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ASYMPTOTIC AND NUMERICAL ANALYSIS OF COHERENT STRUCTURES IN NONLINEAR SCHRODINGER-TYPE EQUATIONS

Abstract
This dissertation concerns itself with coherent structures found in nonlinear Schrödinger-type equations and can be roughly split into three parts. In the first part we study a deformation of the defocusing nonlinear Schrödinger (NLS) equation, the defocusing Camassa-Holm NLS (CH-NLS) equation in both one and two space dimensions. We use asymptotic multiscale expansion methods to reduce this model to a Boussinesq-like equation, which is then subsequently used to obtain approximate solitary wave solutions for both the 1D and 2D CH-NLS equations. We then use direct numerical simulations to investigate the validity of these approximate solutions, their evolution, and their head-on collisions. Using a similar methodology, we also explore a deformation of the derivative nonlinear Schrödinger (DNLS) equation, the Camassa-Holm DNLS (CH-DNLS) equation, in one space dimension. The second part of this thesis involves the construction of numerical methods for identifying steady states of nonlinear wave equations as fixed points. We first introduce two modifications of the so-called accelerated imaginary-time evolution method (AITEM). In our first modification, time integration of the underlying gradient flow is done using exponential time differencing instead of using more standard methods. In the second modification, we present a generalization of the gradient flow model, motivated by the work of Nesterov. Finally, we apply these techniques to the so-called Squared Operator Method, enabling convergence to excited states. The third part consists of the construction of both numerical and analytical methods for finding rogue waves in nonlinear Scrödinger-type equations. First, by identifying rogue wave solutions as fixed points in space-time, we modify a spectrally accurate Newton conjugate gradient method to obtain such solutions for not only the NLS equation but also for equations with a different nonlinearity. We propose a methodology for obtaining rogue wave solutions analytically by considering them as self-similar solutions on a background. Using a number of known equations as case examples, we successfully recover several rogue wave solutions analytically, making this one of the few methods which does not invoke integrability directly.
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