Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves
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| Format: | Preprint |
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2023
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| _version_ | 1866909322431168512 |
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| author | Haragus, Mariana Johnson, Mathew A. Perkins, Wesley R. de Rijk, Björn |
| author_facet | Haragus, Mariana Johnson, Mathew A. Perkins, Wesley R. de Rijk, Björn |
| contents | 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 (orbitally) asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential decay rates of perturbations 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. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in $N$. This in turn yields an improved nonuniform subharmonic stability result providing an $N$-independent ball of initial perturbations which eventually exhibit exponential decay at an $N$-dependent rate. Finally, we argue that our results connect in the limit $N \to \infty$ to previously established stability results against localized perturbations, thereby unifying existing theories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_01176 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves Haragus, Mariana Johnson, Mathew A. Perkins, Wesley R. de Rijk, Björn Analysis of PDEs Mathematical Physics 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 (orbitally) asymptotically stable against $NT$-periodic, i.e. subharmonic, perturbations. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential decay rates of perturbations 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. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in $N$. This in turn yields an improved nonuniform subharmonic stability result providing an $N$-independent ball of initial perturbations which eventually exhibit exponential decay at an $N$-dependent rate. Finally, we argue that our results connect in the limit $N \to \infty$ to previously established stability results against localized perturbations, thereby unifying existing theories. |
| title | Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2307.01176 |