Schrodinger:cubic DNLS on R

Cubic DNLS on $R$

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 references:HaOz1994 HaOz1994.
• GWP for s>1/2 and small L² norm references:CoKeStTkTa2002b CoKeStTkTa2002b. The s=1/2 case remains open.
  • for s>2/3 and small L² norm this was proven in references:CoKeStTkTa2001b CoKeStTkTa2001b.
  • For s > 32/33 with small L² norm this was proven in references:Tk-p Tk-p.
  • For s ³ 1 and small L² norm this was proven in references:HaOz1994 HaOz1994. One can also handle certain pure power additional terms references:Oz1996 Oz1996.
  • The small L² norm condition is required in order to gauge transform the problem; see references:HaOz1993 HaOz1993, references: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 references: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_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 references:KyTs1995 KyTs1995 for arbitrary complex constants a, b. See also references:Ts1994 Ts1994.

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