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2. About the MMT model

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.