We study the nonlinear dynamics of perturbed, spectrally stable T -periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. It is known that for each $N\in\mathbb{N}$, such a $T$-periodic wave train is asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the underlying wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on $N$ and, in fact, tend to zero as $N\to\infty$, leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in $N$. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. ... mehr

Zugehörige Institution(en) am KIT |
Institut für Analysis (IANA) Sonderforschungsbereich 1173 (SFB 1173) |

Publikationstyp |
Forschungsbericht/Preprint |

Publikationsmonat/-jahr |
07.2023 |

Sprache |
Englisch |

Identifikator |
ISSN: 2365-662X KITopen-ID: 1000160233 |

Verlag |
Karlsruher Institut für Technologie (KIT) |

Umfang |
30 S. |

Serie |
CRC 1173 Preprint ; 2023/17 |

Projektinformation |
SFB 1173/3 (DFG, DFG KOORD, SFB 1173/3) |

Externe Relationen |
Siehe auch |

Schlagwörter |
nonlinear stability, periodic waves, subharmonic perturbations, Lugiato-Lefever equation |

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