The aim of this course is to present recent developments in the theory of non-equilibrium statistical physics for nonlinear waves, commonly known as wave  turbulence. A large system of weakly-interacting waves is generally governed by a large number of differential equations which describe the dynamics of each wave. Due to their complexity and the large number of waves, the precise description of an individual wave may be difficult, and it might not be representative of the behavior of the system as a whole. What is sometimes possible is a statistical description of such a system, which provides information about its typical behavior. We will present such a description, in the form of a Wave Kinetic Equation, in the case where the underlying “microscopic” system of nonlinear waves satisfies the cubic Schrödinger equation, and show how to derive it rigorously.

Course contents:
I. Resonances and quasi-resonances. Formal derivation of Wave Kinetic Equation from the cubic Schrödinger equation.
II. Introduction to Gaussian Hilbert Spaces. Isserlis’ theorem. Gaussian hypercontractivity estimates.
III. Picard iteration, remainder, and reduction to a combinatorics problem.
IV. Feynman diagrams, pairings and counting estimates.
V. Well-posedness of WKE in the 3D isotropic setting. Fluxes and special solutions. Long term behavior.

References

[1] Collot, C., and Germain, P. Derivation of the homogeneous kinetic wave equation: longer time scales. Preprint (2020).
[2] Deng, Y., and Hani, Z. On the derivation of the wave kinetic equation for NLS. Forum Math. Pi 9 (2021), Paper No. e6, 37.
[3] Deng, Y., and Hani, Z. Full derivation of the wave kinetic equation. Invent. Math. 233, 2 (2023), 543–724.
[4] Deng, Y., and Hani, Z. Long time justification of wave turbulence theory. Preprint (2023). https://arxiv. org/abs/2311.10082.
[5] Deng, Y., and Hani, Z. Propagation of chaos and higher order statistics in wave kinetic theory. To appear in Journal of European Mathematical Society (JEMS). (2024). https://arxiv.org/abs/2110.04565.
[6] Escobedo, M., and Velazquez, J. J. L.  On the theory of weak turbulence for the nonlinear Schrödinger equation. Mem. Amer. Math. Soc. 238, 1124 (2015), v+107.
[7] Germain, P., Ionescu, A. D., and Tran, M.-B. Optimal local well-posedness theory for the kinetic wave equation. J. Funct. Anal. 279, 4 (2020), 108570, 28.
[8] Grande, R., and Hani, Z. Rigorous derivation of damped-driven wave turbulence theory. Preprint (2024). https://arxiv.org/abs/2407.10711.
[9] Janson, S. Gaussian Hilbert spaces, vol. 129 of Cambridge Tracts in Mathematics. Cambridge University  Press, Cambridge, 1997.
[10] Nazarenko, S. Wave turbulence, vol. 825 of Lecture Notes in Physics. Springer, Heidelberg, 2011.