Testing wave turbulence theory for Gross-Pitaevskii system

Abstract

We test the predictions of the theory of weak wave turbulence by performing numerical simulations of the Gross-Pitaevskii equation (GPE) and the associated wave-kinetic equation (WKE). We consider an initial state localized in Fourier space, and we confront the solutions of the WKE obtained numerically with GPE data for both, the waveaction spectrum and the probability density functions (PDFs) of the Fourier mode intensities. We find that the temporal evolution of GP data is accurately predicted by the WKE, with no adjustable parameters, for about two nonlinear kinetic times. Qualitative agreement between the GPE and the WKE persists also for longer times with some quantitative deviations that may be attributed to the breakdown of the theoretical assumptions underlying the WKE. Furthermore, we study how the wave statistics evolves toward Gaussianity in a time scale of the order of the kinetic time. The excellent agreement between direct numerical simulations of the GPE and the WKE provides a new and solid ground to the theory of weak wave turbulence.

Feynman rules for forced wave turbulence

“…Wave turbulence is both an old topic [6] and one under active study [7]. Questions of current interest include: new physical contexts exhibiting classical wave turbulence, 3 wave turbulence in quantum mechanics and the nonlinear Schrödinger equation [37][38][39][40][41][42][43][44], wave turbulence in quantum field theory and related concepts of prethermalization [45][46][47][48][49][50][51][52][53][54][55][56], mathematical properties of wave turbulence [57][58][59][60][61][62][63][64] including properties of the kinetic equation [65][66][67][68][69] and models such as [70][71][72][73] and [74] .…”