Schrodinger:specific equations

Quadratic NLS

Quadratic NLS on R

• Scaling is s_c = -3/2.
• For any quadratic non-linearity one can obtain LWP for s ³ 0 references:CaWe1990 CaWe1990, references:Ts1987 Ts1987.
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -3/4. references:KnPoVe1996b KnPoVe1996b.
  • This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p]. The X^{s,b} bilinear estimates fail for H^{-3/4} references:NaTkTs-p NaTkTs2001.
• If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. references:KnPoVe1996b KnPoVe1996b.
• Since these equations do not have L² conservation it is not clear whether there is any reasonable GWP result, except possibly for very small data.
• If the non-linearity is |u|u then there is GWP in L² thanks to L² conservation, and ill-posedness below L² by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

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Quadratic NLS on $T$

• For any quadratic non-linearity one can obtain LWP for s ³ 0 references:Bo1993 Bo1993. In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. references:KnPoVe1996b KnPoVe1996b.
• In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L² conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.

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Quadratic NLS on $R^{2}$

• Scaling is s_c = -1.
• For any quadratic non-linearity one can obtain LWP for s ³ 0 references:CaWe1990 CaWe1990, references:Ts1987 Ts1987.
  • In the Hamiltonian case (|u| u) this is sharp by Gallilean invariance considerations [KnPoVe-p]
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -3/4. references:St1997 St1997, references:CoDeKnSt-p CoDeKnSt-p.
  • This can be improved to the Besov space B^{-3/4}_{2,1} [MurTao-p].
• If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. references:Ta-p2 Ta-p2.
• In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L² conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

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Quadratic NLS on T^2

• If the quadratic non-linearity is of u u type then one can obtain LWP for s > -1/2 references#Gr-p2 Gr-p2

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Quadratic NLS on $R^{3}$

• Scaling is s_c = -1/2.
• For any quadratic non-linearity one can obtain LWP for s ³ 0 references:CaWe1990 CaWe1990, references:Ts1987 Ts1987.
• If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. references:St1997 St1997, references:CoDeKnSt-p CoDeKnSt-p.
• If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. references:Ta-p2 Ta-p2.
• In the Hamiltonian case (a non-linearity of type |u| u) we have GWP for s ³ 0 by L² conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.
  • Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.

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Quadratic NLS on $T^{3}$

• If the quadratic non-linearity is of u u type then one can obtain LWP for s > -3/10 references#Gr-p2 Gr-p2

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Cubic NLS

Cubic NLS on R

• Scaling is s_c = -1/2.
• LWP for s ³ 0 references:Ts1987 Ts1987, references:CaWe1990 CaWe1990 (see also references:GiVl1985 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 references:Oz1991 Oz1991.
    • Below s ³0 the solution map was known to be not C² in references:Bo1993 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 L² for which one has LWP. They are defined in terms of the Fourier transform references:VaVe2001 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 ³ 0 thanks to L² 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 u u u or u u u type (as opposed to the usual |u|² u type) then one can obtain LWP for s > -5/12 references#Gr-p2 Gr-p2. If the nonlinearity is of u u u type then one has LWP for s > -2/5 references#Gr-p2 Gr-p2.
• Remark: This equation is sometimes known as the Zakharov-Shabat equation and is completely integrable (see e.g. [[references: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 H¹-stable references:Ws1985 Ws1985, references:Ws1986 Ws1986. Below the energy norm orbital stability is not known, however there are polynomial bounds on the instability references:CoKeStTkTa2003b CoKeStTkTa2003b.
• This equation is related to the evolution of vortex filaments under the localized induction approximation, via the Hasimoto transformation, see e.g. references:Hm1972 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 references:Oz1991 Oz1991, see also [CtCoTa-p]. For small, decaying data one also has asymptotic completeness references:HaNm1998 HaNm1998.
  • For large Schwartz data, these asymptotics can be obtained by inverse scattering methods references:ZkMan1976 ZkMan1976, references:SeAb1976 SeAb1976, references:No1980 No1980, references:DfZx1994 DfZx1994
  • For large real analytic data, these asymptotics were obtained in references:GiVl2001 GiVl2001
  • Refinements to the convergence and regularity of the modified wave operators was obtained in references:Car2001 Car2001
• On the half line R^+, global well-posedness in H^2 was established in references:CrrBu.1991 CrrBu.1991, references:Bu.1992 Bu.1992
• On the interval, the inverse scattering method was applied to generate solutions in [GriSan-p].

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Cubic NLS on $T^{1}$

• LWP for s³0 references:Bo1993 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 ³ 0 thanks to L² conservation references:Bo1993 Bo1993.
  • One also has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) references:Bo1995c Bo1995c. Indeed one has an invariant measure.
• If the cubic non-linearity is of u u u type (instead of |u|² u) then one can obtain LWP for s > -1/3 references#Gr-p2 Gr-p2
• Remark: This equation is completely integrable [[references:AbMa1981 AbMa.1981]]; 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]

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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 ³ 0 references:CaWe1990 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 references:Pl2000 Pl2000, [Pl-p] and Fourier-Lorentz spaces [CaVeVi-p] at the scaling of L². 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 references:CoKeStTkTa2002 CoKeStTkTa2002
  • For s>3/5 this was shown in references:Bo1998 Bo1998.
  • For s>2/3 this was shown in references:Bo1998 Bo1998, references:Bo1999 Bo1999.
  • For s³ 1 this follows from Hamiltonian conservation.
  • For small L² 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 L² norm strictly smaller than the ground state Q references:Me1993 Me1993. If the L² 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 references:Me1993 Me1993, references:Me1992 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 L⁴ 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 L² data? It is known that the only way GWP can fail at L² is if the L² norm concentrates references:Bo1998 Bo1998. Blowup examples with multiple blowup points are known, either simultaneously references:Me1992 Me1992 or non-simultaneously references:BoWg1997 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} (log log|t|)^{1/2} references:LanPapSucSup1988 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 L²(|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 H¹ norm stays bounded. In the non-periodic case it is O(t^{(k-1)+}) references:St1997 St1997, references:St1997b St1997b; this was improved to t^{2/3 (k-1)+} in references:CoDeKnSt-p 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 L² or H¹? (certainly not in the focussing case thanks to solitons). This problem seems of comparable difficulty to the GWP problem for large L² data (indeed, the pseudo-conformal transformation morally links the two problems).
  • For powers slightly higher than cubic, the answer is yes references:Na1999c 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 L² references:Ts1985 Ts1985. This was improved to x^{2/3+} u(0) in L² in references:Bo1998 Bo1998, references:Bo1999 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.

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Cubic NLS on $R\times T$ and $T^{2}$

• Scaling is s_c = 0.
• For RxT one has LWP for s³0 [TkTz-p2].
• For TxT one has LWP for s>0 references:Bo1993 Bo1993.
• In the defocussing case one has GWP for s³1 in both cases by Hamiltonian conservation.
  • On T x 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 TxT 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 S² then uniform local well-posedness fails for 3/20 < s < 1/4 references:BuGdTz2002 BuGdTz2002, [Ban-p], but holds for s>1/4 [BuGdTz-p7].
  • For s > ½ this is in [BuGdTz-p3].
  • These results for the sphere can mostly be generalized to other Zoll manifolds.

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Cubic NLS on $R^{3}$

• Scaling is s_c = 1/2.
• LWP for s ³ 1/2 references:CaWe1990 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 references:FurPlTer2001 FurPlTer2001
    • For s >= 2/3 this is in references:Ka1995 Ka1995.
• GWP and scattering for s > 4/5 in the defocussing case references:CoKeStTkTa-p8 CoKeStTkTa-p8
  • For s > 5/6 GWP is in references:CoKeStTkTa2002 CoKeStTkTa2002
  • For s>11/13 GWP is in references:Bo1999 Bo1999
  • For radial data and s > 5/7 GWP and scattering is in s>5/7 references:Bo1998b Bo1998b, references:Bo1999 Bo1999.
  • For s³ 1 this follows from Hamiltonian conservation. One also has scattering in this case references:GiVl1985 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 L⁵ in spacetime.
  • In the focusing case one has blowup whenever the energy is negative references:Gs1977 Gs1977, references:OgTs1991 OgTs1991, and in particular one has blowup arbitrarily close to the ground state references#BerCa1981 BerCa1981, references:SaSr1985 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^s_loc remaining bounded for all time references:Bo1996c Bo1996c, references:Bo1998b 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]

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Cubic NLS on $T^{3}$

• Scaling is s_c = 1/2.
• LWP is known for s >1/2 references:Bo1993 Bo1993.

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Cubic NLS on $R^{4}$

• Scaling is s_c = 1.
• LWP is known for s ³ 1 references:CaWe1990 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 s³1 in the radial case references:Bo1999 Bo1999. A major obstacle is that the Morawetz estimate only gives L⁴-type spacetime control rather than L⁶.
  • For small non-radial H¹ data one has GWP and scattering. In fact one has scattering whenever the solution has a bounded L⁶ 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 H¹ norm could concentrate at several different places simultaneously.

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Cubic NLS on $T^{4}$

• Scaling is s_c = 1.
• LWP is known for s ³ 2 references:Bo1993d Bo1993d.

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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 H¹ [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.

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Quartic NLS

Quartic NLS on $R$

• Scaling is s_c = -1/6.
• For any quartic non-linearity one can obtain LWP for s ³ 0 references:CaWe1990 CaWe1990
  • Below L^2 we have ill-posedness by Gallilean invariance considerations in both the focusing [KnPoVe-p] and defocusing [CtCoTa-p2] cases.
• If the quartic non-linearity is of u u u u type then one can obtain LWP for s > -1/6. For |u|⁴ one has LWP for s > -1/8, while for the other three types u⁴, u u u u, or u uuu one has LWP for s > -1/6 references#Gr-p2 Gr-p2.
• In the Hamiltonian case (a non-linearity of type |u|^3 u) we have GWP for s ³ 0 by L² conservation. In the other cases it is not clear whether there is any reasonable GWP result, except possibly for very small data.

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Quartic NLS on $T$

• For any quartic non-linearity one has LWP for s>0 references:Bo1993 Bo1993.
• If the quartic non-linearity is of u u u u type then one can obtain LWP for s > -1/6 references#Gr-p2 Gr-p2.
• If the nonlinearity is of |u|³ u type one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) references:Bo1995c Bo1995c. Indeed one has an invariant measure.

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Quartic NLS on $R^{2}$

• Scaling is s_c = 1/3.
• For any quartic non-linearity one can obtain LWP for s ³ s_c references:CaWe1990 CaWe1990.
  • For s<s_c 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.
• In the Hamiltonian case (a non-linearity of type |u|^3 u) we have GWP for s ³ 1 references:Ka1986 Ka1986.
  • This has been improved to s > 1-e in references:CoKeStTkiTa2003c CoKeStTkTa2003c in the defocusing Hamiltonian case. This result can of course be improved further.
  • Scattering in the energy space references:Na1999c Na1999c in the defocusing Hamiltonian case.
  • One also has GWP and scattering for small H^{1/3} data for any quintic non-linearity.

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Quintic NLS

Quintic NLS on $R$

• This equation may be viewed as a simpler version of [#dnls-3_on_R cubic DNLS], and is always at least as well-behaved. It has been proposed as a modifiation of the Gross-Pitaevski approximation for low-dimesional Bose liquids references:KolNewStrQi2000 KolNewStrQi2000
• Scaling is s_c = 0, thus this is an [#L^2-critical_NLS L^2 critical NLS].
• LWP is known for s ³ 0 references:CaWe1990 CaWe1990, references:Ts1987 Ts1987.
  • For s=0 the time of existence depends on the profile of the data as well as the norm.
  • 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>4/9 in the defocussing case [Tzi-p]
  • For s>1/2 this is in references:CoKeStTkTa-p6 CoKeStTkTa-p6
  • For s>2/3 this is in references:CoKeStTkTa-p4 CoKeStTkTa-p4.
  • For s > 32/33 this is implicit in references:Tk-p Tk-p.
  • For s³ 1 this follows from LWP and Hamiltonian conservation.
  • One has GWP and scattering for small L² data for any quintic non-linearity. The corresponding problem for large L² data and defocussing nonlinearity is very interesting, but probably very difficult, perhaps only marginally easier than the corresponding problem for the [#Cubic_NLS_on_R^2 2D cubic NLS]. It would suffice to show that the solution has a bounded L⁶ norm in spacetime.
  • Explicit blowup solutions (with large L² norm) are known in the focussing case references:BirKnPoSvVe1996 BirKnPoSvVe1996. The blowup rate in H¹ is t^-1 in these solutions. This is not the optimal blowup rate; in fact an example has been constructed where the blowup rate is |t|^{-1/2} (log log|t|)^{1/2}[Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least |t|^{-1/2}) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that xu is in L^2.
    • One can modify the explicit solutions from references:BirKnPoSvVe1996 BirKnPoSvVe1996 and in fact create solutions which blow up at any collection of specified points in spacetime references:BoWg1997 BoWg1997, references:Nw1998 Nw1998.
• Remark: This equation is pseudo-conformally invariant. GWP results in H^s automatically transfer to GWP and scattering results in L²(|x|^s) thanks to the pseudo-conformal transformation.
• Solitons are H¹-unstable.

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Quintic NLS on $T$

• This equation may be viewed as a simpler version of cubic DNLS, and is always at least as well-behaved.
• Scaling is s_c = 0.
• LWP is known for s > 0 references:Bo1993 Bo1993.
  • For s < 0 the solution map is not uniformly continuous from C^k to C^{-k} for any k [CtCoTa-p3].
• GWP is known in the defocusing case for s > 4/9 (De Silva, Pavlovic, Staffilani, Tzirakis)
  • For s > 2/3 this is commented upon in [Bo-p2] and is a minor modification of [CoKeStTkTa-p].
  • For s >= 1 one has GWP in the defocusing case, or in the focusing case with small L^2 norm, by Hamiltonian conservation.
    • In the defocusing case one has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) references:Bo1995c Bo1995c; this is roughly of the regularity of H^{1/2}. Indeed one has an invariant measure. In the focusing case the same result holds assuming the L² norm is sufficiently small.

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Quintic NLS on $R^{2}$

• Scaling is s_c = 1/2.
• LWP is known for s ³ 1/2 references:CaWe1990 CaWe1990.
  • For s=1/2 the time of existence depends on the profile of the data as well as the norm.
  • For s<s_c 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 for s ³ 1 by Hamiltonian conservation.
  • This has been improved to s > 1-e in references:CoKeStTkTa2003b CoKeStTkTa2003b. This result can of course be improved further.
  • Scattering in the energy space references:Na1999c Na1999c
  • One also has GWP and scattering for small H^{1/2} data for any quintic non-linearity.

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Quintic NLS on $R^{3}$

• Scaling is s_c = 1.
• LWP is known for s ³ 1 references:CaWe1990 CaWe1990.
  • For s=1 the time of existence depends on the profile of the data as well as the norm.
  • For s<s_c 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 s³1 in the defocusing case [CoKeStTkTa-p]
  • For radial data this is in [Bo-p], references:Bo1999 Bo1999.
  • Blowup can occur in the focussing case from Glassey's virial identity.

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Septic NLS

Septic NLS on $R$

• Scaling is s_c = 1/6.
• LWP is known for s ³ s_c references:CaWe1990 CaWe1990.
  • For s=s_c the time of existence depends on the profile of the data as well as the norm.
  • For s<s_c 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 for s ³ 1 by Hamiltonian conservation.
  • This has been improved to s > 1-e in references:CoKeStTkTa2003b CoKeStTkTa2003b in the defocusing case. This result can of course be improved further.
  • Scattering in the energy space references:Na1999c Na1999c
  • One also has GWP and scattering for small H^{s_c} data for any septic non-linearity.

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Septic NLS on $R^{2}$

• Scaling is s_c = 2/3.
• LWP is known for s ³ s_c references:CaWe1990 CaWe1990.
  • For s=s_c the time of existence depends on the profile of the data as well as the norm.
  • For s<s_c 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 for s ³ 1 by Hamiltonian conservation.
  • This has been improved to s > 1-e in references:CoKeStTkTa2003b CoKeStTkTa2003b in the defocusing case. This result can of course be improved further.
  • Scattering in the energy space references:Na1999c Na1999c
  • One also has GWP and scattering for small H^{s_c} data for any septic non-linearity.

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Septic NLS on $R^{3}$

• Scaling is s_c = 7/6.
• LWP is known for s ³ s_c references:CaWe1990 CaWe1990.
  • For s=s_c the time of existence depends on the profile of the data as well as the norm.
  • For s<s_c 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 small data by Strichartz estimates references:CaWe1990 CaWe1990.
  • For large data one has blowup in the focusing case by the virial identity; in particular one has ill-posedness in the energy space.
  • It is not known (and would be extremely interesting to find out!) what is going on in the defocusing case; for instance, is there blowup from smooth data? Even for radial data nothing seems to be known. This may be viewed as an extremely simplified model problem for the global regularity issue for Navier-Stokes.

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$L^{2}$ critical NLS on $R^{d}$

The L^2 critical situation s_c = 0 occurs when p = 1 + 4/d. Note that the power non-linearity is smooth in dimensions d=1 ([#Quintic_NLS_on_R quintic NLS]) and d=2 ([#Cubic_NLS_on_R^2 cubic NLS]). One always has GWP and scattering in L^2 for small data (see references:Givl1978 GiVl1978, references:GiVl1979 GiVl1979, references:CaWe1990 CaWe1990; the more precise statement in the focusing case that GWP holds when the mass is strictly less than the ground state mass is in references:Ws1983 Ws1983); in the large data defocusing case, GWP is known in H^1 (and slightly below) but is only conjectured in L^2. No scattering result is known for large data, even in the radial smooth case.

In the focusing case, there is blowup for large L^2 data, as can be seen by applying the pseudoconformal transformation to the ground state solution. Up to the usual symmetries of the equation, this is the unique minimal mass blowup solution references:Me1993 Me1993. This solution blows up in H^1 like |t|^{-1} as t -> 0-. However, numerics suggest that there should be solutions that exhibit the much slower blowup |t|^{-1/2} (log log|t|)^{1/2} references:LanPapSucSup1988 LanPapSucSup1988; furthermore, this blowup is stable under perturbations in the energy space [MeRap-p], at least when the mass is close to the critical mass. Note that scaling shows that blowup cannot be any slower than |t|^{-1/2}.

The virial identity shows that blowup must occur when the energy is negative (which can only occur when the mass exceeds the ground state mass).Strictly speaking, the virial identity requires some decay on u – namely that x u lies in L^2, however this restriction can be relaxed (references:OgTs1991 OgTs1991, references:Nw1999 Nw1999,
references:GgMe1995 GgMe1995.

In [#Quintic_NLS_on_R one dimension d=1], the above blowup rate of |t|^{-1/2} (log log|t|)^{1/2} has in fact been achieved [Per-p]. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least |t|^{-1/2}) whenever the energy is negative [MeRap-p], [MeRap-p2], and one either assumes that the mass is close to the critical mass or that xu is in L^2. When the energy is zero, and one is not a ground state, then one has blowup like |t|^{-1/2} (log log |t|)^{1/2} in at least one direction of time (t -> +infinity or t -> -infinity) [MeRap-p], [MeRap-p2].These results extend to higher dimensions as soon as a certain (plausible) spectral condition on the ground state is verified.

The exact nature of the blowup set is not yet fully understood, but there are some partial results.It appears that the generic rate of blowup is |t|^{-1/2} (log log|t|)^{1/2}; the exceptional rate of |t|^{-1} can occur for the self-similar solutions and also for larger solutions references:BoWg1997 BoWg1997, but this seems to be very rare compared to the |t|^{-1/2} (log log|t|)^{1/2} blowup solutions (which are open in H^1 close to the critical mass [MeRap-p]).In fact close to the critical mass, there is a dichotomy, in that the blowup (if it occurs) is either |t|^{-1} or faster, or |t|^{-1/2} (log log |t|)^{1/2} or slower [MeRap-p], [MeRap-p2].Also, near the blowup points the solution should have asymptotically zero energy references:Nw1999 Nw1999 and exhibit mass concentration references:Nw1992 Nw1992.

Conditions on the linearizability of this equation when the dispersion and nonlinearity are both sent to zero at controlled rates has been established in d=1,2 in [CarKer-p] (and in the L^2-supercritical case in [CarFerGal-p].A key role is played by the size of the linear solution in the relevant Strichartz space.

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Higher order NLS

(More discussion later... Ed.)

One can study higher-order NLS equations in which the Laplacian is replaced by a higher power.One class of such examples comes from the

infinite hierarchy of commuting flows arising from the [#Cubic_NLS_on_R 1D cubic NLS].Another is the [kdv:Schrodinger_Airy nonlinear Schrodinger-Airy equation].

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Schrodinger maps

[Many thanks to Andrea Nahmod for help with this section - Ed.]

Schrodinger maps are to the Schrodinger equation as [wave:wm wave maps] are to the wave equation; they are the natural Schrodinger equation when the target space is a complex manifold (such as the sphere S² or hyperbolic space H²). They have the form

iu_t + D u = Gamma(u)( Du, Du )

where Gamma(u) is the second fundamental form. This is the same as the harmonic map heat flow but with an additional "i" in front of the u_t. When the target is S², this equation arises naturally from the Landau-Lifschitz equation for a macroscopic ferromagnetic continuum, see e.g. references:SucSupBds1986 SucSupBds1986; in this case the equation has the alternate form u_t = u x D u, where x is the cross product, and is sometimes known as the Heisenberg model; similar models exist when the target is generalized from a sphere S² to a Hermitian symmetric space (see e.g. [TeUh-p]). The Schrodinger map equation is also related to the Ishimori equation references:Im1984 Im1984 (see references:KnPoVe2000 KnPoVe2000 for some recent results on this equation)

In one dimension local well posedness is known for smooth data by the [#d-nls general theory of derivative nonlinear Schrodinger equations], however this is not yet established in higher dimensions. Assuming this regularity result, there is a gauge transformation (obtained by differentiating the equation, and placing the resulting connection structure in the Coulomb gauge) which creates a null structure in the non-linearity. Roughly speaking, the equation now looks like

iv_t + D v = Dv D^-1(v v) + D^-1(v v) D^-1(v v) v + v³

where v := Du. The cubic term Dv D^-1(v v) has a null structure so that orthogonal interactions (which normally cause the most trouble with derivative
Schrodinger problems) are suppressed.

For certain special targets (e.g. complex Grassmannians) and with n=1, the Schrodinger flow is a completely integrable bi-Hamiltonian system [TeUh-p].In the case of n=1 when the target is the sphere S², the equation is equivalent to the [#Cubic_NLS_on_R cubic NLS] references:ZkTkh1979 ZkTkh1979, references:Di1999 Di1999.

As with [wave:wm wave maps], the scaling regularity is H^{n/2}.

• In one dimension one has global existence in the energy norm references:CgSaUh2000 CgSaUh2000 when the target is a compact Riemann surface; it is conjectured that this is also true for general compact Kahler manifolds.
  • When the target is a complex compact Grassmannian, this is in [TeUh-p].
  • In the periodic case one has local existence and uniqueness of smooth solutions, with global existence if the target is compact with constant sectional curvature references:DiWgy1998 DiWgy1998. The constant curvature assumption was relaxed to non-positive curvature (or Hermitian locally symmetric) in references:PaWghWgy2000 PaWghWgy2000. It is conjectured that one should have a global flow whenever the target is compact Kahler references:Di2002 Di2002.
    • When the target is S² this is in references:ZhGouTan1991 ZhGouTan1991
• In two dimensions there are results in both the radial/equivariant and general cases.
  • With radial or equivariant data one has global existence in the energy norm for small energy references:CgSaUh2000 CgSaUh2000, assuming high regularity LWP as mentioned above.
    • The large energy case may be settled in [CkGr-p], although the status of this paper is currently unclear (as of Feb 2003).
  • In the general case one has LWP in H^s for s > 2 references:NdStvUh2003 NdStvUh2003 (plus later errata), at least when the target manifold is the sphere S². It would be interesting to extend this to lower regularities, and eventually to the critical H¹ case. (Here regularity is stated in terms of u rather than the derivatives v).
  • When the target is S² there are global weak solutions references:KnPoVe1993c KnPoVe1993c, [HaHr-p], and local existence for smooth solutions references:SucSupBds1986 SucSupBds1986.
  • When the target is H^2 one can have blowup in finite time [Di-p].Similarly for higher dimensions.
• In general dimensions one has LWP in H^s for s > n/2+1 references:DiWgy2001 DiWgy2001
  • When the target is is S² this is in references:SucSupBds1986 SucSupBds1986.

Some further discussion on this equation can be found in the survey references:Di2002 Di2002.

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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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Hartree equation

[Sketchy! More to come later. Contributions are of course very welcome and will be acknowledged. - Ed.]

The Hartree equation is of the form

i u_t + D u = V(u) u

where

V(u) = + |x|^{-n} * |u|²

and 0 < n < d. It can thus be thought of as a non-local cubic Schrodinger equation; the cubic NLS is in some sense a limit of this equation as n -> n (perhaps after suitable normalization of the kernel |x|^{-n}, which would otherwise blow up). The analysis divides into the short-range case n > 1, the long-range case 0 < n < 1, and the borderline (or critical) case n=1. Generally speaking, the smaller values of n are the hardest to analyze. The + sign corresponds to defocusing nonlinearity, the - sign corresopnds to focusing.

The H¹ critical value of n is 4, in particular the equation is always subcritical in four or fewer dimensions. For n<4 one has global existence of energy solutions. For n=4 this is only known for small energy.

In the short-range case one has scattering to solutions of the free Schrodinger equations under suitable assumptions on the data. However this is not true in the other two cases references:HaTs1987 HaTs1987. For instance, in the borderline case, at large times t the solution usually resembles a free solution with initial data y, twisted by a Fourier multiplier with symbol exp(i V(hat{y}) ln t). (This can be seen formally by applying the pseudo-conformal transformation, discarding the Laplacian term, and solving the resulting ODE references#GiOz1993 GiOz1993). This creates modified wave operators instead of ordinary wave operators. A similar thing happens when 1/2 < n < 1 but ln t must be replaced by t^{n-1}/(n-1).

The existence and mapping properties of these operators is only partly known:

• When n > 2 and n=1, the wave operators map \hat{H^s} to \hat{H^s} for s > 1/2 and are continuous and open [Nak-p3] (see also references:GiOz1993 GiOz1993)
  • For n>1 and n > 1 this is in references:NwOz1992 NwOz1992
    • In the defocusing case, all solutions in suitable spaces have asymptotic states in L², and one has asymptotic completeness when n > 4/3 references:HaTs1987 HaTs1987.
  • For n < 1, n ³3, and 1 - n/2 < s < 1 this is in [Nak-p4]
    • Many earlier results in references:HaKakNm1998 HaKakNm1998, references:HaKaiNm1998 HaKaiNm1998, references:HaNm2001 HaNm2001, references:HaNm1998b HaNm1998b
  • In the Gevrey and real analytic categories there are some large data results in references:GiVl2000 GiVl2000, references:GiVl2000b GiVl2000b, references:GiVl2001 GiVl2001, covering the cases n< 1 and n > 1.
  • For small decaying data one has some invertibility of the wave operators references:HaNm1998 HaNm1998

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Maxwell-Schrodinger system in $R^{3}$

This system is a partially non-relativistic analogue of the [wave:mkg Maxwell-Klein-Gordon system]., coupling a U(1) connection A_a with a complex scalar field u. The Lagrangian density is

\int F^ab F_ab + 2 Im u D u - D_j u D^j u

giving rise to the system of PDE

i u_t = D_j u D^j u/2 + A u
d^a F_ab = J_b

where the current density J_b is given by

J = |u|^2; J_j = - Im u D_j u

As with the MKG system, there is a gauge invariance for the connection; one can place A in the Lorentz, Coulomb, or Temporal gauges (other choices are of course possible).

Let us place u in H^s, and A in H^sigma x H^{sigma-1}. The lack of scale invariance makes it difficult to exactly state what the critical regularity would be, but it seems to be s = sigma = 1/2.

• In the Lorentz and Temporal gauges, one has LWP for s >= 5/3 and s-1 <= sigma <= s+1, (5s-2)/3 [NkrWad-p]
  • For smooth data (s=sigma > 5/2) in the Lorentz gauge this is in references:NkTs1986 NkTs1986 (this result works in all dimensions)
• Global weak solutions are known in the energy class (s=sigma=1) in the Lorentz and Coulomb gauges references:GuoNkSr1996 GuoNkSr1996. GWP is still open however.
• Modified wave operators have been constructed in the Coulomb gauge in the case of vanishing magnetic field in [GiVl-p3], [GiVl-p5]. No smallness condition is needed on the data at infinity.
  • A similar result for small data is in references:Ts1993 Ts1993
• In one dimension, GWP in the energy class is known references:Ts1995 Ts1995
• In two dimensions, GWP for smooth solutions is known references:TsNk1985 TsNk1985

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