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Chapter 1

INTRODUCTION

1.1 Nonlinear Science

Nonlinear physics, like relativity and quantum mhanicecs contains new
ideas and surprising scientific results. Nonlinear systems cover a scale

ranging from nanometers up to thousands of kilometers, or in other words

ranging from atomic physics up to the motion of Jupiter’s giant Red Spot,
as well as objects moving with any speed. For pedagogical purposes, non-

linear science may be divided into six areas of study, namely, solitons,

fractals, pattern formation, chaos, cellular automata and complex sys-
tems. However, in this thesis we focus our attention on solitonic structure

of low frequency waves in dusty plasmas [1-3].

Plasmas are nonlinear dispersive media. Linear theory depends on a

small amplitude fact in investigating different instabilities types. It de-
scribes all waves growing or damping. As the amplitude becomes enlarge,

the linear approximation breaks down and nonlinearly effects tend to the

limit of instability growth.

Generally, three nonlinearity types are obtained in plasmas:
1. Scalar nonlinearities (solitary wave and envelope solitons, etc.)

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