
The so-called 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.
1. Well-posedness 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 $H^{\alpha /2}$ ?
1. Remark. [Themis Sapsis] Numerically, $\alpha =\frac{1}{2}$ seems to be a significant threshold. This value of $\alpha$ also corresponds to a water-wave model (see Ionescu-Pusateri).
• 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.