Cubic NLS on R

Cubic NLS on $\mathbb {R}$

Description
Equation	$iu_{t}+u_{xx}=\pm |u|^{2}u$
Fields	$u:\mathbb {R} \times \mathbb {R} \to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} )$
Basic characteristics
Structure	completely integrable
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{-1/2}(\mathbb {R} )$
Criticality	mass-subcritical; energy-subcritical; scattering-critical
Covariance	Galilean
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq 0$
GWP	$H^{s}(\mathbb {R} )$ for $s\geq 0$
Related equations
Parent class	cubic NLS
Special cases	-
Other related	KdV, mKdV

The theory of the cubic NLS on one dimension is as follows.

• LWP for $s\geq 0\,$ Ts1987, CaWe1990 (see also GiVl1985).
  • This is sharp for reasons of Gallilean invariance and for soliton solutions in the focussing case KnPoVe-p
    • The result is also sharp in the defocussing case CtCoTa-p, due to Gallilean invariance and the asymptotic solutions in Oz1991.
    • Below $s\geq 0\,$ the solution map was known to be not $C^{2}\,$ in Bo1993
  • For initial data equal to a delta function there are serious problems with existence and uniqueness KnPoVe-p.
  • However, there exist Gallilean invariant spaces which scale below $L^{2}\,$ for which one has LWP. They are defined in terms of the Fourier transform VaVe2001. For instance one has LWP for data whose Fourier transform decays like $|x|^{-1/6-}\,$. Ideally one would like to replace this with $|x|^{0-}\,.$
• GWP for $s\geq 0\,$ thanks to $L^{2}\,$ conservation.
  • GWP can be pushed below to certain of the Gallilean spaces in VaVe2001. For instance one has GWP when the Fourier transform of the data decays like $|x|^{-5/12-}\,$. Ideally one would like to replace this with $0-$.
• If the cubic non-linearity is of ${\underline {uuu}}\,$ or $uuu\,$ type (as opposed to the usual $|u|^{2}u\,$ type) then one can obtain LWP for $s>-5/12\,$ Gr-p2. If the nonlinearity is of ${\underline {uu}}u\,$ type then one has LWP for $s>-2/5\,$ Gr-p2.
• Remark: This equation is sometimes known as the Zakharov-Shabat equation and is completely integrable (see e.g. AbKauNeSe1974; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.
• In the focusing case there are soliton and multisoliton solutions, however the defocusing case does not admit such solutions.
• In the focussing case there is a unique positive radial ground state for each energy $E\,$. By translation and phase shift one thus obtains a four-dimensional manifold of ground states (aka solitons) for each energy. This manifold is $H^{1}\,$-stable Ws1985, Ws1986. Below the energy norm orbital stability is not known, however there are polynomial bounds on the instability CoKeStTkTa2003b.
• This equation is related to the evolution of vortex filaments under the localized induction approximation, via the Hasimoto transformation, see e.g. Hm1972
• 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 Oz1991, see also CtCoTa-p. For small, decaying data one also has asymptotic completeness HaNm1998.
  • For large Schwartz data, these asymptotics can be obtained by inverse scattering methods ZkMan1976, SeAb1976, No1980, DfZx1994
  • For large real analytic data, these asymptotics were obtained in GiVl2001
  • Refinements to the convergence and regularity of the modified wave operators was obtained in Car2001.

Cubic NLS on the half-line and interval

• On the half line $R^{+}\,$, global well-posedness in $H^{2}\,$ was established in CrrBu1991, Bu.1992
• On the interval, the inverse scattering method was applied to generate solutions in GriSan-p.