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A soliton, for example the KdV equation solution, has the profile proportional to a hyperbolic secant squared ${\text{sech}}^{2}(x-ct)$. And since it is hyperbolic it has an exponential dependence, so it has an infinite span, it has tails that extend to infinite.

However the solitons after an interaction do reemerge like nothing have happened, except by an phase shift. What I don't understand is: how can we say that they don't interact after, or even before, the interaction has happened if they have infinite tails? Because I was wondering (to myself) that if the tails are infite so their tails (of two solitons) are always interacting.

[I hope it is not a stupid question]

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An exponentially decaying tail is almost like having no tail for all practical reasons. For example consider the yukawa potential for interaction through exchange of a massive particle, it is $\propto e^{-\mu r}/r$ which is even a stronger tail than the asymptotic behavior of the hyperbolic secant. There we say that the interaction has the the effective distance of $1/\mu$, and is practically zero otherwise.

Said differently, it is the same reason that two neutrons can be considered non interacting if the are father apart than the inverse mass of the pion (the particle exchanged in the effective yukawa description)

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  • $\begingroup$ Ali, I was reading an article by philip rosenau where he says about waves with compact support (compactons) "Two such waves would interact only for a finite time and the, unlike solitons, would be completely oblivious to each other." He points some differences between solitons and compact waves in this sense. So I started to think that if the compactons interact only for a few time because they don't have infinite tails, the solitons could interact for an infinite time. $\endgroup$
    – Poli Tolstov
    Commented Mar 26, 2015 at 20:28
  • $\begingroup$ From what I know about instantons, when they are sufficiently far from each other they are treated as non-interacting. See for example instanton gas, where they are treated like non-interacting particles of an ideal gas. Granted, when density increases so that the mean distance is within the range of their effective interaction you get interesting phenomena like instanton liquid etc.. $\endgroup$
    – Ali Moh
    Commented Mar 26, 2015 at 20:46
  • $\begingroup$ thanks. Ali, i've a different question, if you would help me...I was reading the Drazin's Solitons: an introduction, there he gives in the first chapter pg.14 an tiny explanation about fermi pasta ulam results, and its said "after a very long time, the initial profile - os something very close to it -reappears, a phenomenon requiring the topology of the torus for its explanation". D'you know something about this torus topology and recurrence. I'm looking at internet but i would like something for beginners, if possible. $\endgroup$
    – Poli Tolstov
    Commented Mar 27, 2015 at 23:20

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