Traveling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media
arXiv (Cornell University)(2023)
摘要
Traveling modulating pulse solutions consist of a small amplitude pulse-like
envelope moving with a constant speed and modulating a harmonic carrier wave.
Such solutions can be approximated by solitons of an effective nonlinear
Schrodinger equation arising as the envelope equation. We are interested in a
rigorous existence proof of such solutions for a nonlinear wave equation with
spatially periodic coefficients. Such solutions are quasi-periodic in a
reference frame co-moving with the envelope. We use spatial dynamics, invariant
manifolds, and near-identity transformations to construct such solutions on
large domains in time and space. Although the spectrum of the linearized
equations in the spatial dynamics formulation contains infinitely many
eigenvalues on the imaginary axis or in the worst case the complete imaginary
axis, a small denominator problem is avoided when the solutions are localized
on a finite spatial domain with small tails in far fields.
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关键词
wave equation,pulse solutions,periodic media
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