Employing the theory of nonlinear waves to study rogue modes and stratified fluid flows

Abstract

Rogue waves (RWs) are large amplitude, suddenly appearing waves whose displacements are localized in both space and time. This thesis includes two separate but interrelated research parts about the topic of rogue wave, one is focused on the theoretical direction to work on the features of poles of the exact solutions for rogue wave, the other part is aimed at finding the occurrence of internal rogue wave in stratified flows modeling the realistic density profile in the upper ocean, which makes the theory of rogue wave into real world applications. For the pole research, the locations of the maximum displacements of rogue waves in physical plane may correlate with the movements of the poles of the exact solutions for rogue waves extended to the complex plane through analytic continuation. This conjecture could be verified analytically and readily for the Peregrine breather (lowest order rogue wave) and the derivative nonlinear Schrödinger model. Other more complicated models will also be tested, e.g. an asymmetric second order rogue wave for nonlinear Schrödinger (NLS) equations (an evolution of weakly nonlinear, weakly dispersive slowly varying wave envelope), and a symmetric second order rogue wave of coupled NLS systems. Moreover, the movements of the poles in the complex plane may reverse direction or receive the maximum or minimum points of the trajectories at the time instant of the maximum displacements of rogue waves, which is verified by the models discussed in this thesis. This property will be tested by many other systems in the near future. For the internal rogue wave subject, exact solutions are derived analytically in the search for linear modes governing the spatial structure of small disturbances in a density stratified flow. The Boussinesq approximation is made, where the variation in the background density is neglected except in the buoyancy term. By comparing this linear modal equation with four and five components of the coupled NLS equations from the theory of solitons, exact solutions in terms of hyperbolic functions are derived for special wavenumbers when the buoyancy frequency is quadratic in hyperbolic secant, and the numerical solutions for arbitrary wavenumber can also be derived. The nonlinear evolution of a wave packet can be studied by computing the second harmonic, induced mean flow and eventually third order perturbations, facilitated by these exact expressions of the linear eigenfunctions. The likelihood for the occurrence of internal rogue waves can be assessed by the signs of the product of the quadratic dispersion and cubic nonlinearity of the Schrödinger equation. It is found that the density profile and properties of the carrier envelope play crucial roles in the dynamics of rogue waves.