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We construct and justify leading order weakly nonlinear geometric opticsexpansions for nonlinear hyperbolic initial value problems, including thecompressible Euler equations. The technique of simultaneous Picard iteration isemployed to show approximate solutions tend to the exact solutions in the smallwavelength limit. Recent work [2] by Coulombel, Gues, and Williams studied thecase of reflecting wave trains whose expansions involve only real phases. Wetreat generic boundary frequencies by incorporating into our expansions bothreal and nonreal phases. Nonreal phases introduce difficulties such asapproximately solving complex transport equations and result in the addition ofboundary layers with exponential decay. This also prevents us from doing anerror analysis based on almost-periodic profiles as in [2].

Resonant leading term geometric optics expansions with boundary layers for quasilinear hyperbolic boundary problems