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Standing generalized modulating pulse solutions for a nonlinear wave equation in periodic media. (English) Zbl 1262.35169

Summary: Standing modulating pulse solutions consist of a standing pulse-like envelope modulating an underlying spatially and temporarily oscillating carrier wave. Using spatial dynamics, invariant manifold theory and normal form theory for periodic systems we construct such solutions on large domains in time and space for a nonlinear wave equation with spatially periodic coefficients. Such solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage.

MSC:

35L71 Second-order semilinear hyperbolic equations
35B10 Periodic solutions to PDEs
35C05 Solutions to PDEs in closed form
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems