Difference between revisions of "Cubic DNLS on R"

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* Scaling is <math>s_c  = 0</math>.
 
* Scaling is <math>s_c  = 0</math>.
* LWP for <math>s = 1/2</math> [[Tk-p]].
+
* LWP for <math>s = 1/2</math> [[Tk2001]].
** This is sharp in the uniform sense [[BiLi-p]] (see also [[Tk-p]] for failure of analytic well-posedness below 1/2).
+
** This is sharp in the uniform sense [[BiLi2001]] (see also [[Tk2001]] for failure of analytic well-posedness below 1/2).
 
** For <math>s > 1</math> this was proven in [[HaOz1994]].
 
** For <math>s > 1</math> this was proven in [[HaOz1994]].
 
* GWP for <math>s>1/2</math> and small <math>L^2</math> norm [[CoKeStTkTa2002b]]. The <math>s=1/2</math> case remains open.
 
* GWP for <math>s>1/2</math> and small <math>L^2</math> norm [[CoKeStTkTa2002b]]. The <math>s=1/2</math> case remains open.

Revision as of 01:02, 17 March 2007

Cubic DNLS on

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

  • Scaling is .
  • LWP for Tk2001.
    • This is sharp in the uniform sense BiLi2001 (see also Tk2001 for failure of analytic well-posedness below 1/2).
    • For this was proven in HaOz1994.
  • GWP for and small norm CoKeStTkTa2002b. The case remains open.
    • for and small norm this was proven in CoKeStTkTa2001b.
    • For with small norm this was proven in Tk-p.
    • For and small norm this was proven in HaOz1994. One can also handle certain pure power additional terms Oz1996.
    • The small norm condition is required in order to gauge transform the problem; see HaOz1992, 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, 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 one can obtain GWP for small data KyTs1994 for arbitrary complex constants . See also Ts1994.