Quadratic NLS: Difference between revisions

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{{equation
| name = Quadratic NLS
| equation = <math>iu_t + \Delta u = Q(u, \overline{u})</math>
| fields = <math>u: \R \times \R^d \to \mathbb{C}</math>
| data = <math>u(0) \in H^s(\R^d)</math>
| hamiltonian = non-Hamiltonian
| linear = [[free Schrodinger equation|Schrodinger]]
| nonlinear = [[semilinear]]
| critical = <math>\dot H^{d/2 - 2}(\R^d)</math>
| criticality = N/A
| covariance = N/A
| lwp = varies
| gwp = -
| parent = [[NLS]]
| special = Quadratic NLS on R, T, R^2, T^2, R^3, T^3
| related = -
}}
===Quadratic NLS===
===Quadratic NLS===


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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 \ge 0\,</math> [[CaWe1990]], [[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> [[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> [[MurTao2004]]. The <math>X^{s,b}\,</math> bilinear estimates fail for <math>H^{-3/4}\,</math> [[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> [[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]]
 
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====Quadratic NLS on <math>T</math>====
====Quadratic NLS on <math>T</math>====


* For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#Bo1993|Bo1993]]. In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
* For any quadratic non-linearity one can obtain LWP for <math>s \ge 0\,</math> [[Bo1993]]. In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [[KnPoVe-p]]
* 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. [[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 > -1/2.\,</math> [[KnPoVe1996b]].
* 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.
* 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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[[Category:Equations]]
[[Category:Equations]]
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====Quadratic NLS on <math>R^2</math>====
====Quadratic NLS on <math>R^2</math>====


* Scaling is s<sub>c</sub> = -1.
* Scaling <math>s_c = -1.\,</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 \ge 0\,</math> [[CaWe1990]], [[Ts1987]].
** In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
** In the Hamiltonian case (<math>|u| u\,</math>) this is sharp by Gallilean invariance considerations [[KnPoVe-p]]
* 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#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
* 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> [[St1997]], [[CoDeKnSt2001]].
** This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p].
** This can be improved to the Besov space <math>B^{-3/4}_{2,1}\,</math> [[MurTao2004]].
* 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]].
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Ta2001]].
* 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.
* 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.
** Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
** 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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[[Category:Equations]]
 


====Quadratic NLS on T^2====
====Quadratic NLS on T^2====


* 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]]
* If the quadratic non-linearity is of <math>\underline{uu}\,</math> type then one can obtain LWP for <math>s > -1/2\,</math> [[Gr-p2]]
 
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[[Category:Equations]]


====Quadratic NLS on <math>R^3</math>====
====Quadratic NLS on <math>R^3</math>====


* Scaling is s<sub>c</sub> = -1/2.
* Scaling is <math>s_c = -1/2.\,</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 \ge 0\,</math> [[CaWe1990]], [[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. [[Bibliography#St1997|St1997]], [[references:CoDeKnSt-p CoDeKnSt-p]].
* 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> [[St1997]], [[CoDeKnSt2001]].
* 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]].
* If the quadratic non-linearity is of <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[Ta2001]].
* 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.
* 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.
** Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
** 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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[[Category:Equations]]


====Quadratic NLS on <math>T^3</math>====
====Quadratic NLS on <math>T^3</math>====


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


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

Latest revision as of 01:30, 17 March 2007

Quadratic NLS
Description
Equation
Fields
Data class
Basic characteristics
Structure non-Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity
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

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 MurTao2004. The bilinear estimates fail for 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 Bo1993. In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of or type then one can push LWP to KnPoVe1996b.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by 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
  • For any quadratic non-linearity one can obtain LWP for CaWe1990, Ts1987.
    • In the Hamiltonian case () this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of or type then one can push LWP to St1997, CoDeKnSt2001.
    • This can be improved to the Besov space MurTao2004.
  • If the quadratic non-linearity is of type then one can push LWP to Ta2001.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
    • Below 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 type then one can obtain LWP for Gr-p2

Quadratic NLS on

  • 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 St1997, CoDeKnSt2001.
  • If the quadratic non-linearity is of type then one can push LWP to Ta2001.
  • In the Hamiltonian case (a non-linearity of type ) we have GWP for by conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
    • Below we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on

  • If the quadratic non-linearity is of type then one can obtain LWP for Gr-p2.