The initial formulation of the evolution equation for the leading order approximation in nonlinear elasticity in the weakly nonlinear regime goes back to [Lar83]. Moreover, Lardner identified the appropriate scaling for nonlinear effects to appear in the leading order approximation, which in our case is ε2. This evolution equation is termed the amplitude equation. Hunter derived the analogous results for first order hyperbolic systems in his paper [Hun89]. The amplitude equation for nonlinear elasticity turns out to be a nonlocal Burgers type equation, and the argument to solve it goes back to Benzoni-Gavage. We want to stress that all of this body of work is primarily devoted to constructing approximate leading order solutions to equations, not the exact solution itself. So one of the main goals in geometric optics is to show that the constructed approximate solution is close to the exact solution and that the exact solution exists on a time interval independent of ε.