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Dispersion Relation KdV equation

  1. Jul 30, 2007 #1
    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?
     
  2. jcsd
  3. Aug 8, 2007 #2

    Astronuc

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    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.
     
    Last edited: Aug 8, 2007
  4. Aug 8, 2007 #3

    arildno

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    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.
     
  5. Oct 10, 2007 #4
    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.
     
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