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. The instructor isn't responding to my email queries about this, so any help is most appreciated.