We suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schrödinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analysed with full rigour in Kuksin and Maiocchi (0000), and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the Kolmogorov-Zakharov spectra.

Kuksin, S., Maiocchi, A. (2015). Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation. PHYSICA D-NONLINEAR PHENOMENA, 309, 65-70 [10.1016/j.physd.2015.04.002].

Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation

Maiocchi A.
2015

Abstract

We suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schrödinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analysed with full rigour in Kuksin and Maiocchi (0000), and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the Kolmogorov-Zakharov spectra.
Articolo in rivista - Articolo scientifico
Kolmogorov-Zakharov spectra; Wave kinetic equation; Weak turbulence
English
2015
309
65
70
31621
reserved
Kuksin, S., Maiocchi, A. (2015). Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation. PHYSICA D-NONLINEAR PHENOMENA, 309, 65-70 [10.1016/j.physd.2015.04.002].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/334579
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