Quadratic NLS

Quadratic NLS

Description
Equation	$iu_{t}+\Delta u=Q(u,{\overline {u}})$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{d})$
Basic characteristics
Structure	non-Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{d/2-2}(\mathbb {R} ^{d})$
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	-

Quadratic NLS

Equations of the form

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

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

Quadratic NLS on R

• Scaling is $s_{c}=-3/2\,.$
• For any quadratic non-linearity one can obtain LWP for $s\geq 0\,$ CaWe1990, Ts1987.
• If the quadratic non-linearity is of ${\underline {uu}}\,$ or $uu\,$ type then one can push LWP to $s>-3/4.\,$ KnPoVe1996b.
  • This can be improved to the Besov space $B_{2,1}^{-3/4}\,$ MurTao2004. The $X^{s,b}\,$ bilinear estimates fail for $H^{-3/4}\,$ NaTkTs2001.
• If the quadratic non-linearity is of ${\underline {u}}u\,$ type then one can push LWP to $s>-1/4.\,$ KnPoVe1996b.
• Since these equations do not have $L^{2}\,$ conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
• If the non-linearity is $|u|u\,$ then there is GWP in $L^{2}\,$ thanks to $L^{2}\,$ conservation, and ill-posedness below $L^{2}\,$ by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on $T$

• For any quadratic non-linearity one can obtain LWP for $s\geq 0\,$ Bo1993. In the Hamiltonian case ($|u|u\,$) this is sharp by Gallilean invariance considerations KnPoVe-p
• If the quadratic non-linearity is of ${\underline {uu}}\,$ or $uu\,$ 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 $s\geq 0\,$ by $L^{2}\,$ 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 $R^{2}$

• Scaling $s_{c}=-1.\,$
• For any quadratic non-linearity one can obtain LWP for $s\geq 0\,$ CaWe1990, Ts1987.
  • In the Hamiltonian case ($|u|u\,$) this is sharp by Gallilean invariance considerations KnPoVe-p
• If the quadratic non-linearity is of ${\underline {uu}}\,$ or $uu\,$ type then one can push LWP to $s>-3/4.\,$ St1997, CoDeKnSt-p.
  • This can be improved to the Besov space $B_{2,1}^{-3/4}\,$ MurTao2004.
• If the quadratic non-linearity is of $u{\underline {u}}\,$ type then one can push LWP to $s>-1/4.\,$ Ta2001.
• In the Hamiltonian case (a non-linearity of type $|u|u\,$) we have GWP for $s\geq 0\,$ by $L^{2}\,$ conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • Below $L^{2}\,$ 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 ${\underline {uu}}\,$ type then one can obtain LWP for $s>-1/2\,$ Gr-p2

Quadratic NLS on $R^{3}$

• Scaling is $s_{c}=-1/2.\,$
• For any quadratic non-linearity one can obtain LWP for $s\geq 0\,$ CaWe1990, Ts1987.
• If the quadratic non-linearity is of ${\underline {uu}}\,$ or $uu\,$ type then one can push LWP to $s>-1/2.\,$ St1997, CoDeKnSt-p.
• If the quadratic non-linearity is of $u{\underline {u}}\,$ type then one can push LWP to $s>-1/4.\,$ Ta2001.
• In the Hamiltonian case (a non-linearity of type $|u|u\,$) we have GWP for $s\geq 0\,$ by $L^{2}\,$ conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • Below $L^{2}\,$ we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on $T^{3}$

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