21 Non-Linear Wave Equations and Solitons

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Problem Set 10 (252 KB)

Description

In 1834 the Scottish engineer John Scott Russell observed at the Union Canal at Hermiston a well-localized* and unusually stable disturbance in the water that propagated for miles virtually unchanged. The disturbance was stimulated by the sudden stopping of a boat on the canal. He called it a “wave of translation”; we call it a solitary wave. As it happens, a number of relatively complicated – indeed, non-linear – wave equations can exhibit such a phenomenon. Moreover, these solitary wave disturbances will often be stable in the sense that if two or more solitary waves collide after the collision they will separate and take their original shape. Solitary waves which have this stability property are called solitons. The terminology stems from a combination of the word solitary and the suffix “on” which is used to signify a particle (think of the proton, electron, neutron, etc. ). We shall discuss a little later the sense in which a soliton is like a particle. Solitary waves and solitons have become very important in a variety of physical setting, for example: hydrodynamics, non-linear optics, plasmas, meteorology, and elementary particle physics, to name a few. Our goal in this chapter is to give a very brief — and relatively superficial — introduction to solitonic solutions of non-linear wave equations.

8-2014

Keywords

solitary wave, non-linear system, soliton, chapter 21

Disciplines

Physical Sciences and Mathematics | Physics