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.

#### Quadratic NLS on R

• Scaling is ${\displaystyle s_{c}=-3/2\,.}$
• For any quadratic non-linearity one can obtain LWP for ${\displaystyle s\geq 0\,}$ CaWe1990, Ts1987.
• If the quadratic non-linearity is of ${\displaystyle {\underline {uu}}\,}$ or ${\displaystyle uu\,}$ type then one can push LWP to ${\displaystyle s>-3/4.\,}$ KnPoVe1996b.
• This can be improved to the Besov space ${\displaystyle B_{2,1}^{-3/4}\,}$ [MurTao-p]. The ${\displaystyle X^{s,b}\,}$ bilinear estimates fail for ${\displaystyle H^{-3/4}\,}$ NaTkTs2001.
• If the quadratic non-linearity is of ${\displaystyle {\underline {u}}u\,}$ type then one can push LWP to ${\displaystyle s>-1/4.\,}$ KnPoVe1996b.
• Since these equations do not have ${\displaystyle L^{2}\,}$ conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
• If the non-linearity is ${\displaystyle |u|u\,}$ then there is GWP in ${\displaystyle L^{2}\,}$ thanks to ${\displaystyle L^{2}\,}$ conservation, and ill-posedness below ${\displaystyle L^{2}\,}$ 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 ${\displaystyle s\geq 0\,}$ Bo1993. In the Hamiltonian case (${\displaystyle |u|u\,}$) this is sharp by Gallilean invariance considerations [KnPoVe-p]
• If the quadratic non-linearity is of ${\displaystyle {\underline {uu}}\,}$ or ${\displaystyle uu\,}$ type then one can push LWP to ${\displaystyle s>-1/2.\,}$ KnPoVe1996b.
• In the Hamiltonian case (a non-linearity of type ${\displaystyle |u|u\,}$) we have GWP for ${\displaystyle s\geq 0\,}$ by ${\displaystyle L^{2}\,}$ 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 ${\displaystyle s_{c}=-1.\,}$
• For any quadratic non-linearity one can obtain LWP for ${\displaystyle s\geq 0\,}$ CaWe1990, Ts1987.
• In the Hamiltonian case (${\displaystyle |u|u\,}$) this is sharp by Gallilean invariance considerations [KnPoVe-p]
• If the quadratic non-linearity is of ${\displaystyle {\underline {uu}}\,}$ or ${\displaystyle uu\,}$ type then one can push LWP to ${\displaystyle s>-3/4.\,}$ St1997, CoDeKnSt-p.
• This can be improved to the Besov space ${\displaystyle B_{2,1}^{-3/4}\,}$ [MurTao-p].
• If the quadratic non-linearity is of ${\displaystyle u{\underline {u}}\,}$ type then one can push LWP to ${\displaystyle s>-1/4.\,}$ Ta-p2.
• In the Hamiltonian case (a non-linearity of type ${\displaystyle |u|u\,}$) we have GWP for ${\displaystyle s\geq 0\,}$ by ${\displaystyle L^{2}\,}$ conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
• Below ${\displaystyle L^{2}\,}$ we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

#### Quadratic NLS on T^2

• If the quadratic non-linearity is of ${\displaystyle {\underline {uu}}\,}$ type then one can obtain LWP for ${\displaystyle s>-1/2\,}$ Gr-p2

#### Quadratic NLS on ${\displaystyle R^{3}}$

• Scaling is ${\displaystyle s_{c}=-1/2.\,}$
• For any quadratic non-linearity one can obtain LWP for ${\displaystyle s\geq 0\,}$ CaWe1990, Ts1987.
• If the quadratic non-linearity is of ${\displaystyle {\underline {uu}}\,}$ or ${\displaystyle uu\,}$ type then one can push LWP to ${\displaystyle s>-1/2.\,}$ St1997, CoDeKnSt-p.
• If the quadratic non-linearity is of ${\displaystyle u{\underline {u}}\,}$ type then one can push LWP to ${\displaystyle s>-1/4.\,}$ Ta-p2.
• In the Hamiltonian case (a non-linearity of type ${\displaystyle |u|u\,}$) we have GWP for ${\displaystyle s\geq 0\,}$ by ${\displaystyle L^{2}\,}$ conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
• Below ${\displaystyle L^{2}\,}$ we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

#### Quadratic NLS on ${\displaystyle T^{3}}$

• If the quadratic non-linearity is of ${\displaystyle {\underline {uu}}\,}$ type then one can obtain LWP for ${\displaystyle s>-3/10\,}$ Gr-p2