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Traveling modulating pulse solutions consist of a small amplitude pulse-likeenvelope moving with a constant speed and modulating a harmonic carrier wave.Such solutions can be approximated by solitons of an effective nonlinearSchrodinger equation arising as the envelope equation. We are interested in arigorous existence proof of such solutions for a nonlinear wave equation withspatially periodic coefficients. Such solutions are quasi-periodic in areference frame co-moving with the envelope. We use spatial dynamics, invariantmanifolds, and near-identity transformations to construct such solutions onlarge domains in time and space. Although the spectrum of the linearizedequations in the spatial dynamics formulation contains infinitely manyeigenvalues on the imaginary axis or in the worst case the complete imaginaryaxis, a small denominator problem is avoided when the solutions are localizedon a finite spatial domain with small tails in far fields.

Traveling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media