The essential spectrum of periodically stationary solutions of the complex Ginzburg–Landau equation

Abstract

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.

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Notes

  1. 1.

    We observe that the third condition in Hypothesis 3.1 that \(\varvec{\psi }\) decays exponentially can be significantly relaxed. In particular, we do not even require that \(\varvec{\psi }\) or \(\varvec{\psi }_x\) belongs to \(L^2(\mathbb R,\mathbb C^2)\).

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Acknowledgements

We thank the reviewer for their careful reading of the paper.

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Correspondence to John Zweck.

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To Matthias Hieber with best wishes

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J.Z. was supported by the NSF under Grant DMS 1620293 and thanks the Department of Mathematics at UNC Chapel Hill for hosting his Fall 2018 sabbatical, during which this work began.

Y.L. was supported by the NSF under Grant DMS 1710989 and thanks the Courant Institute of Mathematical Sciences and, especially, Prof. Lai-Sang Young, for the opportunity to visit the Institute where this work was conducted.

J.L.M. was supported in part by NSF CAREER Grant DMS 1352353 and NSF Grant DMS 1909035.

C.K.R.T.J. was supported by the US Office of Naval Research under Grant N00014-18-1-2204.

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Zweck, J., Latushkin, Y., Marzuola, J.L. et al. The essential spectrum of periodically stationary solutions of the complex Ginzburg–Landau equation. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00640-8

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Keywords

  • Nonlinear waves
  • Breather solutions
  • Essential spectrum
  • Analytic semigroups
  • Fiber lasers

Mathematics Subject Classification

  • 35B10
  • 35Q56
  • 37L15
  • 47D06
  • 78A60