On solutions to integrable and nonintegrable nonlinear wave equations

Abstract

All linear wave equations with constant coefficients have analytical solutions that are superpositions of plane waves which satisfy the dispersion relation of the linear wave equation. In general, however, nonlinear wave equations do not necessarily have closed form analytical solutions. Numerical solutions can be computed, but these solutions are not necessarily "nice" or easy to understand. However, some very special nonlinear wave equations can be solved using the Inverse Scattering Transform (IST), which is an integral transform similar to the Fourier Transform. These equations are called integrable because they have known closed form solutions.

Second, the linear part of the NLS has a quadratic dispersion relation. I have found that when continuous radiation dominates solutions to the NLS, they exhibit an effective dispersion relation of a shifted parabola. This indicates some degree of linear behavior over long time average. I will first show that this effective dispersion relation minimizes the effective nonlinear behavior in both the PDE and the Hamiltonian, and will conclude by showing that as we allow the solution to formally be more nonlinear, the relative effectively nonlinear part of the Hamiltonian actually decreases and the energy behaves effectively more linearly.

In this thesis I studied two aspects of the NLS. First, working in a laboratory reference frame, I was able to extend the solution to the initial-boundary value problem on the quarter plane to the solution on the half plane, obtaining solutions of reflected solitons in this regime.

The Nonlinear Schröedinger Equation has a multitude of applications, including super-fast lasers, surface gravity waves, and pulse propagation in optical fibers. It frequently arises to describe the behavior of the slowly varying envelope of an underlying carrier wave. While the integrability and soliton solutions to the NLS on the half plane have been widely studied, recent interest has arisen in terms of a long-time statistical average of general solutions.

The Maxwell Bloch Equations were derived to model the transitions of electrons from a ground state to a single excited energy state in an active optical medium interacting with light. In the physically realistic model, two additional damping terms are included; with these extra terms, the system is no longer integrable via the IST. I have studied behavior of a soliton-like input pulse when damping is present in the system, for which I will show analytical and numerical results.

Two well known integrable nonlinear wave equations that I will discuss are the undamped Maxwell Bloch Equations (MBE) and the Nonlinear Schröedinger Equation (NLS). Both have soliton solutions: single wave pulses that retain their shape, area, and group velocity as they propagate.