Mathematical methods for nonlinear wave interactions

Abstract, DMS 0506495, R Goodman, New Jersey Inst of technology

Title: Mathematical methods for nonlinear wave interactions

This project studies interactions between waves in models from optics

and mathematical physics, and the relative stability of waves trapped

by localized potentials. Wave interactions in many applications have

been known for many years to display resonance structures in which wave

capture and reflection alternate in a surprising fractal-like manner.

Methods from the theory of dynamical systems, including Melnikov

methods, matched asymptotics, and transport theory of iterated maps,

are to be applied to ordinary differential equations models of such

wave interactions, extending recently published work of the PI. In a

second thread of research, we look at the nonlinear interaction between

trapped modes at potentials engineered into optical fibers, modeled by

the nonlinear Schrodinger equation (NLS), or the nonlinear coupled mode

equations (NLCME) in the case of fiber Bragg gratings. The NLCME

system lacks an energy-minimization principle (all bound states are

saddle points of the energy), yet somehow a ground state is chosen. We

study this using numerical analysis, derivation of simplified ordinary

differential equation models, and ideas from the spectral theory of

Hamiltonian systems.

Nonlinear wave phenomena are ubiquitous in physics. An important

engineering example is in optical communications, where information is

sent as pulses of light through glass fibers. It is important to

understand how these waves interact with each other, and with local

structures in the medium through which they travel, in order to better

design devices that exploit their novel properties. The aim of this

research is to extend mathematical theory from dynamical systems to

explain phenomena seen in wave interaction, such as wave-trapping. It

will use sophisticated computational and analytical methods, some

developed in the study of Bose-Einstein condensates, to study the

behavior of light in novel physical configurations.