2. About the MMT model
The socalled MMT model is a simple 1D equation introduced by Majda, McLaughlin and Tabak in the nineties. Numerical simulations show the existence of weak turbulence for this model.
Wellposedness issues
Problem 2.1.
[Alexandru Ionescu] For which values of $\alpha<1$ is it possible to find global solutions to $$ i\partial_t u = D^\alpha u+u^2u,\quad (t,x)\in \mathbb{R}\times \mathbb{T},$$ in the energy space $H^{\alpha /2}$ ?
Remark. [Themis Sapsis] Numerically, $\alpha =\frac{1}{2}$ seems to be a significant threshold. This value of $\alpha$ also corresponds to a waterwave model (see IonescuPusateri).


Other modifications of NLS
Problem 2.2.
[Themis Sapsis] Can you find solutions to other modified NLS equations, such as $$ u_t +\frac{1}{2}\frac{\partial u}{\partial x}+i\frac{\partial^2 u}{\partial x^2}+\frac{i}{2}u^2u\frac{1}{16}\frac{\partial^3 u}{\partial x^3}+\frac{3}{2}u^2\frac{\partial u}{\partial x}+\frac{1}{4}u^2\frac{\partial \overline{u}}{\partial x}=0 ,$$ where dissipation is involved ?
Cite this as: AimPL: Mathematical questions in wave turbulence theory, available at http://aimpl.org/waveturb.