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Rewriting the Fermi-Pasta-Ulam problem as a Korteveg-de Vries equation

  1. Apr 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Hello, I've been given an assignment of solving the Fermi-Pasta-Ulam problem
    (involving a chain of oscillators, details found here: http://en.wikipedia.org/wiki/Fermi%E...93Ulam_problem [Broken]). I have a problem with a few details, as described below.

    2. Relevant equations
    I have to rewrite
    q_tt=c^2*q_xx + e*q_x*q_xx + b*q_xxxx (1) (c, e and b are constants)
    into
    u_t' + u*u_x' + d^2*u_x'x'x' = 0 (2) (d is a constant)
    using the substitution
    x' = x - c_1*t, t'=c_2*t, q_x=c_3*u(x',t')
    and a smart choice of constants c_i.

    I also have to find the implicit solution of (2) without the
    dissipative term, that is
    u_t' + u*u_x' = 0 (3)

    3. The attempt at a solution
    To rewrite the derivatives in (1) with respect to x is pretty straightforward:
    q_x=c_3*u, q_xx=c_3*u_x', q_xxxx=c_3*u_x'x'x'
    if I'm not mistaken. What I don't get is how to rewrite q_tt, since both
    x' and t' involve the variable t.

    As for solving (3), I really have no clue. The instructions claim that the implicit
    solution is of the form
    u(x',t') = f(x - u(x',t')t) (4)
    where f is any differentiable function, but I'm pretty sure this is a typo, since (4)
    involves a function u referring to itself.

    The instructor isn't responding to my email queries about this, so any help is most
    appreciated.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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