5 - Model equations for small amplitude waves and solitons; weakly nonlinear theory

Summary

Introduction

Some physical equations ask for surgery

Classical physicists usually agree on their equations. In this respect they are very fortunate and can feel rewarded for not working in more fashionable fields such as the frontiers of high energy physics. However, these established equations often compound many different physical effects and can be difficult to solve.

Once we have derived as realistic a set of equations as possible for a given situation, we can try to reduce the number of terms or otherwise simplify by some logical process. Only very good scientists can get away with formulating equations that model chosen phenomena from the start, say by ignoring some physical effects or chopping off terms they consider to be insignificant. Lesser mortals are well advised to develop a systematic scheme for simplifying model equations. To do this one should look for at least one small dimensionless parameter and use it as a surgical tool.

The above remarks concern a theoretical treatment. Computer scientists, on the other hand, will increasingly welcome elaborate mathematical models, embracing more and more rather than less and less physics.

There can be two broad justifications for introducing a small parameter scheme to simplify a system of physical equations (other than being fed up with not being able to solve it). One is that a dimensionless parameter is always small. (An example of such a parameter is the ratio of the centre of mass velocity of a massive heavenly body to c, the velocity of light.