Schrodinger:quadratic NLS

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 ³ 0 references:CaWe1990 CaWe1990, references:Ts1987 Ts1987.
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -3/4. references:KnPoVe1996b KnPoVe1996b.
  • This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p]. The X^{s,b} bilinear estimates fail for H^{-3/4} references:NaTkTs-p NaTkTs2001.
• If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. references:KnPoVe1996b 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.

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Quadratic NLS on $T$

• For any quadratic non-linearity one can obtain LWP for s ³ 0 references:Bo1993 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. references:KnPoVe1996b 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.

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Quadratic NLS on $R^{2}$

• Scaling is s_c = -1.
• For any quadratic non-linearity one can obtain LWP for s ³ 0 references:CaWe1990 CaWe1990, references:Ts1987 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. references:St1997 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.

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Quadratic NLS on T^2

• If the quadratic non-linearity is of u u type then one can obtain LWP for s > -1/2 references#Gr-p2 Gr-p2

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Quadratic NLS on $R^{3}$

• Scaling is s_c = -1/2.
• For any quadratic non-linearity one can obtain LWP for s ³ 0 references:CaWe1990 CaWe1990, references:Ts1987 Ts1987.
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. references:St1997 St1997, references:CoDeKnSt-p CoDeKnSt-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.

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Quadratic NLS on $T^{3}$

• If the quadratic non-linearity is of u u type then one can obtain LWP for s > -3/10 references#Gr-p2 Gr-p2

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