# Quadratic NLS

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### Quadratic NLS

Equations of the form

which a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.

#### Quadratic NLS on R

- Scaling is
- For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
- If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
- This can be improved to the Besov space MurTao-p. The bilinear estimates fail for NaTkTs2001.

- If the quadratic non-linearity is of type then one can push LWP to KnPoVe1996b.
- Since these equations do not have conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
- If the non-linearity is then there is GWP in thanks to conservation, and ill-posedness below by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

#### Quadratic NLS on

- For any quadratic non-linearity one can obtain LWP for Bo1993. In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p
- If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
- In the Hamiltonian case (a non-linearity of type ) we have GWP for by 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

- Scaling
- For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
- In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p

- If the quadratic non-linearity is of or type then one can push LWP to St1997, CoDeKnSt-p.
- This can be improved to the Besov space MurTao-p.

- If the quadratic non-linearity is of type then one can push LWP to Ta-p2.
- In the Hamiltonian case (a non-linearity of type ) we have GWP for by 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 T^2

- If the quadratic non-linearity is of type then one can obtain LWP for Gr-p2

#### Quadratic NLS on

- Scaling is
- For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
- If the quadratic non-linearity is of or type then one can push LWP to St1997, CoDeKnSt-p.
- If the quadratic non-linearity is of type then one can push LWP to Ta-p2.
- In the Hamiltonian case (a non-linearity of type ) we have GWP for by 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

- If the quadratic non-linearity is of type then one can obtain LWP for Gr-p2