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3. Understanding the random phase approximation

The Random Phase Approximation (RPA) is a key assumption in Zakharov’s theory for wave turbulence ; it consists in assuming that the wave spectrum is chaotic.
    1. About the kinetic timescale

          Start from the cubic NLS equation $$ i\partial_tu+\Delta u=\lambda^2|u|^2u$$ on a large box of typical size $L$, say $[0,L\beta_1]\times[0,L\beta_2]\times \dots \times [0,L\beta_d]$. The initial data is taken to be $$ u(t=0)=\frac 1 {L^d} \sum_{k\in \mathbb{Z}^d/L} \rho_ke^{i\theta_k}e^{ik\cdot x} ,$$ where $\theta_k$ are independent and identically distributed random variables.

      Conjecture 3.1.

      [Tristan Buckmaster] Writing $$u(t)=\frac 1 {L^d} \sum_{k\in \mathbb{Z}^d/L} a_k(t)e^{i(k\cdot x+t|k|^2)},$$ the averaged amplitude $\mathbb{E}(|a_k|^2)$ solves the kinetic wave equation up to the so-called kinetic timescale $\tau \sim \frac {L^{2d}} {\lambda^4}$.
        • Validity of the RPA

          Problem 3.2.

          In the preceding setting, up to what time can we assume that the phases remain independent ?

              Cite this as: AimPL: Mathematical questions in wave turbulence theory, available at http://aimpl.org/waveturb.