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Solution of a differential equation?

  1. Sep 7, 2007 #1
    Hi all. I am getting confused about the notion of "a solution" to a differential equation.
    Let's consider the KdV equation, ut+uux+uxxx=0.
    So, if the initial condition is a Sech^2 pulse, then the solution would be a travelling wave solution and this is the well known solitary wave.

    So, what if I arbitrarily use another initial condition? say, a Sech^3 pulse or anything? This initial profile shall be also governed by the KdV equation and the evolution of this strange initial profile shall be still a solution of the KdV equation, right? just that we cannot find the analytical or exact solution?

    Say if I use an excellent numerical scheme to see the evolution of this Sech^3, theoretically, the evolution generated using this arbitrary initial condition shall be called a solution of the KdV, right?
     
  2. jcsd
  3. Sep 7, 2007 #2

    arildno

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    Certainly.

    Solitons are not the only solution to the KdV equation.

    What IS surprising is that soliton solutions DO exist..:smile:

    The soliton solution is gained by hypothesizing the existence of a non-dispersive solution of KdV; calculations then reveal that:

    Insofar as such solutions exist, they need to have a Sech^2-profile.
     
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