Publication Date

Fall 11-12-2020

Abstract

The present work offers an investigation of dynamics and stability of nonlinear waves in Hamiltonian systems. The first part of the manuscript discusses the classical problem of water waves on the surface of an ideal fluid in 2D. We demonstrate how to construct the Stokes waves, and how to apply a continuation method to find waves in close vicinity to the limiting Stokes wave. We provide new insight into the stability of the Stokes waves by identifying previously inaccessible branches of instability in the equations of motion for the fluid. We provide numerical evidence that pairs of unstable eigenvalues of linearized dynamical equations appear as a result of collision of pairs of neutrally stable eigenvalues at extrema of the Hamiltonian. Moreover, we find that eigenvalues of the linearized problem that become unstable follow a self-similar law as they approach the instability threshold, and a power law is suggested for unstable eigenvalues in the immediate vicinity of the limiting wave.

A related problem of formation of Stokes waves from a generic plane wave is considered. It is determined that over long time a plane wave tends to a solution that is effectively described by a Stokes wave with a perturbation moving in the opposite direction to the Stokes wave. This perturbation to the Stokes wave may be described by an effective Hamiltonian, that has quadratic and cubic terms with respect to the perturbations.

A train of Stokes waves can be studied assuming a slowly-varying envelope, with dynamics of the envelope subject to the nonlinear Schroedinger equation (NLSE). In the second part of the present work we provide comparison of two numerical methods to solve NLSE. The first one is the standard second order split-step method based on an operator splitting approach. The second method is the Hamiltonian-conserving method referred to as the Hamiltonian integration method (HIM). HIM allows exact conservation of the Hamiltonian and wave action but requires implicit time stepping. We find that the NLSE can benefit from the Hamiltonian-conserving method compared to the split step method in particular for such solutions as the Akmediev and the Kuznetsov-Ma solitons as well as multisoliton solutions. We find that numerical error for HIM is systematically smaller than for the split-step scheme for the same timestep. At the same time, one can take orders of magnitude larger time steps in HIM, compared to split step, while still ensuring numerical stability. We propose the Hamiltonian-conserving method for the Majda-Maclaughlin-Tabak (MMT) model, which is a generalization of NLSE.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Alexander Korotkevich

Second Committee Member

Pavel Lushnikov

Third Committee Member

Deborah Sulsky

Fourth Committee Member

Peter Vorobieff

Fifth Committee Member

Evangelos A. Coutsias

Language

English

Keywords

Nonlinear waves, pseudospectral methods, numerical methods, computational physics, nonlinear Schroedinger equation

Document Type

Dissertation

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