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Nonlinearity and dispersion in Kdv equation?

  1. Mar 28, 2007 #1
    Hi all.
    I am referring to the Kdv equtaion as follows:
    u_t = u u_x + u_(xxx)
    A fundamental point concerning the KdV equation is that it exhibits two
    opposing tendencies:

    1. "nonlinear convection", uu_x, which tends to -steepen- wavecrests,

    2. "dispersion", u_(xxx), which tends to -flatten- wave crests.

    However, I don't quite understand why the nonlinear term and the disperion term would tend to steepen the wave profile?

    Could I see this through the geometrical meaning of uu_x and u_(xxx) or what? how could one make such a statment?
     
  2. jcsd
  3. Mar 29, 2007 #2

    arildno

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    By the method of characteristics you may show that the non-linear term typically will steepen the wave crest until the point of wave-breaking.
     
  4. Mar 29, 2007 #3
    how about the dispersive term? I know dispersive wave flatten the wave profile but why the third derivative of the dimensionaless amplitude implies dispersivity?
     
  5. Mar 30, 2007 #4
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    I would say the nonlinear + dispersion terms tend to maintain the waveform as opposed to steepening as u propose... maybe you check up references on solitons/solitary waves and stuff - start with basic texts like Agrawal's Nonlinear Effects in Optical Fibers...

    good luck and please do come back to let us know what u find
     
    Last edited: Mar 30, 2007
  6. Mar 30, 2007 #5

    arildno

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    First of all, call it the LINEAR dispersive term, not dispersive!

    Let us look at the linearized equation:
    [tex]u_{t}=u_{xxx}[/tex]
    Let us consider a monochrome wave train as our trial solution,
    [tex]u(x,t)=Ae^{i(kx-wt)}[/tex]

    The dispersion relation is of the form w(k), and we have non-dispersive situation if the phase velocity c=w/k is independent of k.

    Let us insert our trial solution in our diff.eq; in order to have NON-TRIVIAL solutions, we must have a specific relation between w and k, i.e the disp. relation.
    We get
    [tex]-iwAe^{i(kx-wt)}=-ik^{3}Ae^{i(kx-wt)}[/tex]
    If this is to be satisfied for all t and x, A non-zero, we must have:
    [tex]w=w(k)=k^{3}[/tex]
    The phase velocity c is therefore a function of the wave-number:
    [tex]c=k^{2}[/tex]

    This shows that high wave-number components of a complex wave will rush off from the low wave-number components, leaving long flattened-out waves moving slowly behind them.

    (Since most of the energy will typically be contained in the low/mid-range wavenumbers, it folllows that the amplitude of the fast-moving ripples is so small that most of them won't be seen, since the amplitude is typically proportional to the root of the energy contained in the wave-component.)
     
    Last edited: Mar 30, 2007
  7. Mar 30, 2007 #6
    You can find a nice treatment of the approximation (viscocity=0, the term multiplied to the third derivative) in Whitham's book on nonlinear waves. There are proofs of all statements made above.
     
  8. Apr 2, 2007 #7
    thanks so much for the explantation Arildno...
     
  9. Apr 8, 2007 #8
    May you please show me how to solve the following NLSE pde:

    [tex]iU_{z} + dU_{tt} = 0 [/tex]
    where
    1. [tex] d [\tex] = constant, and
    2. [tex]U(z=0,t)=e^{(-t^{2})}[/tex]

    It's the NLSE, the nonlinear terms and the loss terms are here considered negligible.
     
    Last edited: Apr 8, 2007
  10. Apr 8, 2007 #9
    Why don't you propose a traveling wave?

    Let [itex]U(z,t)=f(z-ct)[/itex], find an ode for [itex]f[/itex] and the value of [itex]c[/itex].

    P.S. The TeX command \qquad works like the indent you are using.
     
  11. Apr 8, 2007 #10

    Chris Hillman

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    I think the OP is probably reading Drazin and Johnson, Solitons: an Introduction, which IMO explains all these issues very clearly. I'd add that of course not all interesting exact solutions of the KdV are obtained by assuming that the solution we want is a traveling wave! Jacobi elliptic functions are your friend, and all that. In addition to the book I just mentioned, I'd also highly recommend Olver, Applications Of Lie Groups To Differential Equations.
     
  12. Apr 9, 2007 #11
  13. Apr 9, 2007 #12

    Chris Hillman

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    Consolidate current KdV threads?

    It seems to me that Mute answered your basic question (and your later question about how [itex]u u_x[/itex] drops out to give the "linearized KdV").

    You imply that you are disappointed in the response to your questions. Part of the problem might be that you seem to have started several distinct threads on the KdV equation. Another might be that you didn't explain what book(s) you are reading (not just Drazin and Johnson?) or why the author was discussing perturbation expansions; rather, you in effect asked us to guess. We can probably make some guesses based on knowledge of the literature on solitons, but it would be helpful if you provided more context, I think. Thanks for mentioning your background, since this is also helpful.

    But to answer an easy question: yes, in [itex]u_t + 6 u u_x + u_{xxx} =0[/itex] the nonlinearity is due to the term [itex]u u_x[/itex], the term which drops out when we form the linearized KdV [itex]u_t + u_{xxx} = 0[/itex].
     
    Last edited: Apr 9, 2007
  14. Apr 10, 2007 #13
    Chris Hillman,
    I am grateful for Mute's nice reply but I have got some follow-up questions for Mute. However, it has been a long time that my follow-up questions are not replied by Mute. So I am wondering if some other people would kindly help answer the questions.
     
  15. Apr 11, 2007 #14
    Where does all that sudden interest in solitons come from?
    Earthquakes,tsunami,what?
    2-3 years ago you could rarely even hear of a word "soliton".
    Now,solitons this solitons that!
     
  16. Apr 11, 2007 #15

    Mute

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    Sorry, I was going to reply, but I wanted to make sure I didn't say anything that wasn't quite correct, so I delayed the response, but I've hit exam season and so studying for that has taken up the bulk of my time and I hadn't had time to carefully think about what to say.

    In essence, though, you can think of "weakly nonlinear" as meaning the nonlinear term in a given PDE as being more like a perturbation to the linear equation, so it's the linear terms that dominate the behaviour of the solution. Typically this means that the order of the nonlinear term is lower than the order of the nonlinear terms. (This point was the one I wanted to think about more, since the scale on which the time and spatial derivatives change can cause u_t and u_xxx to be small - so, depending on the x scaling, the dispersive term could be quite small, but this isn't necessarily the case in the weakly nonlinear regime, I think). But I think that is essentially the point - in the weakly nonlinear regime, the order of the nonlinear terms in a PDE is smaller than the linear terms, and so the nonlinearity acts more like a perturbation to the linear equation.

    To answer the last question from that other thread, you assume a perturbation expansion like that because you expect the first few terms of the expansion to dominate the dynamics, with further terms just contributing higher order corrections. As I should in the other thread, the leading order term of the expansion obeyed the linearized KdV (assuming time and spatial derivates did not change the order of the derivative terms), which is obviously easier to solve than the nonlinear PDE. This was another thing I wanted to think about before replying - the way we've done things in the course I took this semester was to assume a Fourier series expansion for u(x,t) in which two or three of the modes dominated the dynamics, and the rest were negligible. I think the same sorts of things go for the kind of expansion you were using, but I wasn't quite certain so I had neglected to say anything about it before now.
     
    Last edited: Apr 11, 2007
  17. Apr 14, 2007 #16

    Chris Hillman

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    Actually, people have always been very interested in solitons. However, if you are reading the arxiv, eprints in the math section are often motivated by issues arising from better understanding the beauteous inverse scattering transform, or generalized symmetries and infinite sequences of conserved quantities, or connections (no pun intended) between Lax pairs and differential geometry.
    Some interesting equations currently touted in mathematical physics turn out to have the same exciting properties as the KdV and friends. There are some excellent review papers you can look for which discuss all these points. You might also see the lovely picture book by by E. Atlee Jackson, Perspectives of nonlinear dynamics, Cambridge University Press, 1989.
     
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