Quadratic NLS: Difference between revisions

**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	-

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Latest revision as of 01:30, 17 March 2007

Quadratic NLS

Equations of the form

$i\partial _{t}u+\Delta u=Q(u,{\overline {u}})$

which $Q(u,{\overline {u}})$ a quadratic function of its arguments are quadratic nonlinear Schrodinger equations.

Quadratic NLS on R

Scaling is $s_{c}=-3/2\,.$
For any quadratic non-linearity one can obtain LWP for $s\geq 0\,$ CaWe1990, Ts1987.
If the quadratic non-linearity is of ${\underline {uu}}\,$ ${\underline {uu}}\,$ or $uu\,$ $uu\,$ type then one can push LWP to $s>-3/4.\,$ $s>-3/4.\,$ KnPoVe1996b.
- This can be improved to the Besov space $B_{2,1}^{-3/4}\,$ MurTao2004. The $X^{s,b}\,$ bilinear estimates fail for $H^{-3/4}\,$ NaTkTs2001.
If the quadratic non-linearity is of ${\underline {u}}u\,$ type then one can push LWP to $s>-1/4.\,$ KnPoVe1996b.
Since these equations do not have $L^{2}\,$ conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
If the non-linearity is $|u|u\,$ then there is GWP in $L^{2}\,$ thanks to $L^{2}\,$ conservation, and ill-posedness below $L^{2}\,$ by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on $T$

For any quadratic non-linearity one can obtain LWP for $s\geq 0\,$ Bo1993. In the Hamiltonian case ( $|u|u\,$ ) this is sharp by Gallilean invariance considerations KnPoVe-p
If the quadratic non-linearity is of ${\underline {uu}}\,$ or $uu\,$ type then one can push LWP to $s>-1/2.\,$ 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 $L^{2}\,$ $L^{2}\,$ 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 $uu\,$ type then one can push LWP to $s>-1/2.\,$ St1997, CoDeKnSt2001.
If the quadratic non-linearity is of $u{\underline {u}}\,$ type then one can push LWP to $s>-1/4.\,$ Ta2001.
In the Hamiltonian case (a non-linearity of type $|u|u\,$ $|u|u\,$ ) we have GWP for $s\geq 0\,$ $s\geq 0\,$ by $L^{2}\,$ $L^{2}\,$ conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
- Below $L^{2}\,$ we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.

Quadratic NLS on $T^{3}$

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

@@ Line 1: / Line 1: @@
+{{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===
@@ Line 11: / Line 29: @@
 * Scaling is <math>s_c=-3/2\,.</math>
-* For any quadratic non-linearity one can obtain LWP for <math>s \ge 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.
-* 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]]
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
 ====Quadratic NLS on <math>T</math>====
-* 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]
+* 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 <math>\underline{uu}\,</math> or <math>uu\,</math> type then one can push LWP to <math>s > -1/2.\,</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 > -1/2.\,</math> [[KnPoVe1996b]].
 * 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.
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
 [[Category:Equations]]
@@ Line 39: / Line 47: @@
 * Scaling <math>s_c = -1.\,</math>
-* For any quadratic non-linearity one can obtain LWP for <math>s \ge 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]].
-** In the Hamiltonian case (<math>|u| u\,</math>) 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 <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]].
+* 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 <math>B^{-3/4}_{2,1}\,</math> [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 <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[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 <math>|u| u\,</math>) we have GWP for <math>s>= 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.
+* 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 <math>L^2\,</math> 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.
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
-[[Category:Equations]]
 ====Quadratic NLS on T^2====
-* 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]]
+* 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]]
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
-[[Category:Equations]]
 ====Quadratic NLS on <math>R^3</math>====
 * Scaling is <math>s_c = -1/2.\,</math>
-* For any quadratic non-linearity one can obtain LWP for <math>s \ge 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>u u\,</math> type then one can push LWP to <math>s > -1/2.\,</math> [[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 <math>u \underline{u}\,</math> type then one can push LWP to <math>s > -1/4.\,</math> [[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 <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 <math>L^2\,</math> 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.
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
-[[Category:Equations]]
 ====Quadratic NLS on <math>T^3</math>====
-* 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]]
+* 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]].
-<div class="MsoNormal" style="text-align: center"><center>
-----
-</center></div>
 [[Category:Equations]]

Description
Equation	$iu_{t}+\Delta u=Q(u,{\overline {u}})$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{d})$
Basic characteristics
Structure	non-Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{d/2-2}(\mathbb {R} ^{d})$
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: Difference between revisions

Latest revision as of 01:30, 17 March 2007

Contents

Quadratic NLS

Quadratic NLS on R

Quadratic NLS on $T$

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}

Quadratic NLS on T^2

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}

Quadratic NLS on $T^{3}$

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Quadratic NLS: Difference between revisions

Latest revision as of 01:30, 17 March 2007

Quadratic NLS

Quadratic NLS on R

Quadratic NLS on T {\displaystyle T}

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}

Quadratic NLS on T^2

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}

Quadratic NLS on T 3 {\displaystyle T^{3}}

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Quadratic NLS on $T$

Quadratic NLS on $T^{3}$