Cubic NLS

Cubic NLS on R

• Scaling is $s_{c}\,=-1/2$.
• 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 [VaVe-p]. 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. [[Bibliography#AbKauNeSe1974|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
• On the half line $R^{+}\,$, global well-posedness in $H^{2}\,$ was established in CrrBu1991, Bu1992
• On the interval, the inverse scattering method was applied to generate solutions in [GriSan-p].

Cubic NLS on $T^{1}$

• LWP for $s\geq 0\,$ Bo1993.
  • For $s<0\,$ one has failure of uniform local well-posedness [CtCoTa-p], [BuGdTz-p].In fact, the solution map is not even continuous from $H^{s}\,$ to $H^{\sigma }\,$ for any $\sigma$, even for small times and small data [CtCoTa-p3].
• GWP for $s\geq 0\,$ thanks to $L^{2}\,$ conservation Bo1993.
  • One also has GWP for random data whose Fourier coefficients decay like $1/|k|\,$ (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
• If the cubic non-linearity is of ${\underline {uuu}}\,$ type (instead of $|u|^{2}u\,$) then one can obtain LWP for $s>-1/3\,$ Gr-p2
• Remark: This equation is completely integrable AbMa1981; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.
• Methods of inverse scattering have also been successfully applied to cubic NLS on an interval [FsIt-p]

Cubic NLS on $R^{2}$

• Scaling is $s_{c}=0\,$, thus this is an [#L^2-critical_NLS $L^{2}\,$ critical NLS].
• LWP for $s\geq 0\,$ CaWe1990.
  • For $s=0\,$ the time of existence depends on the profile of the data as well as the norm.
  • LWP has also been obtained in Besov spaces Pl2000, [Pl-p] and Fourier-Lorentz spaces [CaVeVi-p] at the scaling of $L^{2}\,$. This is also connected with the construction of self-similar solutions to NLS (which are generally not in the usual Sobolev spaces globally in space).
  • Below $L^{2}\,$ we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
• GWP for $s>4/7\,$ in the defocussing case CoKeStTkTa2002
  • For $s>3/5\,$ this was shown in Bo1998.
  • For $s>2/3\,$ this was shown in Bo1998, Bo1999.
  • For $s\geq 1\,$ this follows from Hamiltonian conservation.
  • For small $L^{2}\,$ data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More precisely, one has global well-posedness whenever the data has an $L^{2}\,$ norm strictly smaller than the ground state Q Me1993. If the $L^{2}\,$ norm is exactly equal to that of Q then one has blow-up if and only if the data is a pseudo-conformal transformation of the ground state Me1993, Me1992. In particular, the ground state is unstable.
    • Scattering is known whenever the solution is sufficiently small in $L^{2}\,$ norm, or more generally whenever the solution is $L^{4}\,$ in spacetime.Presumably one in fact has scattering whenever the mass is strictly smaller than the ground state, though this has not yet been established.
  • The $s>4/7\,$ result is probably improvable by correction term methods.
  • Remark: $s=1/2\,$ is the least regularity for which the non-linear part of the solution has finite energy (Bourgain, private communication).
  • Question: What happens for large $L^{2}\,$ data? It is known that the only way GWP can fail at $L^{2}\,$ is if the $L^{2}\,$ norm concentrates Bo1998. Blowup examples with multiple blowup points are known, either simultaneously Me1992 or non-simultaneously BoWg1997. It is conjectured that the amount of energy which can go into blowup points is quantized. The $H^{1}\,$ norm in these examples blows up like $|t|^{-1}\,.$ It is conjectured that slower blow-up examples exist, in particular numerics suggest a blowup rate of $|t|^{-1/2}(loglog|t|)^{1/2}\,$ LanPapSucSup1988; interestingly, however, if we perturb NLS to the [misc:Zakharov-2 Zakharov system] then one can only have blowup rates of $|t|^{-1}\,.$
• Remark: This equation is pseudo-conformally invariant. Heuristically, GWP results in $H^{s}\,$ transfer to GWP and scattering results in $L^{2}(|x|^{2s})\,$ thanks to the pseudo-conformal transformation. Thus for instance GWP and scattering occurs this weighted space for $s>2/3\,$ (the corresponding statement for, say, $s>4/7\,$ has not yet been checked).
• In the periodic case the $H^{k}\,$ norm grows like $O(t^{2(k-1)+})\,$ as long as the $H^{1}\,$ norm stays bounded. In the non-periodic case it is $O(t^{(k-1)+})\,$ St1997, St1997b; this was improved to $t^{2/3(k-1)+}\,$ in CoDeKnSt-p, and also generalized to higher order multilinearity. A preliminary analysis suggests that the I-method can push the growth bounds down to $t^{(k-1)+/2}\,.$
• Question: Is there scattering in the cubic defocussing case, in $L^{2}\,$ or $H^{1}\,$? (certainly not in the focussing case thanks to solitons). This problem seems of comparable difficulty to the GWP problem for large $L^{2}\,$ data (indeed, the pseudo-conformal transformation morally links the two problems).
  • For powers slightly higher than cubic, the answer is yes Na1999c, and indeed we have bounded $H^{k}\,$ norms in this case [Bourgain?].
  • If the data has sufficient decay then one has scattering. For instance if $xu(0)\,$ is in $L^{2}\,$ Ts1985. This was improved to $x^{2/3+}u(0)\in L^{2}\,$ in Bo1998, Bo1999; the above results on GWP will probably also extend to scattering.
• This equation has also been studied on bounded domains, see [BuGdTz-p]. Sample results: blowup solutions exist close to the ground state, with a blowup rate of $(T-t)^{-1}\,$. If the domain is a disk then uniform LWP fails for $1/5<s<1/3\,$, while for a square one has LWP for all $s>0\,.$ In general domains one has LWP for $s>2\,.$

Cubic NLS on RxT and T2

• Scaling is $s_{c}\,$= 0.
• For $R\times T$ one has LWP for $s>0\,$ [TkTz-p2].
• For $T\times T$ one has LWP for $s>0\,$ Bo1993.
• In the defocussing case one has GWP for $s>1\,$ in both cases by Hamiltonian conservation.
  • On $T\times T$ one can improve this to $s>2/3\,$ by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).
• In the focusing case on $T\times T$ one has blowup for data close to the ground state, with a blowup rate of $(T^{*}-t)^{-1}\,$ [BuGdTz-p]
• If instead one considers the sphere $S^{2}\,$ then uniform local well-posedness fails for $3/20<s<1/4\,$ BuGdTz2002, [Ban-p], but holds for $s>1/4\,$ [BuGdTz-p7].
  • For $s>1/2\,$ this is in [BuGdTz-p3].
  • These results for the sphere can mostly be generalized to other Zoll manifolds.

Cubic NLS on $R^{3}$

• Scaling is $s_{c}=1/2\,$.
• LWP for $s\geq 1/2\,$ CaWe1990.
  • For $s=1/2\,$ the time of existence depends on the profile of the data as well as the norm.
  • For $s<1/2\,$ we have ill-posedness, indeed the $H^{s}\,$ norm can get arbitrarily large arbitrarily quickly [CtCoTa-p2]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
  • For $s>1/2\,$ there is unconditional well-posedness FurPlTer2001
    • For $s>=2/3\,$ this is in Ka1995.
• GWP and scattering for $s>4/5\,$ in the defocussing case CoKeStTkTa-p8
  • For $s>5/6\,$ GWP is in CoKeStTkTa2002
  • For $s>11/13\,$ GWP is in Bo1999
  • For radial data and $s>5/7\,$ GWP and scattering is in $s>5/7\,$ Bo1998b, Bo1999.
  • For $s\geq 1\,$ this follows from Hamiltonian conservation. One also has scattering in this case GiVl1985.
  • For small $H^{1/2}\,$ data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is $L^{5}\,$ in spacetime.
  • In the focusing case one has blowup whenever the energy is negative Gs1977, OgTs1991, and in particular one has blowup arbitrarily close to the ground state BerCa1981, SaSr1985.If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms $H_{loc}^{s}\,$ remaining bounded for all time Bo1996c, Bo1998b.Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component [Ta-p7]

Cubic NLS on $T^{3}$

• Scaling is $s_{c}=1/2\,$.
• LWP is known for $s>1/2\,$ Bo1993.

Cubic NLS on $R^{4}$

• Scaling is $s_{c}=1\,$.
• LWP is known for $s\geq 1\,$ CaWe1990.
  • For $s=1\,$ the time of existence depends on the profile of the data as well as the norm.
  • For $s<1\,$ we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly [CtCoTa-p2]. In the focusing case we have instantaneous blowup from the virial identity and scaling.
• GWP and scattering for $s\geq 1\,$ in the radial case Bo1999. A major obstacle is that the Morawetz estimate only gives $L^{4}\,$-type spacetime control rather than $L^{6}\,.$
  • For small non-radial $H^{1}\,$ data one has GWP and scattering. In fact one has scattering whenever the solution has a bounded $L^{6}\,$ norm in spacetime.

The large data non-radial case is still open, and very interesting. The main difficulty is infinite speed of propagation and the possibility that the $H^{1}\,$ norm could concentrate at several different places simultaneously.

Cubic NLS on $T^{4}$

• Scaling is $s_{c}=1\,$.
• LWP is known for $s\geq 2\,$ Bo1993d.

Cubic NLS on $S^{6}$

• Scaling is $s_{c}=2\,$.
• Uniform LWP holds in $H^{s}\,$ for $s>5/2\,$ [BuGdTz-p3].
• Uniform LWP fails in the energy class $H^{1}\,$ [BuGdTz-p2]; indeed we have this failure for any NLS on $S^{6}$, even ones for which the energy is subcritical. This is in contrast to the Euclidean case, where one has LWP for powers $p<2\,$.