On the derivation of the homogeneous kinetic wave equation - the role of the dispersion relation

On the derivation of the homogeneous kinetic wave equation - the role of the dispersion relation

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Charles Collot, CNRS-Cergy Paris Université

Zoom link: https://princeton.zoom.us/j/92147928280

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The kinetic wave equation arises in weak wave turbulence theory. In this talk we are interested in its derivation as an effective equation from the nonlinear Schrodinger equation (NLS) for the microscopic description of a system. More precisely, we will consider (NLS) in a weakly nonlinear regime on a torus in any dimension greater than two, and for highly oscillatory random Gaussian fields as initial data. A conjecture in statistical physics is that there exists a kinetic time scale on which, statistically, the Fourier modes evolve according to the kinetic wave equation.

   There are two parameters in this problem: the oscillation length of the field, and the strength of the nonlinearity. It is still unknown in which regimes is the kinetic wave equation rigorously valid. The dispersion relation - alternatively, the geometry of the torus - seems to play a key role, since the distribution properties of the associated quadratic form on integer points are directly related to the structure of the resonant terms in the dynamics.

   In the case where the torus is the standard one, we prove that only one particular regime allows for the convergence of the Dyson series up to the kinetic time scale. We also show that, for generic quadratic dispersion relations (non rectangular tori), the Dyson series converges on significantly longer time scales. We are able to control the full solution up to the kinetic time, in the particular regime for the standard torus, and for a larger set of regimes for generic tori, up to an arbitrarily small polynomial error. This is joint work with P. Germain.