Spectral instabilities in water wave models

Abstract

This thesis investigates the spectral instabilities of various nonlinear water wave models through rigorous analytical techniques. Focusing on three fundamental types of instabilities, modulational instability, transverse instability, and high-frequency instability, the work provides a unified spectral framework to study how small perturbations evolve and potentially destabilize wave solutions in dispersive systems. We begin by analyzing modulational instability, wherein a periodic traveling wave becomes unstable to long-wavelength perturbations. Using perturbation theory and spectral analysis, we characterize conditions under which modulational instability arises in generalized Ostrovsky equations. The effect of dispersion, nonlinearity, and surface tension is examined in detail. The study then turns to transverse instability, where planar wave trains destabilize due to perturbations in the transverse direction. We consider rotation-modified and surface tension-influenced variants of the Kadomtsev–Petviashvili (KP) equation, the rotation-modified KP equation, and the KD equation, and identify parameter regimes leading to transverse spectral instabilities. Finally, we explore high-frequency instability, focusing on the behavior of the spectrum. We demonstrate how high-frequency perturbations can induce instabilities in small-amplitude periodic traveling waves. Altogether, the results contribute to a deeper understanding of how wave coherence is affected by perturbations of various scales and directions. The insights gained have potential implications for the stability of wave patterns in physical settings such as oceanography, fluid mechanics, and nonlinear optics.