Speaker: Jani Lukkarinen (University of Helsinki)

Title: On the rigorous derivation of wave turbulence in spatially homogeneous discrete nonlinear Schrödinger equation

Abstract:
 Over the last year, significant advance has been made towards the rigorous derivation of kinetic theory, i.e., wave turbulence, for the nonlinear Schrödinger equation (NLS). For example, in the recent preprints by Yu Deng and Zaher Hani [arXiv:2104.11204 and arXiv:2110.04565] the derivation is completed for the continuum NLS on a torus of size L->infinity, assuming that the coupling is proportional to 1/L (subkinetic torus size); the proof relies on further developing the techniques from [Buckmaster, T., Germain, P., Hani, Z. et al. Invent. math. 225, 787–855 (2021)].  Using a different, noise induced, regularization scheme, A. Dymov, S. Kuksin, A. Maiocchi, and S. Vladuts [arXiv:2104.11967 and arXiv:2110.13873] consider the continuum NLS with Langevin noice terms.  In this talk, we apply Wick polynomial representations to the discrete nonlinear Schrödinger evolution (DNLS) with random, spatially homogeneous, Gaussian initial data, and show how they can be employed to simplify the cumulant hierarchy of the system. This allows correctly identifying the kinetic collision term of the Boltzmann-Peierls equation for the DNLS, and hence to study its wave turbulence.  I will also discuss a joint ongoing work with Aleksis Vuoksenmaa where we propose a method which should lead to bounds of the remaining error terms in the cumulant hierarchy, and hence prove the validity of the kinetic equation in the discrete model up to a finite kinetic time, for all small enough couplings and sufficiently large volumes.  The talk is based on joint works with Matteo Marcozzi, Alessia Nota, and Herbert Spohn.