Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals

Chapter 3: Theoretical Approaches To Nonlinear Wave Evolution In Higher Dimensions

Antonmaria A. Minzoni

Fenomenos Nonlineales y Mecánica, Department of Mathematics and Mechanics Universidad Nacional Autónoma de México, Mexico D.F., Mexico

Noel F. Smyth

School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, Scotland, United Kingdom

3.1 Simple Example of Multiple Scales Analysis

One of the main ideas used to understand the evolution of coherent structures is that of modulations, provided by Whitham [1, 2], which is related to the perturbation theory technique of multiple scales [3]. Modulation theory was developed in the context of slowly varying waves, both linear and nonlinear, which are governed by partial differential equations. It is illustrative to understand the basic principles of modulation theory in terms of a linear oscillator as it is governed by an ordinary differential equation. The basic ideas and concepts of modulation theory will then become clear, without the complexity of the solutions of partial differential equations.

Let us begin by recalling the ordinary differential equation for a linear, simple harmonic oscillator

3.1

where ω is the constant frequency of the oscillator and y is some characteristic displacement of the oscillatory motion. The solution of this equation is

3.2

Here, A and δ are constants related ...