Difference between revisions of "Cubic DNLS on R"

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* Scaling is s<sub>c</sub> = 0.
 
* Scaling is s<sub>c</sub> = 0.
* 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 [[Bibliography#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 [[Bibliography#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 [[Bibliography#HaOz1994|HaOz1994]].
 
** 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 [[Bibliography#CoKeStTkTa2002b|CoKeStTkTa2002b]]. The s=1/2 case remains open.
 
* 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 [[Bibliography#CoKeStTkTa2001b|CoKeStTkTa2001b]].
 
** 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 [[Bibliography#Tk-p |Tk-p]].
 
** 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]].
 
** 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 [[Bibliography#HaOz1993|HaOz1993]], [[Bibliography#Oz1996|Oz1996]].
 
** The small L<sup>2</sup> norm condition is required in order to gauge transform the problem; see [[Bibliography#HaOz1993|HaOz1993]], [[Bibliography#Oz1996|Oz1996]].

Revision as of 15:58, 31 July 2006

Cubic DNLS on

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

  • Scaling is sc = 0.
  • LWP for s ³ 1/2 Tk-p.
    • This is sharp in the C uniform sense [BiLi-p] (see also 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 L2 norm CoKeStTkTa2002b. The s=1/2 case remains open.
    • for s>2/3 and small L2 norm this was proven in CoKeStTkTa2001b.
    • For s > 32/33 with small L2 norm this was proven in Tk-p.
    • For s ³ 1 and small L2 norm this was proven in HaOz1994. One can also handle certain pure power additional terms Oz1996.
    • The small L2 norm condition is required in order to gauge transform the problem; see HaOz1993, 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 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 ux u).

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