Equations of the form

${\displaystyle i\partial _{t}u+\Delta u=Q(u,{\overline {u}})}$

which ${\displaystyle Q(u,{\overline {u}})}$ a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.

• Scaling is sc = -3/2.
• For any quadratic non-linearity one can obtain LWP for s ³ 0 CaWe1990, Ts1987.
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -3/4. KnPoVe1996b.
• If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. KnPoVe1996b.
• Since these equations do not have L2 conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
• If the non-linearity is |u|u then there is GWP in L2 thanks to L2 conservation, and ill-posedness below L2 by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

Quadratic NLS on ${\displaystyle T}$

• For any quadratic non-linearity one can obtain LWP for s ³ 0 Bo1993. In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. KnPoVe1996b.
• In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L2 conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.

Quadratic NLS on ${\displaystyle R^{2}}$

• Scaling is sc = -1.
• For any quadratic non-linearity one can obtain LWP for s ³ 0 CaWe1990, Ts1987.
• In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -3/4. St1997, references:CoDeKnSt-p CoDeKnSt-p.
• This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p].
• If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. references:Ta-p2 Ta-p2.
• In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L2 conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
• Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.