# Homework Help: Rewriting the Fermi-Pasta-Ulam problem as a Korteveg-de Vries equation

1. Apr 23, 2009

### furryscience

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–Pasta–Ulam_problem). 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.