# Difference between revisions of "Schrodinger:cubic DNLS on R"

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* LWP for s <font face="Symbol">³</font> 1/2 [[references:Tk-p Tk-p]]. | * LWP for s <font face="Symbol">³</font> 1/2 [[references:Tk-p Tk-p]]. | ||

** This is sharp in the C uniform sense [BiLi-p] (see also [[references:Tk-p Tk-p]] for failure of analytic well-posedness below 1/2). | ** This is sharp in the C uniform sense [BiLi-p] (see also [[references:Tk-p Tk-p]] for failure of analytic well-posedness below 1/2). | ||

− | ** For s <font face="Symbol">³</font> 1 this was proven in [[ | + | ** For s <font face="Symbol">³</font> 1 this was proven in [[Bibliography#HaOz1994|HaOz1994]]. |

− | * GWP for s>1/2 and small L<sup>2</sup> norm [[ | + | * GWP for s>1/2 and small L<sup>2</sup> norm [[Bibliography#CoKeStTkTa2002b|CoKeStTkTa2002b]]. The s=1/2 case remains open. |

− | ** for s>2/3 and small L<sup>2</sup> norm this was proven in [[ | + | ** for s>2/3 and small L<sup>2</sup> norm this was proven in [[Bibliography#CoKeStTkTa2001b|CoKeStTkTa2001b]]. |

** For s > 32/33 with small L<sup>2</sup> norm this was proven in [[references:Tk-p Tk-p]]. | ** For s > 32/33 with small L<sup>2</sup> norm this was proven in [[references:Tk-p Tk-p]]. | ||

− | ** For s <font face="Symbol">³</font> 1 and small L<sup>2</sup> norm this was proven in [[ | + | ** For s <font face="Symbol">³</font> 1 and small L<sup>2</sup> norm this was proven in [[Bibliography#HaOz1994|HaOz1994]]. One can also handle certain pure power additional terms [[Bibliography#Oz1996|Oz1996]]. |

− | ** The small L<sup>2</sup> norm condition is required in order to gauge transform the problem; see [[ | + | ** The small L<sup>2</sup> norm condition is required in order to gauge transform the problem; see [[Bibliography#HaOz1993|HaOz1993]], [[Bibliography#Oz1996|Oz1996]]. |

− | * Solutions do not scatter to free Schrodinger solutions. In the focussing case this can be easily seen from the existence of solitons. But even in the defocussing case wave operators do not exist, and must be replaced by modified wave operators (constructed in [[ | + | * Solutions do not scatter to free Schrodinger solutions. In the focussing case this can be easily seen from the existence of solitons. But even in the defocussing case wave operators do not exist, and must be replaced by modified wave operators (constructed in [[Bibliography#HaOz1994|HaOz1994]] for small data). |

This equation has the same scaling as the [#Quintic_NLS_on_R quintic NLS], and there is a certain gauge invariance which unifies the two (together with an additional nonlinear term u <u>u</u><sub>x</sub> u). | This equation has the same scaling as the [#Quintic_NLS_on_R quintic NLS], and there is a certain gauge invariance which unifies the two (together with an additional nonlinear term u <u>u</u><sub>x</sub> u). | ||

− | For non-linearities of the form f = a (u <u>u</u>)<sub>x</sub> u + b (u <u>u</u>)<sub>x</sub> u<sub>x</sub> one can obtain GWP for small data [[ | + | For non-linearities of the form f = a (u <u>u</u>)<sub>x</sub> u + b (u <u>u</u>)<sub>x</sub> u<sub>x</sub> one can obtain GWP for small data [[Bibliography#KyTs1995|KyTs1995]] for arbitrary complex constants a, b. See also [[Bibliography#Ts1994|Ts1994]]. |

<div class="MsoNormal" style="text-align: center"><center> | <div class="MsoNormal" style="text-align: center"><center> |

## Revision as of 20:24, 28 July 2006

### Cubic DNLS on

Suppose the non-linearity has the form f = i (u __u__ u)_{x}. Then:

- Scaling is s
_{c}= 0. - LWP for s ³ 1/2 references:Tk-p Tk-p.
- This is sharp in the C uniform sense [BiLi-p] (see also references:Tk-p Tk-p for failure of analytic well-posedness below 1/2).
- For s ³ 1 this was proven in HaOz1994.

- GWP for s>1/2 and small L
^{2}norm CoKeStTkTa2002b. The s=1/2 case remains open.- for s>2/3 and small L
^{2}norm this was proven in CoKeStTkTa2001b. - For s > 32/33 with small L
^{2}norm this was proven in references:Tk-p Tk-p. - For s ³ 1 and small L
^{2}norm this was proven in HaOz1994. One can also handle certain pure power additional terms Oz1996. - The small L
^{2}norm condition is required in order to gauge transform the problem; see HaOz1993, Oz1996.

- for s>2/3 and small L
- Solutions do not scatter to free Schrodinger solutions. In the focussing case this can be easily seen from the existence of solitons. But even in the defocussing case wave operators do not exist, and must be replaced by modified wave operators (constructed in HaOz1994 for small data).

This equation has the same scaling as the [#Quintic_NLS_on_R quintic NLS], and there is a certain gauge invariance which unifies the two (together with an additional nonlinear term u __u___{x} u).

For non-linearities of the form f = a (u __u__)_{x} u + b (u __u__)_{x} u_{x} one can obtain GWP for small data KyTs1995 for arbitrary complex constants a, b. See also Ts1994.