Re: Nonlinear term in the NLS

From: Alwyn C. Scott (rover_at_theriver.com)
Date: 10/15/04

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    Date: 14 Oct 2004 18:45:42 -0700

    Dear Ilusha,

    Considering the solution

    U(x,t) = A exp(ikx – iwt)

    to have a finite amplitude, the dispersion equation is

    w = k^2 – A^2

    This means that for |k| < |A| the group velocity (dw/dk) is in the
    opposite direction as the phase velocity (w/k), which leads to
    instability.

    If you send me your email address, I'll send you a pdf file with more
    details.

    Al
    rover@theriver.com

    ===
    ===

    "Ilusha" <eshliz@yahoo.com> wrote in message news:<ckm0lm$do4$1@news.iucc.ac.il>...
    > Dear Prof. Alwyn Scott,
    >
    > Thank you for your answer, the valuable information and references.
    > I am familiar with the "Benjamin-Feir" stability of the SIS solution
    > of the focusing/defocusing NLS. However, I still cannot
    > understand what is the effect of nonlinearity and how it
    > balances dispersion.
    >
    > As I understand, to show the effect of dispersion we consider the linearized
    > equation:
    >
    > iUt + Uxx = 0
    >
    > Then the solution can be expressed as a traveling wave solution:
    >
    > U = Aexp( i(kx-wt) )
    >
    > We submit the solution into the linearized equation and get the dispersion
    > relation:
    >
    >
    > w = |k|^2
    >
    > >From which we induce the group velocity:
    >
    > Vg = dw/dk = 2k
    >
    > Which means that each Fourier component travels with different speed - the
    > effect of
    > dispersion.
    >
    >
    > But how can I show the effect of nonlinearity? I thought to consider the
    > dispersionless equation
    >
    > iUt - |U|U^2 = 0
    >
    > And show somehow that the components of the solution also travel with
    > different
    > speed.... Such that
    >
    > the dispersion can be balanced....
    >
    > Also, why in the defocusing case the balance will happen only when there is
    > a "dark"
    > soliton
    >
    > (as I understand: negative amplitude of the bright soliton)?
    >
    > Thank you very much for your help,I"Alwyn C. Scott" <rover@theriver.com>
    > wrote in message news:35fc3b08.0410101106.11f78659@posting.google.com...
    > > eli@gteko-dot-com.no-spam.invalid (Ilusha) wrote in message
    > news:<41682d9a$1_2@127.0.0.1>...
    > > > Hello,
    > > >
    > > > How can I show for a nonlinear schrodinger equation that the
    > > > nonlinearity
    > > > term leads to the concentration of a wave (in contrast to
    > > > dispersion).
    > > >
    > > > I know that I should start by dropping the dispersion term....
    > > >
    > > > Thanks,
    > > > I
    > >
    > > ===
    > > ===
    > >
    > > Dear I,
    > >
    > > Consider the NLS equation in the form
    > >
    > > iUt + Uxx + s|U|^2U = 0,
    > >
    > > where s is either +1 or -1.
    > >
    > > For either value of s, this equation evidently has a space-independent
    > > solution (SIS) of the form
    > >
    > > U = A exp(i s A^2 t),
    > >
    > > where A is a real constant.
    > >
    > > If s = -1, this SIS is stable. If s = +1, the SIS is unstable to the
    > > formation of solitons.
    > >
    > > This instability was first noticed in the context of nonlinear optics
    > > by Ostrovosky and shortly thereafter in the context of water waves by
    > > Benjamin and Feir. In the US and Europe, it is usually called the
    > > "Benjamin-Feir insatability".
    > >
    > > L.A. Ostrovsky. Propagation of wave packets and space-time
    > > self-focusing in a nonlinear medium. Soviet physics JETP 24 (1967)
    > > 797-800.
    > >
    > > T. Brooke Benjamin and J.E. Feir. The disintegration of wave trains in
    > > deep water. J. Fluid Mechanics 27 (1967) 417-430.
    > >
    > > See my recent book Nonlinear Science: Emergence and Dynamics of
    > > Coherent Structures (2nd edition, 2003) at
    > > http://www.oup.co.uk/isbn/0-19-852852-3 for additional details.
    > >
    > > Alwyn Scott
    > > http://personal.riverusers.com/~rover

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