Re: Nonlinear term in the NLS

From: Ilusha (eshliz_at_yahoo.com)
Date: 10/14/04

  • Next message: Alwyn C. Scott: "Re: Nonlinear term in the NLS" 
    Date: Thu, 14 Oct 2004 16:56:42 +0200

    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

  • Next message: Alwyn C. Scott: "Re: Nonlinear term in the NLS"

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