T. Dohnal, D. Pelinovsky, and G. Schneider
Traveling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media
Nonlinearity 37 (2024) 055005 (36pp)
Abstract:
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.
Keywords:
nonlinear wave equation; traveling modulating pulses; spatial dynamics; invariant manifolds; amplitude equations;