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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.

    u′′=au3+bu2+cu

    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.
    Thanks
     
  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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