Dispersion Relation KdV equation

[hanson]

Hi all.
I have some questions about the dispersion relation in the study of waves.
First of all, why do we always assume a plane wave solution when we want to obtain a dispersion relation?

Second, is "assuming a plane wave solution" a general way to obtian all dispersion relations? for both linear and nonlinear wave equations? So, what is the dispersion relation for the KdV equation? I can hardly see anyone deriving the dispersio relation for nonlinear equations like KdV and NLS etc, why?

 

[Astronuc]

Try this link for excerpts from "Oscillations and Waves"
http://books.google.com/books?id=ge...6VB&sig=pDI-xn5gOrVd_qmFKMPIsIl3xEA#PPA413,M1


I think we tend to start with the simplest form for the mathematical description of a physical phenonmenon and work up to more complex situations.

Working in one dimension is simple compared to two or more, and where possible, one would prefer a linear system to a nonlinear one.

Solutions to 'idealized' models are a first step before introducing more complex and difficult structures.

 

[arildno]

First, concerning LINEAR diff.eqs:

A typical feature of the solutions to these is that the SUM of two solutions is ALSO a solution.

Furthermore, it is generally true that ALL solutions of a given diff.eq can be regarded as generated by a sum of base solutions.

This is what is called the superposition principle, and it is immensely important in all of physics.

However, with NON-linear diff.eqs, the superposition principle does not in general hold, i.e it is NOT normally true that the sum of two solutions is also a solutions, nor can we assume there exists a set of basis solutions out of which we can generate all solutions.

But that means, esssentially, that for the non-linear case, "every particular problem has its own solution", rather than this solution being derivable from solution to other problem.
The problem with non-linear diff.eqs is that neither the solutions methods nor the solutions themselves for a particular problem has any generalizable value.

 

[kanne]

Plane wave solutions are handy to work with since they of course are plane, hence if the wave is progressing in one direction, for example parallel to an x-axis, then for every point on x the wave motion can be described by a single number denoting the amplitude in the entire yz-plane. Now all periodic functions and following all waves can be described by a Fourier transform which in essence is a long combination of plane waves weighted by a coefficient function. Therefore the dispersion relation can be found by considering the plane component waves of this Fourier transform.

For a linear equation all component waves will be independent of each other due to the superposition principle and therefore it is relatively easy to find the dispersion realtion. For non-linear equation like the KdV the superposition principle does not apply and there is mixing between the different component plane waves. This makes it difficult to determine a exact dispersion relation.

An approximation to the dispersion relation can be deduced by inserting a plane wave solution in the linearized version( just dropping the non-linear term) and this yields:

w(k)=ck-ek^3

where c denotes the coefficient before the term involving the first derivative with respect to x and e is the coefficient of the dispersive term.