Abstract
In this Chapter, Wave Turbulence (WT) in Bose-Einstein condensates (BEC) and nonlinear optics is considered. Models considered include 2D and 3D Gross-Pitaevskii equations, Nordheim quantum kinetic equation, Boltzmann equation and 1D nonlinear equation for optical WT in nematic liquid crystals. Roles of the thermodynamic Rayleigh- Jeans, cascading Kolmogorov-Zakharov, mixed warm cascade and critical balance states are discussed. A picture of WT life cycle where a coherent uniform condensate component arises from an inverse cascade process. In this setting the WT starts as a four-wave process followed by breakdown of weakly nonlinear description and creation of a gas of strongly nonlinear vortices which undergo annihilations resulting in a vortex-free coherent condensate with random acoustic Bogoliubov waves engaged in mutual three-wave interactions
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Notes
- 1.
Paper [7] filtered out high k’s at the data processing, but not at the computational stage. Obviously, this could not prevent the backscatter/bottleneck effect discussed here.
- 2.
To be precise, this is a special case of the Boltzmann equation for rigid-sphere particles.
- 3.
The opposite statement about the negative energy flux was made in [17] based on the spectrum \(n=({c}/{\omega}) \ln^{1/3} ({ \omega} )\) thus effectively assuming an IR (rather than an UV) cutoff, which is unnatural because one should not assume a zero spectrum at the forcing scale.
- 4.
The normal component in Helium is representing, on a course-grained level, motion of the gas of phonons.
- 5.
In 1D NLS the solitons pass each other almost freely,—without change of their amplitudes or shapes but with small shifts of the original trajectories.
- 6.
Index x = 0 corresponds to a formal solution with particle equipartition in the momentum space. However, this solution is not normalizable.
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Nazarenko, S.V. (2011). Bose-Einstein Condensation. In: Wave Turbulence. Lecture Notes in Physics, vol 825. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15942-8_15
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