Validity of Whitham’s Equations for the Modulation of Periodic Traveling Waves in the NLS Equation

We prove that slow modulations in time and space of periodic wave trains of the NLS equation can be approximated via solutions of Whitham’s equations associated with the wave train. The error estimates are based on a suitable choice of polar coordinates, a Cauchy–Kowalevskaya-like existence and uniqueness theorem, and energy estimates.

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