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| ===Quadratic NLS===
| | #REDIRECT [[Quadratic NLS]] |
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| Equations of the form
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| <math> i \partial_t u + \Delta u = Q(u, \overline{u})</math>
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| which <math>Q(u, \overline{u})</math> a quadratic function of its arguments are [[Schrodinger:quadratic NLS|quadratic nonlinear Schrodinger equations]].
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| ====Quadratic NLS on R====
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| * Scaling is s<sub>c</sub> = -3/2.
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| * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
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| * If the quadratic non-linearity is of <u>u</u> <u>u</u> or u u type then one can push LWP to s > -3/4. [[references:KnPoVe1996b KnPoVe1996b]].
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| ** 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]].
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| * If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[references:KnPoVe1996b KnPoVe1996b]].
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| * Since these equations do not have L<sup>2</sup> conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
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| * If the non-linearity is |u|u then there is GWP in L<sup>2</sup> thanks to L<sup>2</sup> conservation, and ill-posedness below L<sup>2</sup> by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
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| [[Category:Equations]]
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| ====Quadratic NLS on <math>T</math>====
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| * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:Bo1993 Bo1993]]. In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
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| * If the quadratic non-linearity is of <u>u</u> <u>u</u> or u u type then one can push LWP to s > -1/2. [[references:KnPoVe1996b KnPoVe1996b]].
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| * In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s <font face="Symbol">³</font> 0 by L<sup>2</sup> 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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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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| ====Quadratic NLS on <math>R^2</math>====
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| * Scaling is s<sub>c</sub> = -1.
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| * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
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| ** In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
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| * If the quadratic non-linearity is of <u>u</u> <u>u</u> or u u type then one can push LWP to s > -3/4. [[references:St1997 St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
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| ** This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p].
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| * If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[references:Ta-p2 Ta-p2]].
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| * In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s <font face="Symbol">³</font> 0 by L<sup>2</sup> 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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| ** 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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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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| ====Quadratic NLS on T^2====
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| * If the quadratic non-linearity is of <u>u</u> <u>u</u> type then one can obtain LWP for s > -1/2 [[references#Gr-p2 Gr-p2]]
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]] | |
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| ====Quadratic NLS on <math>R^3</math>====
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| * Scaling is s<sub>c</sub> = -1/2.
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| * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[references:CaWe1990 CaWe1990]], [[references:Ts1987 Ts1987]].
| |
| * If the quadratic non-linearity is of <u>u</u> <u>u</u> or u u type then one can push LWP to s > -1/2. [[references:St1997 St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
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| * If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[references:Ta-p2 Ta-p2]].
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| * In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s <font face="Symbol">³</font> 0 by L<sup>2</sup> 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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| ** 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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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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| ====Quadratic NLS on <math>T^3</math>====
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| * If the quadratic non-linearity is of <u>u</u> <u>u</u> type then one can obtain LWP for s > -3/10 [[references#Gr-p2 Gr-p2]]
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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