Schrodinger:quadratic NLS: Difference between revisions

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====Quadratic NLS on R====
====Quadratic NLS on R====


* Scaling is s<sub>c</sub> = -3/2.
* Scaling is <math>s_c=-3/2\,.</math>
* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]].
* For any quadratic non-linearity one can obtain LWP for <math>s>=0\,</math> [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#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 > -3/4. [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
* 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]].
** 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]].
** 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]].
* If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[Bibliography#KnPoVe1996b|KnPoVe1996b]].
* 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]].
* 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.
* 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.
* 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.
* 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.


[[Category:Equations]]
[[Category:Equations]]

Revision as of 21:04, 29 July 2006

Quadratic NLS

Equations of the form

which a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.


Quadratic NLS on R

  • Scaling is
  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
  • If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
    • This can be improved to the Besov space [MurTao-p]. The bilinear estimates fail for references:NaTkTs-p NaTkTs2001.
  • If the quadratic non-linearity is of type then one can push LWP to KnPoVe1996b.
  • Since these equations do not have conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • If the non-linearity is then there is GWP in thanks to conservation, and ill-posedness below by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

Quadratic NLS on

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

Quadratic NLS on

  • Scaling is sc = -1.
  • For any quadratic non-linearity one can obtain LWP for s ³ 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 u u or u u type then one can push LWP to s > -3/4. 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.


Quadratic NLS on T^2


Quadratic NLS on

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

Quadratic NLS on