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| ===Cubic DNLS on <math>R</math>===
| | #REDIRECT [[Cubic DNLS on R]] |
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| Suppose the non-linearity has the form f = i (u <u>u</u> u)<sub>x</sub>. Then:
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| * Scaling is s<sub>c</sub> = 0.
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| * LWP for s <font face="Symbol">³</font> 1/2 [[references:Tk-p Tk-p]].
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| ** 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).
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| ** For s <font face="Symbol">³</font> 1 this was proven in [[Bibliography#HaOz1994|HaOz1994]].
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| * GWP for s>1/2 and small L<sup>2</sup> norm [[Bibliography#CoKeStTkTa2002b|CoKeStTkTa2002b]]. The s=1/2 case remains open.
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| ** for s>2/3 and small L<sup>2</sup> norm this was proven in [[Bibliography#CoKeStTkTa2001b|CoKeStTkTa2001b]].
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| ** For s > 32/33 with small L<sup>2</sup> norm this was proven in [[references:Tk-p Tk-p]].
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| ** 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]].
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| ** The small L<sup>2</sup> norm condition is required in order to gauge transform the problem; see [[Bibliography#HaOz1993|HaOz1993]], [[Bibliography#Oz1996|Oz1996]].
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| * 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).
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| 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).
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| 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]].
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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