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

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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. 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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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, 75080, USA

    John Zweck

  2. Department of Mathematics, The University of Missouri, Columbia, MO, 65211, USA

    Yuri Latushkin

  3. Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA

    Jeremy L. Marzuola & Christopher K. R. T. Jones

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  1. John Zweck
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  2. Yuri Latushkin
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  4. Christopher K. R. T. Jones
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Correspondence to John Zweck.

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To Matthias Hieber with best wishes

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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. 21, 3313–3329 (2021). https://doi.org/10.1007/s00028-020-00640-8

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