Justification of Multidimensional Single Phase Semilinear Geometric Optics

Abstract

For semilinear strictly hyperbolic systems $Lu= f(x,u)$, we construct and justify high frequency nonlinear asymptotic expansions of the form \[ {u^\varepsilon }(x)\sim \sum \limits _{j \geq 0} {{\varepsilon ^j}{U_j}(x,\varphi (x)/\varepsilon }, \quad L{u^\varepsilon } - f(x,{u^\varepsilon })\sim 0 .\] The study of the principal term of such expansions is called nonlinear geometric optics in the applied literature. We show (i) formal expansions with periodic profiles ${U_j}$ can be computed to all orders, (ii) the equations for the profiles from (i) are solvable, and (iii) there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion.