
## 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.