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4. The cubic Szegő equation with random initial data

The cubic Szegő equation is a toy model for wave turbulence, which reads $$ i\partial_t u=\Pi(|u|^2u),\quad (t,x)\in \mathbb{R}\times\mathbb{T},$$ where $\Pi$ is the Szegő projector onto $L^2_+$, the subspace of $L^2(\mathbb{T})$ of functions with only nonnegative Fourier modes.
    1.     Gérard and Grellier constructed a nonlinear Fourier transform $$ u\mapsto (\{I_k\},\{\varphi_k\})\in \Omega\times \mathbb{T}^\infty ,$$ which reduces the Szegő evolution to the following system of equations \begin{aligned} \dot{I_k}&=0, \\ \dot{\varphi_k}&=\varphi_k(0)+t\omega(I_k). \end{aligned}

      Problem 4.1.

      [Patrick Gérard] Assume $\mathbb{T}^\infty$ is embedded with the uniform probability measure. Find a suitable fixed set of actions $\{I_k\}\in\Omega $ for which the $H^s$-norms of solutions to the cubic Szegő equation ($s> 1/2$) remain almost surely bounded.

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