Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials

Huang, Guan; Kuksin, Sergei; Maiocchi, Alberto

doi:10.1007/978-1-4939-2950-4_11

Cited by 19 publications

(32 citation statements)

References 15 publications

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“…Here δ(ω 12 3s ) = 1 if ω 12 3s = 0 and δ(ω 12 3s ) = 0 otherwise. The following result is proven in [19,16]. It is known that, under assumptions of Theorem 1.1, equations (1.15) and (1.17) are mixing, so they have unique stationary measures.…”

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confidence: 91%

“…It is known that, under assumptions of Theorem 1.1, equations (1.15) and (1.17) are mixing, so they have unique stationary measures. In [19,16] it is also proven that assertions of the theorem remain true if a ε (τ ) and a 0 (τ ) denote stationary solutions of equations (1.15) and (1.17).…”

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confidence: 95%

“…the proof we start with the result obtained in [19,16], where the limiting as ν → 0 behaviour of equation (1.6) is examined (while L and ρ are kept fixed). Then we pass to the limit as L → ∞, following the approach of [6,7] and using the developed there tools, such as a specific Feynman diagram presentation.…”

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confidence: 99%

“…It is known as the limit of discrete turbulence (see [23,Section 10]) and has been successfully studied in [19,16]. To explain the result let us take the Fourier transform of equation (1.6):…”

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confidence: 99%

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The large-period limit for equations of discrete turbulence

Andrey¹,

Kuksin²,

Maiocchi³

et al. 2021

Preprint

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We consider the damped/driven cubic NLS equation on the torus of a large period L with a small nonlinearity of size λ, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first λ → 0 and then L → ∞. The first limit, called the limit of discrete turbulence, is known to exist, and in this work we study the second limit L → ∞ for solutions to the equations of discrete turbulence. Namely, we decompose the solutions to formal series in amplitude and study the second order truncation of this series. We prove that the energy spectrum of the truncated solutions becomes close to solutions of a damped/driven nonlinear wave kinetic equation. Kinetic nonlinearity of the latter is similar to that which usually appears in works on wave turbulence, but is different from it (in particular, it is non-autonomous). Apart from tools from analysis and stochastic analysis, our work uses two powerful results from the number theory.

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confidence: 91%

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confidence: 95%

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confidence: 99%

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confidence: 99%

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The large-period limit for equations of discrete turbulence

Andrey¹,

Kuksin²,

Maiocchi³

et al. 2021

Preprint

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“…The limit ν → 0 with L fixed was rigorously treated in a number of publications, e.g. see [9]. The only mathematical work, addressing the limit (1.4), is paper [7].…”

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Formal expansions in stochastic model for wave turbulence 1: kinetic limit

Dymov¹,

Kuksin²

2019

Preprint

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We consider the damped/driver (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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The Effective Equation Method

Kuksin

Maiocchi

2016

Lecture Notes in Physics

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In this chapter we present a general method of constructing the effective equation which describes the behaviour of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behaviour of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three-and fourwave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. In the case of the NLS equation we use next some heuristic approximation from the arsenal of wave turbulence to show that under the iterated limit "the volume goes to infinity", taken after the limit "the amplitude of oscillations goes to zero", the energy spectrum of solutions for the effective equation is described by a Zakharov-type kinetic equation. Evoking the Zakharov ansatz we show that stationary in time and homogeneous in space solutions for the latter equation have a power law form. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanology.

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Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials

Cited by 19 publications

References 15 publications

The large-period limit for equations of discrete turbulence

The large-period limit for equations of discrete turbulence

Formal expansions in stochastic model for wave turbulence 1: kinetic limit

The Effective Equation Method

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