Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

Buckmaster, Tristan; Germain, Pierre; Hani, Zaher; Shatah, Jalal

doi:10.1007/s00222-021-01039-z

Cited by 54 publications

(53 citation statements)

References 37 publications

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“…The main novelty of this work is in the first component, which is the hardest. The second component follows similar lines to those in [7]. Regarding the third component, the main novelty of this work is to complement the number-theoretic results in [7] (which dealt only with the generically irrational torus) by the cases of general tori (in the admissible range of time ≪ 2 ).…”

Section: Ideas Of the Proofmentioning

confidence: 70%

“…For other scaling laws, we identify significant absolute divergences in the power-series expansion for E | ( , )| 2 at much earlier times. We can therefore only justify this approximation at such shorter times (which are still better than those in [7]). In these cases, whether or not formula (1.1) holds up to time scales − kin depends on whether such series converge conditionally instead of absolutely, and thus would require new methods and ideas, as we explain later.…”

Section: Statement Of the Resultsmentioning

confidence: 93%

“…This manuscript continues the investigation initiated in [7], aimed at providing a rigorous justification of the wave kinetic equation corresponding to the nonlinear Schrödinger equation,…”

Section: Introductionmentioning

confidence: 78%

“…As will be clear later, no similar results can hold for ≫ 2 in the case of a rational torus, as this would require rational quadratic forms to be equidistributed on scales ≪ 1, which is impossible. However, if the aspect ratios are assumed to be generically irrational, then one can access equidistribution scales that are as small as − +1 for the resulting irrational quadratic forms [4,7]. This allows us to consider scaling laws for which kin can be as big as − on generically irrational tori.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…The importance of such an endeavour stems from the fact that it allows an understanding of the exact regimes and the limitations of the kinetic theory, which has long been a matter of scientific interest (see [20,1]). A few mathematical investigations have recently been devoted to studying problems in this spirit [23,7,35], yielding some partial results and useful insights.…”

Section: Introductionmentioning

confidence: 99%

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On the derivation of the wave kinetic equation for NLS

Deng

Hani

2021

Forum of Mathematics, Pi

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A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale $T_{\mathrm {kin}} \gg 1$ and in a limiting regime where the size L of the domain goes to infinity and the strength $\alpha $ of the nonlinearity goes to $0$ (weak nonlinearity). For the cubic nonlinear Schrödinger equation, $T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$ and $\alpha $ is related to the conserved mass $\lambda $ of the solution via $\alpha =\lambda ^2 L^{-d}$ . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the $(\alpha , L)$ limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when $\alpha $ approaches $0$ like $L^{-\varepsilon +}$ or like $L^{-1-\frac {\varepsilon }{2}+}$ (for arbitrary small $\varepsilon $ ), we exhibit the wave kinetic equation up to time scales $O(T_{\mathrm {kin}}L^{-\varepsilon })$ , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales $T_*\ll T_{\mathrm {kin}}$ and identify specific interactions that become very large for times beyond $T_*$ . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond $T_*$ toward $T_{\mathrm {kin}}$ for such scaling laws seems to require new methods and ideas.

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Section: Ideas Of the Proofmentioning

confidence: 70%

Section: Statement Of the Resultsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 78%

Section: Statement Of the Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the derivation of the wave kinetic equation for NLS

Deng

Hani

2021

Forum of Mathematics, Pi

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Large deviations principle for the cubic NLS equation

Garrido

Grande

Kurianski

et al. 2023

Comm Pure Appl Math

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In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.

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Inhomogeneous turbulence for the Wick Nonlinear Schrödinger equation

Hani,

Shatah,

Zhu

2024

Comm Pure Appl Math

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We introduce a simplified model for wave turbulence theory—the Wick nonlinear Schrödinger equation, of which the main feature is the absence of all self‐interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.

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Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

Cited by 54 publications

References 37 publications

On the derivation of the wave kinetic equation for NLS

On the derivation of the wave kinetic equation for NLS

Large deviations principle for the cubic NLS equation

Inhomogeneous turbulence for the Wick Nonlinear Schrödinger equation

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