# Difference between revisions of "Schrodinger:quadratic NLS"

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* Scaling is s<sub>c</sub> = -3/2. | * Scaling is s<sub>c</sub> = -3/2. | ||

− | * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[ | + | * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[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. [[ | + | * 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]]. |

** 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 B^{-3/4}_{2,1} [MurTao-p]. The X^{s,b} bilinear estimates fail for H^{-3/4} [[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. [[ | + | * If the quadratic non-linearity is of u <u>u</u> type then one can push LWP to s > -1/4. [[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 L<sup>2</sup> 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 |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. | ||

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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 [[ | + | * 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] |

− | * 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. [[ | + | * 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]]. |

* 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 |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. | ||

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* Scaling is s<sub>c</sub> = -1. | * Scaling is s<sub>c</sub> = -1. | ||

− | * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[ | + | * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[Bibliography#CaWe1990|CaWe1990]], [[Bibliography#Ts1987|Ts1987]]. |

** In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p] | ** In the Hamiltonian case (|u| u) 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. [[ | + | * 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]]. |

** This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-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>u</u> type then one can push LWP to s > -1/4. [[references:Ta-p2 Ta-p2]]. | * 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]]. | ||

Line 66: | Line 66: | ||

* Scaling is s<sub>c</sub> = -1/2. | * Scaling is s<sub>c</sub> = -1/2. | ||

− | * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[ | + | * For any quadratic non-linearity one can obtain LWP for s <font face="Symbol">³</font> 0 [[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 > -1/2. [[ | + | * 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 u <u>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 <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. |

## Revision as of 20:03, 28 July 2006

## Contents

### Quadratic NLS

Equations of the form

which 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 ³ 0 CaWe1990, Ts1987.
- If the quadratic non-linearity is of
__u____u__or u u type then one can push LWP to s > -3/4. 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.

- If the quadratic non-linearity is of 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

- 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 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

- Scaling is s
_{c}= -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 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^2

- If the quadratic non-linearity is of
__u____u__type then one can obtain LWP for s > -1/2 references#Gr-p2 Gr-p2

#### Quadratic NLS on

- Scaling is s
_{c}= -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 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

- If the quadratic non-linearity is of
__u____u__type then one can obtain LWP for s > -3/10 references#Gr-p2 Gr-p2