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 $s=1/2$ Tk2001.
  • This is sharp in the uniform sense BiLi2001 (see also Tk2001 for failure of analytic well-posedness below 1/2).
  • For $s>1$ this was proven in HaOz1994.
• GWP for $s>1/2$ and small $L^{2}$ norm CoKeStTkTa2002b. The $s=1/2$ case remains open. The existence of finite time blowup solutions emerging from smooth data with large $L^{2}$ size is also unknown.
  • 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 Tk-p.
  • For $s>1$ and small norm this was proven in HaOz1994. One can also handle certain pure power additional terms Oz1996.
  • The small $L^{2}$ norm condition is used to obtain a priori $H^{1}$ control on the gauge transformed solution; see HaOz1992, Oz1996.
• Solutions do not scatter to free Schrodinger solutions. In the focusing 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 u_x u).

For non-linearities of the form $f=a(u{\overline {u}})_{x}u+b(u{\overline {u}})_{x}u_{x}$ one can obtain GWP for small data KyTs1994 for arbitrary complex constants $a,b$. See also Ts1994.

Problem: Maximal-in-time behavior for large $L^{2}$ data?

For smooth (say $H^{1}$) initial data with large $L^{2}$, what happens? Do there exist finite time blowup solutions? This problem is gauge transforms into quintic focusing NLS on $\mathbb {R}$ which is known to have blowup solutions. However, the extra term in the equation $u{u}{\overline {u}}_{x}$ scales the same way so could conceivably counteract the focusing quintic nonlinearity. Also, the equation is completely integrable so blowup may possibly be ruled out by looking at other conservation properties.

Problem: Global well-posedness in $H^{1/2}$ under the mass constraint?

For $s=1/2$, we have LWP Tk2001. For $s>1/2$ and under a mass upper bound, we have GWP CoKeStTkTa2002b. Does GWP hold also for the endpoint $s=1/2$?

Problem: LWP for $s<1/2$?

The fixed-point approach to proving LWP Tk2001, BiLi2001 fails for $s<1/2$. Does LWP with merely continuous dependence upon the initial data hold for $0\leq s<1/2$?