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The essential spectrum of periodically stationary solutions of the complex Ginzburg-Landau equation. (English) Zbl 1483.35234

Summary: We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically stationary pulses in ultrafast fiber lasers.

MSC:

35Q56 Ginzburg-Landau equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q41 Time-dependent Schrödinger equations and Dirac equations
35B10 Periodic solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
47D06 One-parameter semigroups and linear evolution equations
78A60 Lasers, masers, optical bistability, nonlinear optics
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