) on ( L2(0, 1) )4 endowed with a special scalar product. After constructing a suitable Hilbert basis for this state space we project the dynamical system to the span of 4N + 4 basis vectors and thus obtain a sequence of truncated problems (EVPN ) which are ordinary differential equations on finite-dimensional spaces. Assuming the prototypical exponential density stratification, Theorem II states that the truncated problems (EVPN ) permit the construction of Evans functions DN (κ) which allow to decide whether these systems have bounded solutions for some κ ∈ C+. We currently know no way to generally decide whether the Evans functions possess roots with positive real part. For small-amplitude waves, however, we show in Theorem III that the Evans function does not have roots with positive real part in a neighbourhood of the origin; one part of the proof is based on the relationship between small-amplitude internal solitary waves and soliton solutions of the Korteweg-deVries equation (KdV) and exploits the spectral stability of KdV solitons.