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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 <math>s_c=-3/2\,.</math>
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| * For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
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| * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
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| ** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [MurTao-p]. The <math>X^{s,b}\,</math> bilinear estimates fail for <math>H^{-3/4}\,</math> [[references:NaTkTs-p NaTkTs2001]].
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| * If the quadratic non-linearity is of <math>\underline{u}u\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
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| * Since these equations do not have <math>L^2\,</math> 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 <math>|u|u\,</math> then there is GWP in <math>L^2\,</math> thanks to <math>L^2\,</math> conservation, and ill-posedness below <math>L^2\,</math> 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 <math>s \ge 0\,</math> [[Bibliography#Bo1993|Bo1993]]. In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [KnPoVe-p]
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| * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
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| * In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s \ge 0\,</math> by <math>L^2\,</math> 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 <math>s_c = -1.\,</math>
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| * For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
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| ** In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [KnPoVe-p]
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| * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
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| ** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [MurTao-p].
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| * If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[references:Ta-p2 Ta-p2]].
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| * In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s \ge 0\,</math> by <math>L^2\,</math> 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 <math>L^2\,</math> 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 <math>\underline{uu}\,</math> type then one can obtain LWP for <math>s > -1/2\,</math> [[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 <math>s_c = -1/2.\,</math>
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| * For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
| |
| * If the quadratic non-linearity is of <math>\underline{uu}\,</math> or <math>u u\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
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| * If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[references:Ta-p2 Ta-p2]].
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| * In the Hamiltonian case (a non-linearity of type <math>|u| u\,</math>) we have GWP for <math>s \ge 0\,</math> by <math>L^2\,</math> 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 <math>L^2\,</math> 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 <math>\underline{uu}\,</math> type then one can obtain LWP for <math>s > -3/10\,.</math> [[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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