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Formal Expansions in Stochastic Model for Wave Turbulence 1: Kinetic Limit

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Abstract

We consider the damped/driven (modified) cubic NLS equation on a large torus with a properly scaled forcing and dissipation, and decompose its solutions to formal series in the amplitude. We study the second order truncation of this series and prove that when the amplitude goes to zero and the torus’ size goes to infinity the energy spectrum of the truncated solutions becomes close to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.

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Notes

  1. Note in addition that if u(t, x) satisfies eq. (1.2), then \(u'= e^{ 2it\lambda \Vert u \Vert ^2}u\) is a solution of (1.1).

  2. The symmetric form of the Fourier transform which we use—with the same scaling factor \( L^{-d/2}\) for the direct and the inverse transformations—is convenient for the heavy calculation below in the paper.

  3. i.e. \(\beta _s = \beta _s^1 + i\beta _s^2\), where \(\{\beta _s^j, s\in {\mathbb {Z}}^d_L, j=1,2 \}\) are standard independent real Wiener processes.

  4. For example, if \(\gamma _s = (1+ |s|^2)^{r_*}\), then \(\mathfrak {A}= (1-\Delta )^{r_*}\).

  5. Indeed, denoting \(\xi =(x,\bar{\eta })\) we see that the first fundamental form \(I^\xi \) of \(\Sigma ^x\) is given by \(I^\xi _{ij} =(\delta _{i,j} +\theta _i\theta _j)\), where \(\theta = \partial _\xi \varphi \in \mathbb {R} ^{2d-1}\). So for \(X\in \mathbb {R} ^{2d-1}\),

    $$\begin{aligned} I^\xi (X,X) = \sum X_j^2 + \sum _{i,j} X_i\theta _i X_j\theta _j = \sum X_j^2 +(X\cdot \theta )^2. \end{aligned}$$

    Choosing in \( \mathbb {R} ^{2d-1} \) a coordinate system with the first basis vector \(\theta /|\theta |\) we find that \( I^\xi (X,X) = X_1^2(1+|\theta |^2) + \sum _{j\ge 2} X_j^2\). So det\(I^\xi = 1+| \partial _\xi \varphi |^2\), which implies the formula for the density \(\bar{p}\).

  6. This is true for the most of Wick-pairings, while for some of them the affine space of variables may become empty because of the restrictions of the type \(\{s_1,s_2\}\ne \{s_3,s\}\) imposed by \(\delta '^{12}_{3s}\).

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Acknowledgements

AD was supported by the Grant of the President of the Russian Federation (Project MK-1999.2021.1.1) and by the Russian Foundation for Basic Research (Project 18-31-20031), and SK – by Agence Nationale de la Recherche through the grant 17-CE40-0006. We thank Johannes Sjöstrand for discussion and an anonymous referee for careful reading of the paper and pointing out some flaws.

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Authors and Affiliations

  1. Steklov Mathematical Institute of RAS, Moscow, Russia, 119991

    Andrey Dymov

  2. National Research University Higher School of Economics, Moscow, Russia, 119048

    Andrey Dymov

  3. UFR de Mathématiques - Batiment Sophie Germain, Université Paris-Diderot (Paris 7), 5 rue Thomas Mann, 75205, Paris, France

    Sergei Kuksin

  4. School of Mathematics, Shandong University, Jinan, People’s Republic of China

    Sergei Kuksin

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  1. Andrey Dymov
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  2. Sergei Kuksin
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Correspondence to Sergei Kuksin.

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Communicated by C. Liverani.

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Dymov, A., Kuksin, S. Formal Expansions in Stochastic Model for Wave Turbulence 1: Kinetic Limit. Commun. Math. Phys. 382, 951–1014 (2021). https://doi.org/10.1007/s00220-021-03955-w

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