Date of Degree

5-2018

Document Type

Dissertation

Degree Name

Ph.D.

Program

Physics

Advisor

Tobias Schaefer

Committee Members

Joel Gersten

Carlo Lancellotti

Richard Moore

Vadim Oganesyan

Subject Categories

Applied Statistics | Dynamical Systems | Fluid Dynamics | Non-linear Dynamics | Numerical Analysis and Scientific Computing | Optics | Ordinary Differential Equations and Applied Dynamics | Partial Differential Equations | Statistical Models | Statistical, Nonlinear, and Soft Matter Physics

Keywords

Stochastic Methods, Numerical Methods, Stochastic Partial Differential Equations, Ito Calculus

Abstract

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are shown to be in good agreement with the numerics.

Lastly this thesis investigates the relationship between such PDEs and associated ordinary differential equation (ODE) reductions. We find that while there is good agreement for certain properties of the two systems, the ODE model can have difficulty reproducing some aspects of the PDE solution.

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