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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* Scaling is <math>s_c=-3/2\,.</math>  
* Scaling is <math>s_c=-3/2\,.</math>  
* For any quadratic non-linearity one can obtain LWP for <math>s>=0\,</math> [[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 <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -3/4.\,</math> [[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 <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]].
** 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  <math>\underline{u}u\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[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 <math>L^2\,</math> 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 <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.
* 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle iu_t + \Delta u = Q(u, \overline{u})}
Fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u: \R \times \R^d \to \mathbb{C}}
Data class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(0) \in H^s(\R^d)}
Basic characteristics
Structure non-Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot H^{d/2 - 2}(\R^d)}
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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \partial_t u + \Delta u = Q(u, \overline{u})}

which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q(u, \overline{u})} a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.


Quadratic NLS on R

  • Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c=-3/2\,.}
  • For any quadratic non-linearity one can obtain LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} CaWe1990, Ts1987.
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uu}\,} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle uu\,} type then one can push LWP to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -3/4.\,} KnPoVe1996b.
    • This can be improved to the Besov space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B^{-3/4}_{2,1}\,} MurTao2004. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X^{s,b}\,} bilinear estimates fail for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{-3/4}\,} NaTkTs2001.
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{u}u\,} type then one can push LWP to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -1/4.\,} KnPoVe1996b.
  • Since these equations do not have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • If the non-linearity is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u|u\,} then there is GWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} thanks to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} conservation, and ill-posedness below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T}

  • For any quadratic non-linearity one can obtain LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} Bo1993. In the Hamiltonian case (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u| u\,} ) this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uu}\,} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle uu\,} type then one can push LWP to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -1/2.\,} KnPoVe1996b.
  • In the Hamiltonian case (a non-linearity of type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u| u\,} ) we have GWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^2}

  • Scaling Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = -1.\,}
  • For any quadratic non-linearity one can obtain LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} CaWe1990, Ts1987.
    • In the Hamiltonian case (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u| u\,} ) this is sharp by Gallilean invariance considerations KnPoVe-p
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uu}\,} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u u\,} type then one can push LWP to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -3/4.\,} St1997, CoDeKnSt2001.
    • This can be improved to the Besov space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B^{-3/4}_{2,1}\,} MurTao2004.
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u \underline{u}\,} type then one can push LWP to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -1/4.\,} Ta2001.
  • In the Hamiltonian case (a non-linearity of type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u| u\,} ) we have GWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
    • Below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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

  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uu}\,} type then one can obtain LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -1/2\,} Gr-p2

Quadratic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3}

  • Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = -1/2.\,}
  • For any quadratic non-linearity one can obtain LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} CaWe1990, Ts1987.
  • If the quadratic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uu}\,} 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.
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