# Difference between revisions of "Quadratic NLS"

Description
Equation ${\displaystyle iu_{t}+\Delta u=Q(u,{\overline {u}})}$
Fields ${\displaystyle u:\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {C} }$
Data class ${\displaystyle u(0)\in H^{s}(\mathbb {R} ^{d})}$
Basic characteristics
Structure non-Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity ${\displaystyle {\dot {H}}^{d/2-2}(\mathbb {R} ^{d})}$
Criticality N/A
Covariance N/A
Theoretical results
LWP varies
GWP -
Related equations
Parent class NLS
Special cases Quadratic NLS on R, T, R^2, T^2, R^3, T^3
Other related -

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 ${\displaystyle s_{c}=-3/2\,.}$
• 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 ${\displaystyle B_{2,1}^{-3/4}\,}$ MurTao2004. 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, CoDeKnSt2001.
• This can be improved to the Besov space ${\displaystyle B_{2,1}^{-3/4}\,}$ MurTao2004.
• If the quadratic non-linearity is of ${\displaystyle u{\underline {u}}\,}$ type then one can push LWP to ${\displaystyle s>-1/4.\,}$ Ta2001.
• 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.

• 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, CoDeKnSt2001.
• If the quadratic non-linearity is of ${\displaystyle u{\underline {u}}\,}$ type then one can push LWP to ${\displaystyle s>-1/4.\,}$ Ta2001.
• 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.