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Solitons=Only waves?

  1. Jul 20, 2007 #1
    Solitons=Only waves??

    given any wave equation (linear or not) could we always find a solution.

    [tex] \Psi (x,t)= f(x-ct) [/tex]

    is this the so-called only wave or soliton ?? , and what would be the shape of f(r) r=x-ct given a certain wave equation ?.. does the Schröedinguer equation admit such solutions or only if this SE is non-linear ??
     
  2. jcsd
  3. Jul 21, 2007 #2

    olgranpappy

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    no.

    67890
     
  4. Jul 23, 2007 #3

    Claude Bile

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    I would say that the term "wave-equation" implies the existence of wave-like solutions (i.e. functions of the form f(x-c.t)). Olgranpappy, I'm a little confused by your post - I would think that you would at least attempt to qualify such a statement! (either that or I am missing something clever).

    Functions of the form f(x-c.t) are not solitons, though they must satisfy this condition, as they are propagating waves. A solition is a wave that does not change its shape as it propagates, hence not only must it be of the form f(x-c.t), its profile in the y and z directions cannot vary with x.

    The Schrodinger equation being a linear equation does not have any soliton solutions for finite waves (i.e. not including the artificial examples of infinite plane waves and so forth). The non-linear Schordinger equation does have soliton solutions. As I understand it, nonlinearity is crucial for soliton formation and propagation.

    Claude.
     
  5. Jul 23, 2007 #4

    olgranpappy

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    What?...

    The form [tex]\psi(x,t)=f(x-ct)[/tex] necessarily implys a relation between the space and time derivatives. Namely,

    [tex]
    \frac{\partial \psi}{\partial x}=\frac{-1}{c}\frac{\partial \psi}{\partial t}
    [/tex]

    Why would such a thing be true for a solution of a general differential equation? For example, I could explicitly forbid such solutions in the differential equation.
     
  6. Jul 24, 2007 #5

    Claude Bile

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    Ah, I see where you are coming from now - though I think the OP was talking specifically about wave-equations, not DEs in general.

    Claude.
     
  7. Jul 24, 2007 #6

    olgranpappy

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    oh, he did say "wave equation." But then what exactly is the difference between a "wave equation" and some other type of differential equation? Are "wave equations" hyperbolic? I don't get it.
     
  8. Jul 25, 2007 #7

    Claude Bile

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    Yes, this is a little confusing for me too - I thought that the term wave-equation is defined on the basis of having solutions of the form f(x-vt) because such functions describe a propagating disturbance, though this is more of a physicists classification than one a mathematician would use.

    Claude.
     
  9. Jul 25, 2007 #8

    olgranpappy

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    In that case, the answer to the original question is: "Yes, by definition."
     
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