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A Asymptotic form of soliton-antisoliton solution for S.G eqn.

  1. Nov 23, 2017 #1
    I'm not sure this is the best place for this question, and apologize if it isn't. I'm studying the classical field solutions on the first few chapter of Rajaraman's Solitons and instantons : an introduction to solitons and instantons in quantum field theory. Well, my question is about one of those intermediate steps that authors love to skip on the grounds that the reader should be able to do it without much difficulty. It may be silly, but I just can't see it. Anyway, we know that soliton-antisoliton scattering is allowed for the sine-gordon equation, and this solution has the form:
    \phi(x,t)=4arctan\left( \frac{sinh(ut/\sqrt{1-v^2})}{u \ cosh(x/\sqrt{1-v^2})}\right)
    He argues that in the limit that t goes to minus infinity, for example, this becomes
    \phi\rightarrow 4arctan\left[exp\left(\frac{x+v(t+\Delta/2)}{\sqrt{1-v^2}}\right) \right] -
    4arctan\left[exp\left(\frac{x-v(t+\Delta/2)}{\sqrt{1-v^2}}\right) \right]
    \Delta\equiv \frac{1-v^2}{v}lnv
    and a similar solution for the positive infinity case(Page 38 of Rajaraman's forementioned book). I tried to put the solution in a form in which I can use the arctangent addiction formula, but thus far no success. Closer I got was
    4arctan\left \{\left[\frac{exp\left(x + \gamma v(t+(lnv)/(v\gamma))\right)}{1+e^{2\gamma x}} \right] - \left[\frac{exp\left(x - \gamma v(t-(lnv)/(v\gamma))\right)}{1+e^{2\gamma x}} \right] \right \}
    Thanks in advance!
  2. jcsd
  3. Nov 29, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
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