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Initial condition effect in Nonlinear PDE of a wave

  1. Jun 23, 2014 #1
    Hey there,

    I have modeled a propagating wave in a 1D dispersive media, in which square and cubic nonlinear terms are present.


    the propagating pulse starts to steepen with time which is the effect of nonlinearity, but there is an effect which I can't understand.

    so here is the effect,

    case a)
    u(x,0)=sech(x-ct) and v(x,0)=0
    from d'alambert solution, I have 2 equal propagating wave in (+) and (-) directions carrying the effect of nonlinearity. which make sense.

    case b)
    u(x,0)=sech(x-ct) and v(x,0)=g(x,t)
    where the initial velocity,g, is the first time derivative of initial displacement, sech(x-ct).
    In this case I expect to have pulse propagating in one direction (+), but I get two waves, one with big amplitude in main direction (+), and another one with small amplitude in opposite direction (-).

    I can't understand how does the small amplitude wave develop and propagate. any explanation?

    I would appreciate any hint or comment.
  2. jcsd
  3. Jul 4, 2014 #2
    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
  4. Jul 30, 2014 #3
    I am not even sure I understand the question. In the first equation is the differentiation with respect to [itex]\xi = x-ct[/itex]? And what is v(x,t) defined as? I am not used to this solitonic language so it is hard to follow what is going on.
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