Non-linear Schrodinger equations

Overview

The free Schrodinger equation

i u_t + D u = 0

where u is a complex-valued function in R^{d+1}, describes the evolution of a free non-relativistic quantum particle in d spatial dimensions. This equation can be modified in many ways, notably by adding a potential or an obstacle, but we shall be interested in non-linear perturbations such as

u_t - i D u = f(u, u, Du, Du)

where D denotes spatial differentiation. In such full generality, we refer to this equation as a derivative non-linear Schrodinger equation (D-NLS). If the non-linearity does not contain derivatives then we refer to this equation as a semilinear Schrodinger equation (NLS). These equations (particularly the cubic NLS) arise as model equations from several areas of physics.

Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms L^q_t L^r_x or L^r_x L^q_t, or in X^{s,b} spaces, defined by

|| u ||_s,b = || <x>^s <t-|x|²>^b \hat{u} ||₂.

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates. They are useful for NLS without derivatives, but are much less useful for derivative non-linearities. Other linear estimates include smoothing estimates and maximal function estimates. The X^{s,b} spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear. These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities. See also [Bo1993b], [KlMa1993].

Linear estimates

[More references needed here!]

On R^d:

If f is in X^{0,1/2+}, then

(Energy estimate) f is in L^¥_t L²_x
(Strichartz estimates) f is in L^{2(d+2)/d}_x,t [Sz1977].

More generally, f is in L^q_t L^r_x whenever 1/q+n/2r = n/4, r < ¥, and q > 2.

The endpoint q=2, r = 2d/(d-2) is true for d³ 3 [KeTa1998]. When d=2 it fails even in the BMO case [Mo1998], although it still is true for radial functions [Ta2000b], [Stv-p]. In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable [Ta2000b], although there is a limit as to low little regularity one can impose [MacNkrNaOz-p].
In the radial case there are additional weighted smoothing estimates available [Vi2001]
When d=1 one also has f in L⁴_t L^¥_x.
When d=1 one can refine the L² assumption on the data in rather technical ways on the Fourier side, see e.g. [VaVe2001].
When d=1, the L^6_{t,x} estimate has a maximizer [Kz-p2]. This maximizer is in fact given by Gaussian beams, with a constant of 12^{-1/12} [Fc-p4]. Similarly when d=2 with the L^4 estimate, which is also given by Gaussian beams with a constant of 2^{-1/2}

(Kato estimates) D^{1/2} f is in L²_{x,loc} L²_t[Sl1987], [Ve1988]

When d=1 one can improve this to D^{1/2} f in L^¥_x L²_t

(Maximal function estimates) In all dimensions one has D^{-s} f is in L²_x,loc L^¥_t for all s > 1/2.

When d=1 one also has D^{-1/4} f in L⁴_x L^¥_t.
When d=2 one also has D^{-1/2} f in L⁴_x L^¥_t. The -1/2 can be raised to -1/2+1/32+e [TaVa2000b], with the corresponding loss in the L⁴ exponent dictated by scaling. Improvements are certainly possible.

Variants of some of these estimates exist for manifolds, see [BuGdTz-p]

Fixed time estimates for free solutions:

(Energy estimate) If f(0) is in L², then f(t) is also in L².
(Decay estimate) If f(0) is in L¹, then f(t) has an L^¥ norm of O(t^{-d/2}).
Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

X^{0,3/8} embds into L⁴_x,t [Bo1993] (see also [HimMis2001]).
X^{0+,1/2+} embeds into L⁶_x,t [Bo1993]. One cannot remove the + from the 0+ exponent, however it is conjectured in [Bo1993] that one might be able to embed X^{0,1/2+} into L^{6-}_x,t.

On T^d:

When d >= 1, X^{d/4 - 1/2+,1/2+} embeds into L⁴_x,t (this is essentially in [Bo1993])

The endpoint d/4 - 1/2 is probably false in every dimension.

Strichartz estimates are also available on more general manifolds, and in the presence of a potential. Inhomogeneous estimates are also available off

the line of duality; see [Fc-p2] for a discussion.

Bilinear estimates

On R² we have the bilinear Strichartz estimate [Bo1999]:

|| uv ||_{1/2+, 0} <~ || u ||_1/2+, 1/2+ || v ||_{0+, 1/2+}

On R² [St1997], [CoDeKnSt-p], [Ta-p2] we have the sharp estimates

|| u v ||_{0, -1/2+} <~ || u ||_{-1/2+, 1/2+} || v ||_{-1/2+, 1/2+}

|| u v ||_{-1/2-, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}

|| u v ||_{-1/2-, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_-3/4+, 1/2+

|| u v ||_{-1/4+, -1/2+} <~ || u ||_{-1/4+, 1/2+} || v ||_{-1/4+, 1/2+}

On R [KnPoVe1996b] we have

|| u v ||_{-3/4-, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}

|| u v ||_{-3/4+, -1/2+} <~ || u ||_-3/4+, 1/2+ || v ||_{-3/4+, 1/2+}

|| u v ||_{-1/4+, -1/2+} <~ || u ||_-1/4+, 1/2+ || v ||_{-1/4+, 1/2+}

and [BkOgPo1998]

|| u v ||_{L^¥_tH^1/3_x} <~ || u ||_0, 1/2+ || v ||_{0, 1/2+}

Also, if u has frequency |x| ~ R and v has frequency |x| << R then we have (see e.g. [CoKeStTkTa-p4])

|| u v ||_1/2,0 <~ || u ||_{0, 1/2+} || v ||_{0, 1/2+}

and similarly for uv, uv, uv.

The s indices on the right cannot be lowered, but perhaps the s indices on the left can be raised in analogy with the R² estimates. The analogues on T are also known [KnPoVe1996b]:

|| u v ||_{-1/2-, -1/2+} <~ || u ||_{-1/2+, 1/2+} || v ||_{-1/2+, 1/2+}

|| u v ||_{-3/4+, -1/2+} <~ || u ||_-1/2+, 1/2+ || v ||_{-1/2+, 1/2+}

|| u v ||_{0, -1/2+} <~ || u ||_0, 1/2+ || v ||_{0, 1/2+}

Trilinear estimates

On R we have the following refinement to the L^6 Strichartz inequality [Gr-p2]:

|| u v w ||_{0, 0} <~ || u ||_{0, 1/2+} || v ||_{-1/4, 1/2+} || w ||_{1/4, 1/2+}

Multilinear estimates

In R² we have the variant

|| u_1 ... u_n ||_{1/2+, 1/2+} <~ || u_1 ||_1+,1/2+ ... || u_n ||_1+,1/2+

where each factor u_i is allowed to be conjugated if desired. See [St1997b], [CoDeKnSt-p].

Semilinear Schrodinger (NLS)

[Many thanks to Kenji Nakanishi with valuable help with the scattering theory portion of this section. However, we are still missing many references and results, e.g. on NLS blowup. - Ed.]

The semilinear Schrodinger equation is

i u_t + D u + l |u|^{p-1} u = 0

for p>1. (One can also add a potential term, which leads to many physically interesting problems, however the field of Schrodinger operators with potential is far too vast to even attempt to summarize here). In order to consider this problem in H^s one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume

p is an odd integer, or p > [s]+1.

This is a Hamiltonian flow with the Hamiltonian

H(u) = ò |Ñ u|²/2 - l |u|^{p+1}/(p+1) dx

and symplectic form

{u, v} = Im ò u v dx.

From the phase invariance u -> exp(i q) u one also has conservation of the L² norm. The case l > 0 is focussing; l < 0 is defocussing.

The scaling regularity is s_c = d/2 - 2/(p-1). The most interesting values of p are the L²-critical or pseudoconformal power p=1+4/d and the H¹-critical power p=1+4/(d-2) for d>2 (for d=1,2 there is no H¹ conformal power). The power p = 1 + 2/d is also a key exponent for the scattering theory (as this is when the non-linearity |u|^{p-1} u has about equal strength with the decay t^{-d/2}). The cases p=3,5 are the most natural for physical applications since the non-linearity is then a polynomial. The cubic NLS in particular seems to appear naturally as a model equation for many different physical contexts, especially in dispersive, weakly non-linear perturbations of a plane wave. For instance, it arises as a model for dilute Bose-Einstein condensates.

Dimension d	Scattering power 1+2/d	L²-critical power 1+4/d	H¹-critical power 1+4/(d-2)
1	3	5	N/A
2	2	3	infinity
3	5/3	7/3	5
4	3/2	2	3
5	7/5	9/5	7/3
6	4/3	5/3	2

The pseudoconformal transformation of the Hamiltonian gives that the time derivative of

|| (x + 2it Ñ) u ||²_2 - 8 l t²/(p+1) || u ||_{p+1}^{p+1}

is equal to

4dtl(p-(1+4/d))/(p+1) ||u||_{p+1}^{p+1}.

This law is useful for obtaining a priori spacetime estimates on the solution given sufficient decay in space (e.g. xu(0) in L²), especially in the L²-critical case p=1+4/d (when the above derivative is zero). The L² norm of xu(0) is sometimes known as the pseudoconformal charge.

The equation is invariant under Gallilean transformations

u(x,t) -> exp(i (v.x/2 - |v|² t)) u(x-vt, t).

This can be used to show ill-posedness below L² in the focusing case [KnPoVe-p], and also in the defocusing case [CtCoTa-p2]. (However if the non-linearity is replaced by a non-invariant expression such as u², then one can go below L²).

From scaling invariance one can obtain Morawetz inequalities, which usually estimate quantities of the form

ò ò |u|^{p+1}/|x| dx dt

in the defocussing case in terms of the H^{1/2} norm. These are useful for limiting the number of times the solution can concentrate at the origin; this is especially handy in the radially symmetric case.

In the other direction, one has LWP for s ³ 0, s_c [CaWe1990]; see also [Ts1987]; for the case s=1, see [GiVl1979]. In the L²-subcritical cases one has GWP for all s³0 by L² conservation; in all other cases one has GWP and scattering for small data in H^s, s ³ s_c. These results apply in both the focussing and defocussing cases. At the critical exponent one can prove Besov space refinements [Pl2000], [Pl-p4]. This can then be used to obtain self-similar solutions, see [CaWe1998], [CaWe1998b], [RiYou1998], [MiaZg-p1], [MiaZgZgx-p], [MiaZgZgx-p2], [Fur2001].

Now suppose we remove the regularity assumption that p is either an odd integer or larger than [s]+1. Then some of the above results are still known to hold:

? In the H^1 subcritical case one has GWP in H^1, assuming the nonlinearity is smooth near the origin [Ka1986]

In R^6 one also has Lipschitz well-posedness [BuGdTz-p5]

In the periodic setting these results are much more difficult to obtain. On the one-dimensional torus T one has LWP for s > 0, s_c if p > 1, with the endpoint s=0 being attained when 1 <= p <= 4 [Bo1993]. In particular one has GWP in L^2 when p < 4, or when p=4 and the data is small norm. For 6 > p ³ 4 one also has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [Bo1995c]. (For p=6 one needs to impose a smallness condition on the L² norm or assume defocusing; for p>6 one needs to assume defocusing).

For the defocussing case, one has GWP in the H¹-subcritical case if the data is in H¹. To improve GWP to scattering, it seems that needs p to be L² super-critical (i.e. p > 1 + 4/d). In this case one can obtain scattering if the data is in L²(|x|² dx) (since one can then use the pseudo-conformal conservation law).

In the d ³ 3 cases one can remove the L²(|x|² dx) assumption [GiVl1985] (see also [Bo1998b]) by exploiting Morawetz identities, approximate finite speed of propagation, and strong decay estimates (the decay of t^{-d/2} is integrable). In this case one can even relax the H¹ norm to H^s for some s<1 [CoKeStTkTa-p7].
For d=1,2 one can also remove the L²(|x|² dx) assumption [Na1999c] by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.

In the L^2-supercritical focussing case one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality

d²_t ò x² |u|² dx ~ H(u);

see e.g. [OgTs1991]. By scaling this implies that we have instantaneous blowup in H^s for s < s_c in the focusing case. In the defocusing case blowup
is not known, but the H^s norm can still get arbitrarily large arbitrarily quickly for s < s_c [CtCoTa-p2]

Suppose we are in the L^2 subcritical case p < 1 + 2/d, with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy E. By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is H¹-stable [Ws1985], [Ws1986]. Below the H^1 norm, this is not known, but polynomial upper bounds on the instability are in [CoKeStTkTa2003b]. Multisolitons are also asymptotically stable under smooth decaying perturbations [Ya1980], [Grf1990], [Zi1997], [RoScgSf-p], [RoScgSf-p2], provided that p is betweeen 1+2/d and 1+4/d.

One can go beyond scattering and ask for asymptotic completeness and existence of the wave operators. When p £ 1 + 2/d this is not possible due to the poor decay in time in the non-linear term [Bb1984], [Gs1977b], [Sr1989], however at p = 1+2/d one can obtain modified wave operators for data with suitable regularity, decay, and moment conditions [Oz1991], [GiOz1993], [HaNm1998], [ShiTon2004], [HaNmShiTon2004]. In the regime between the L² and H¹ critical powers the wave operators are well-defined in the energy space [LnSr1978], [GiVl1985], [Na1999c]. At the L² critical exponent 1 + 4/d one can define wave operators assuming that we impose an L^p_x,t integrability condition on the solution or some smallness or localization condition on the data [GiVl1979], [GiVl1985], [Bo1998] (see also [Ts1985] for the case of finite pseudoconformal charge). Below the L² critical power one can construct wave operators on certain spaces related to the pseudo-conformal charge [CaWe1992], [GiOz1993], [GiOzVl1994], [Oz1991]; see also [GiVl1979], [Ts1985]. For H^s wave operators were also constructed in [Na2001]. However in order to construct wave operators in spaces such as L²(|x|² dx) (the space of functions with finite pseudoconformal charge) it is necessary that p is larger than or equal to the rather unusual power

1 + 8 / (sqrt(d² + 12d + 4) + d - 2);

see [NaOz2002] for further discussion.

Many of the global results for H^s also hold true for L²(|x|^{2s} dx). Heuristically this follows from the pseudo-conformal transformation, although making this rigorous is sometimes difficult. Sample results are in [CaWe1992], [GiOzVl1994], [Ka1995], [NkrOz1997], [NkrOz-p]. See [NaOz2002] for further discussion.

NLS on manifolds and obstacles

The NLS has also been studied on non-flat manifolds. For instance, for smooth two-dimensional compact surfaces one has LWP in H¹ [BuGdTz-p3], while for smooth three-dimensional compact surfaces and p=3 one has LWP in H^s for s>1, together with weak solutions in H¹ [BuGdTz-p3]. In the special case of a sphere one has LWP in H^{d/2 + 1/2} for d³3 and p < 5 [BuGdTz-p3].

· For the cubic equation on two-dimensional surfaces one has LWP in H^s for s > ½ [BuGdTz-p3]

o For s >= 1 one has GWP [Vd1984], [OgOz1991] and regularity [BrzGa1980]

o For s < 0 uniform ill-posedness can be obtained by adapting the argument in [BuGdTz2002] or [CtCoTa-p]

o For the sphere, cylinder, or torus more precise results are known

A key tool here is the development of Strichartz estimates on curved space. For general manifolds one has all the L^q_t L^r_x Strichartz estimates (locally in time), but with a loss of 1/q derivatives, see [BuGdTz-p3]. (This though compares favorably to Sobolev embedding, which would require a loss of 2/q derivatives). When the manifold is flat outside of a compact set and obeys a non-trapping condition, the optimal Strichartz estimates (locally in time) were obtained in [StTt-p].
When instead the manifold is decaying outside of a compact set and obeys a non-trapping condition, the Strichartz estimates (locally in time) with an epsilon loss were obtained by Burq [Bu-p3]; in the special case of L^4 estimates on R^3, and for non-trapping asymptotically conic manifolds, the epsilon was removed in [HslTaWun-p]

Outside of a non-trapping obstacle (with Dirichlet boundary conditions), the known results are as follows.

If (p-1)(d-2) < 2 then one has GWP in H^1 assuming a coercivity condition (e.g. if the nonlinearity is defocusing) [BuGdTz-p4].

Note there is a loss compared with the non-obstacle theory, where one expects the condition to be (p-1)(d-2) < 4.
The same is true for the endpoint d=3, p=3 if the energy is sufficiently small [BuGdTz-p4].
If d <= 4 then the flow map is Lipschitz [BuGdTz-p4]
For d=2, p <= 3 this is in [BrzGa1980], [Vd1984], [OgOz1991]

If p < 1 + 2/d then one has GWP in L^2 [BuGdTz-p4]

For d=3 GWP for smooth data is in [Jor1961]
Again, in the non-obstacle theory one would expect p < 1 + 4/d
if p < 1 + 1/d then one also has strong uniqueness in the class L^2 [BuGdTz-p4]

On a domain in R^d, with Dirichlet boundary conditions, the results are as follows.

Local well-posedness in H^s for s > d/2 can be obtained by energy methods.
In two dimensions when p <=3, global well-posedness in the energy class (assuming energy less than the ground state, in the p=3 focusing case) is in [BrzGa1980], [Vd1984], [OgOz1991], [Ca1989]. More precise asymptotics of a minimal energy blowup solution in the focusing p=3 case are in [BuGdTz-p], [Ban-p3]
When p > 1 + 4/d blowup can occur in the focusing case [Kav1987]

GWP and scattering for defocusing NLS on Schwarzchild manifolds for radial data is in [LabSf1999]

NLS with potential

(Thanks to Remi Carles for much help with this section. - Ed.)

One can ask what happens to the NLS when a potential is added, thus

i u_t + D u + l|u|^{p-1} u = V u

where V is real and time-independent. The behavior depends on whether V is positive or negative, and how V grows as |x| -> infinity. In the following results we suppose that V grows like some sort of power of x (this can be made precise with estimates on the derivatives of V, etc.) A particularly important case is that of the quadratic potential V = +- |x|^2; this can be used to model a confining magnetic trap for Bose-Einstein condensation. Most of the mathematical research has gone into the isotropic quadatic potentials, but anisotropic ones (given by quadratic forms other than |x|^2) are also of physical interest.

If V is linear, i.e. V(x) = E.x, then the potential can in fact be eliminated by a change of variables [CarNky-p]
If V is smooth, positive, and has bounded derivatives up to order 2 (i.e. is quadratic or subquadratic), then the theory is much the same as when there is no potential - one has decay estimates, Strichartz estimates, and the usual local and global well posedness theory (see [Fuj1979], [Fuj1980], [Oh1989])
When V is exactly a positive quadratic potential V = w^2 |x|^2, then one has blowup for the focusing nonlinearity for negative energy in the L^2 supercritical or critical, H^1 subcritical case [Car2002b].

In the L^2 critical case one can in fact eliminate this potential by a change of variables [Car2002c]. One consequence of this change of variables is that one can convert the usual solitary wave solution for NLS into a solution that blows up in finite time (cf. how the pseudoconformal transform is used to achieve a similar effect without the potential).

When V is exactly a negative quadratic potential, one can prevent blowup even in the focusing case if the potential is sufficiently strong [Car-p]. Indeed, one has a scattering theory in this case [Car-p]
If V grows faster than quadratic, then there are significant problems due to the failure of smoothness of the fundamental solution; if V is also negative, then even the linear theory fails (for instance, the Hamiltonian need not be essentially self-adjoint on test functions). However for positive superquadratic potentials partial results are still possible [YaZgg2001].

Much work has also been done on the semiclassical limit of these equations; see for instance [BroJer2000], [Ker2002], [CarMil-p], [Car2003]. For work on standing waves for NLS with quadratic potential, see [Fuk2001], [Fuk2003], [FukOt2003], [FukOt2003b].

One component of the theory of NLS with potential is that of Strichartz estimates with potential, which in turn may be derived from dispersive estimates with potential, although it is possible to obtain Strichartz estimates without a dispersive inequality. Both types of estimates can only be expected to hold after first projecting to the absolutely continuous part of the spectrum (although this is not necessary if the potential is positive).

The situation for dispersive estimates (which imply Strichartz), and related estimates such as local L^2 decay, and of L^p boundedness of wave operators (which implies both the dispersive inequality and Strichartz) is as follows. Here we consider potentials which could oscillate; relying mostly on magnitude bounds on V rather than on symbol-type bounds.

When d=1 one has dispersive estimates whenever <x> V is L^1 [GbScg-p].

For potentials such that <x>^{3/2+} V is in L^1, this is essentially in [Wed2000]; the stronger L^p boundedness of wave operators in this case was established in [Wed1999], [ArYa2000].

When d=2, relatively little is known.

L^p boundedness of wave operators for potentials decaying like <x>^{-6-}, assuming 0 is not a resonance nor eigenvalue, is in [Ya1999], [JeYa2002]. The method does not quite extend to p=1,infinity and thus does not directly imply the dispersive estimate although it does give Strichartz estimates for 1 < p < infinity.
Local L^2 decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in [JeNc2001]

When d=3 one has dispersive estimates whenever V decays like <x>^{-3-} and 0 is neither a eigenvalue nor resonance [GbScg-p]

For potentials which decay like <x>^{-7-} and whose Fourier transform is in L^1, a version of this estimate is in [JouSfSo1991]
A related local L^2 decay estimate was obtained for exponentially decaying potentials in [Ra1978]; this was refined to polynomially decaying potentials (e.g. <x>^{-3-}) (with additional resolvent estimates at low frequencies) in [JeKa1980].
L^p boundedness of wave operators was established in [Ya1995] for potentials decaying like <x>^{-5-} and for which 0 is neither an eigenvalue nor a resonance.
If 0 is a resonance one cannot expect to obtain the optimal decay estimate; at best one can hope for t^{-1/2} (see [JeKa1980]).
Dispersive estimates have also been proven for potentials whose Rollnik and global Kato norms are both smaller than the critical value of 4pi [RoScg-p]. Indeed their arguments partly extend to certain time-dependent potentials (e.g. quasiperiodic potentials), once one also imposes a smallness condition on the L^{3/2} norm of V
If the potential is in L^2 and has finite global Kato norm, then one has dispersive estimates for high frequencies at least [RoScg-p].
Strichartz estimates have been obtained for potentials decaying like <x>^{-2-} if 0 is neither a zero nor a resonance [RoScg-p]

This has been extended to potentials decaying exactly like |x|^2 and d >= 3 assuming some radial regularity and if the potential is not too negative [BuPlStaTv-p], [BuPlStaTv-p2]; in particular one has Strichartz estimates for potentials V = a/|x|^2, d >= 3, and a > -(n-2)^2/4 (this latter condition is necessary to avoid bound states).

For d > 3, most of the d=3 results should extend. For instance, the following is known.

For potentials which decay like <x>^{-d-4-} and whose Fourier transform is in L^1, dispersive estimates are in [JouSfSo1991]
Local L^2 decay and resolvent estimates for low frequencies for polynomially decaying potentials are obtained in [Je1980], [Je1984].

For finite rank perturbations of the Laplacian, where each rank one perturbation is generated by a bump function and the bump functions are sufficiently far apart in physical space, decay and Schrodinger estimates were obtained in [NieSf2003]. The bounds obtained grow polynomially in the number of rank one perturbations.

Local smoothing estimates seem to be more robust than dispersive estimates, holding in a wider range of situations. For instance, in R^d, any potential in L^p for p >= d/2, as well as inverse square potentials 1/|x|^2, and linear combinations of these, have local smoothing [RuVe1994]. Note one does not need to project away the bound states for such estimates (as the bound states tend to already be rather smooth). However, for p < d/2, one can have breakdown of local smoothing (and also dispersive and Strichartz estimates) [Duy-p]

For time-dependent potentials, very little is known. If the potential is small and quasiperiodic in time (or more generally, has a highly concentrated Fourier transform in time) then dispersive and Strichartz estimates were obtained in [RoScg-p]; the smallness is used to rule out bound states, among other things. In the important case of the charge transfer model (the time-dependent potential that arises in the stability analysis of multisolitons) see [Ya1980], [Grf1990], [Zi1997], [RoScgSf-p], [RoScgSf-p2], where energy, dispersive, and Strichartz estimates are obtained, with application to the asymptotic stability of multisolitons.

The nonlinear interactions between the bound states of a Schrodinger operator with potential (which have no dispersion or decay properties in time) and the absolutely continuous portion of the spectrum (where one expects dispersion and Strichartz estimates) is not well understood. A preliminary result in this direction is in [GusNaTsa-p], which shows in R^3 that if there is only one bound state, and Strichartz estimates hold in the remaining portion of the spectrum, and the non-linearity does not have too high or too low a power (say 4/3 <= p <= 4, or a Hartree-type nonlinearity) then every small H^1 solution will asymptotically decouple into a dispersive part evolving like the linear flow (with potential), plus a nonlinear bound state, with the energy and phase of this bound state eventually stabilizing at a constant. In [SfWs-p] the interaction of a ground state and an excited state is studied, with the generic behavior consisting of collapse to the ground state plus radiation.

Unique continuation

A question arising by analogy from the theory of unique continuation in elliptic equations, and also in control theory, is the following: if u is a solution to a nonlinear Schrodinger equation, and u(t_0) and u(t_1) is specified on a domain D at two different times t_0, t_1, does this uniquely specify the solution everywhere at all other intermediate times?

For the 1D cubic NLS, with D equal to a half-line, and u assumed to vanish on D, this is in [Zg1997].
For general NLS with analytic non-linearity, and with u assumed compactly supported, this is in [Bo1997b].
For D the complement of a convex cone, and arbitrary NLS of polynomial growth with a bounded potential term, this is in [KnPoVe2003]
For D in a half-plane, and allowing potentials in various Lebesgue spaces, this is in [IonKn-p]
A local unique continuation theorem (asserting that a non-zero solution cannot vanish on an open set) is in [Isk1993]

Quadratic NLS on R

Scaling is s_c = -3/2.
For any quadratic non-linearity one can obtain LWP for s ³ 0 [CaWe1990], [Ts1987].
If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -3/4. [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} [NaTkTs2001].

If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. [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.

Quadratic NLS on T

For any quadratic non-linearity one can obtain LWP for s ³ 0 [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. [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.

Quadratic NLS on R²

Scaling is s_c = -1.
For any quadratic non-linearity one can obtain LWP for s ³ 0 [CaWe1990], [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. [St1997], [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. [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.

Quadratic NLS on T²

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

Quadratic NLS on R³

Scaling is s_c = -1/2.
For any quadratic non-linearity one can obtain LWP for s ³ 0 [CaWe1990], [Ts1987].
If the quadratic non-linearity is of u u or u u type then one can push LWP to s > -1/2. [St1997], [CoDeKnSt-p].
If the quadratic non-linearity is of u u type then one can push LWP to s > -1/4. [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.

Quadratic NLS on T³

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

Cubic NLS on R

Scaling is s_c = -1/2.
LWP for s ³ 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 ³0 the solution map was known to be not C² 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 L² 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 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 [Gr-p2]. If the nonlinearity is of u u 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 H¹-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 [CrrBu.1991], [Bu.1992]
On the interval, the inverse scattering method was applied to generate solutions in [GriSan-p].

Cubic NLS on T

LWP for s³0 [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 [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 u u u type (instead of |u|² u) then one can obtain LWP for s > -1/3 [Gr-p2]
Remark: This equation is completely integrable [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]

Cubic NLS on R²

Scaling is s_c = 0, thus this is an L^2 critical NLS.
LWP for s ³ 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 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 [CoKeStTkTa2002]

For s>3/5 this was shown in [Bo1998].
For s>2/3 this was shown in [Bo1998], [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 [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 [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 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 [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} (log log|t|)^{1/2} [LanPapSucSup1988]; interestingly, however, if we perturb NLS to the 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)+}) [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 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 [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 [Ts1985]. This was improved to x^{2/3+} u(0) in L² 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 RxT one has LWP for s³0 [TkTz-p2].
For TxT one has LWP for s>0 [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 [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.

Cubic NLS on R³

Scaling is s_c = 1/2.
LWP for s ³ 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 s³ 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 L⁵ 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^s_loc 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 T³

Scaling is s_c = 1/2.
LWP is known for s >1/2 [Bo1993].

Cubic NLS on R⁴

Scaling is s_c = 1.
LWP is known for s ³ 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 s³1 in the radial case [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.

Cubic NLS on T⁴

Scaling is s_c = 1.
LWP is known for s ³ 2 [Bo1993d].

Cubic NLS on S⁶

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.

Quartic NLS on R

Scaling is s_c = -1/6.
For any quartic non-linearity one can obtain LWP for s ³ 0 [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 [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.

Quartic NLS on T

For any quartic non-linearity one has LWP for s>0 [Bo1993].
If the quartic non-linearity is of u u u u type then one can obtain LWP for s > -1/6 [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) [Bo1995c]. Indeed one has an invariant measure.

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 [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 [Ka1986].

This has been improved to s > 1-e in [CoKeStTkTa2003c] in the defocusing Hamiltonian case. This result can of course be improved further.
Scattering in the energy space [Na1999c] in the defocusing Hamiltonian case.
One also has GWP and scattering for small H^{1/3} data for any quintic non-linearity.

Quintic NLS on R

This equation may be viewed as a simpler version of 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 [KolNewStrQi2000]
Scaling is s_c = 0, thus this is an L^2 critical NLS.
LWP is known for s ³ 0 [CaWe1990], [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 [CoKeStTkTa-p6]
For s>2/3 this is in [CoKeStTkTa-p4].
For s > 32/33 this is implicit in [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 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 [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 [BirKnPoSvVe1996] and in fact create solutions which blow up at any collection of specified points in spacetime [BoWg1997], [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.

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 [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) [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.

Quintic NLS on R²

Scaling is s_c = 1/2.
LWP is known for s ³ 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<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 [CoKeStTkTa2003b]. This result can of course be improved further.
Scattering in the energy space [Na1999c]
One also has GWP and scattering for small H^{1/2} data for any quintic non-linearity.

Quintic NLS on R³

Scaling is s_c = 1.
LWP is known for s ³ 1 [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], [Bo1999].
Blowup can occur in the focussing case from Glassey's virial identity.

Septic NLS on R

Scaling is s_c = 1/6.
LWP is known for s ³ s_c [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 [CoKeStTkTa2003b] in the defocusing case. This result can of course be improved further.
Scattering in the energy space [Na1999c]
One also has GWP and scattering for small H^{s_c} data for any septic non-linearity.

Septic NLS on R^2

Scaling is s_c = 2/3.
LWP is known for s ³ s_c [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 [CoKeStTkTa2003b] in the defocusing case. This result can of course be improved further.
Scattering in the energy space [Na1999c]
One also has GWP and scattering for small H^{s_c} data for any septic non-linearity.

Septic NLS on R^3

Scaling is s_c = 7/6.
LWP is known for s ³ s_c [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 [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.

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) and d=2 (cubic NLS). One always has GWP and scattering in L^2 for small data (see [GiVl1978], [GiVl1979], [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 [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 [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} [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 ([OgTs1991], [Nw1999],
[GgMe1995].

In 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 [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 [Nw1999] and exhibit mass concentration [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.

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 1D cubic NLS. Another is the nonlinear Schrodinger-Airy equation.

Derivative non-linear Schrodinger

By derivative non-linear Schrodinger (D-NLS) we refer to equations of the form

u_t - i D u = f(u, u, Du, Du)

where f is an analytic function of u, its spatial derivatives Du, and their complex conjugates which vanishes to at least second order at the origin. We often assume the natural gauge invariance condition

f(exp(i q) u, exp(-i q) u, exp(i q) Du, exp(-i q) Du) = exp(i q) f(u, u, Du, Du).

The main new difficulty here is the loss of regularity of one derivative in the non-linearity, which causes standard techniques such as the energy method to fail (since the energy estimate does not recover any regularity in the case of the Schrodinger equation). However, there are other estimates which can recover a full derivative for the inhomogeneous Schrodinger equation providing there is sufficient decay in space, and so one can still obtain well-posedness results for sufficiently smooth and regular data. In the analytic category some existence results can be found in [SnTl1985], [Ha1990].

An alternative strategy is to apply a suitable transformation in order to place the non-linearity in a good form. For instance, a term such as u Du is preferable to u Du (the Fourier transform is less likely to stay near the upper paraboloid, and these terms are more likely to disappear in energy estimates). One can often "gauge transform" the equation (in a manner dependent on the solution u) so that all bad terms are eliminated. In one dimension this can be done by fairly elementary methods (e.g. the method of integrating factors), but in higher dimensions one must use some pseudo-differential calculus.

In order to quantify the decay properties needed, we define H^{s,m} denote the space of all functions u for which <x>^m D^s u is in L²; thus s measures regularity and m measures decay. It is a well-known fact that the Schrodinger equation trades decay for regularity; for instance, data in H^{m,m'} instantly evolves to a solution locally in H^{m+m'} for the free Schrodinger equation and m, m' ³ 0.

If m ³ [d/2] + 4 is an integer then one has LWP in H^m \cap H^{m-2,2} [Ci1999]; see also [Ci1996], [Ci1995], [Ci1994].

If f is cubic or better then one can improve this to LWP in H^m [Ci1999]. Furthermore, if one also has gauge invariance then data in H^{m,m'} evolves to a solution in H^{m+m'} for all non-zero times and all positive integers m' [Ci1999].
If d=1 and f is cubic or better then one has LWP in H³ [HaOz1994b].

For special types of cubic non-linearity one can in fact get GWP for small data in H^{0,4} \cap H^{4,0} [Oz1996].

LWP in H^s \cap H^{0,m} for small data for sufficiently large s, m was shown in [KnPoVe1993c]. The solution was also shown to have s+1/2 derivatives in L²_{t,x,loc}.

If f is cubic or better one can take m=0 [KnPoVe1993c].
If f is quartic or better then one has GWP for small data in H^s. [KnPoVe1995]
For large data one still has LWP for sufficiently large s, m [Ci1995], [Ci1994].

If the non-linearity consists mostly of the conjugate wave u, then one can do much better. For instance [Gr-p], when f = (Du)^k one can obtain LWP when s > s_c = d/2 + 1 - 1/(k-1), s³1, and k ³ 2 is an integer; similarly when f = D(u^k) one has LWP when s > s_c = d/2 - 1/(k-1), s ³0, and k ³ 2 is an integer. In particular one has GWP in L² when d=1 and f = i(u²)_x and GWP in H¹ when d=1 and f = i({u}_x)². These results apply in both the periodic and non-periodic setting.

Non-linearities such as t^{-\alpha} |u_x|² in one dimension have been studied in [HaNm2001b], with some asymptotic behaviour obtained.

In d=2 one has GWP for small data when the nonlinearities are of the form u Du + u Du [De2002].

Schrodinger maps

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

Schrodinger maps are to the Schrodinger equation as 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. [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 [Im1984] (see [KnPoVe2000] for some recent results on this equation)

In one dimension local well posedness is known for smooth data by the 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 [ZkTkh1979], [Di1999].

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

In one dimension one has global existence in the energy norm [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 [DiWgy1998]. The constant curvature assumption was relaxed to non-positive curvature (or Hermitian locally symmetric) in [PaWghWgy2000]. It is conjectured that one should have a global flow whenever the target is compact Kahler [Di2002].

When the target is S² this is in [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 [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 [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 [KnPoVe1993c], [HaHr-p], and local existence for smooth solutions [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 [DiWgy2001]

When the target is is S² this is in [SucSupBds1986].

Some further discussion on this equation can be found in the survey [Di2002].

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 [Tk-p].

This is sharp in the C⁰ uniform sense [BiLi-p] (see also [Tk-p] for failure of analytic well-posedness below 1/2).
For s ³ 1 this was proven in [HaOz1994].

GWP for s>1/2 and small L² norm [CoKeStTkTa2002b]. The s=1/2 case remains open.

for s>2/3 and small L² norm this was proven in [CoKeStTkTa2001b].
For s > 32/33 with small L² norm this was proven in [Tk-p].
For s ³ 1 and small L² norm this was proven in [HaOz1994]. One can also handle certain pure power additional terms [Oz1996].
The small L² norm condition is required in order to gauge transform the problem; see [HaOz1993], [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 [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 u)_x u + b (u u)_x u_x one can obtain GWP for small data [KyTs1995] for arbitrary complex constants a, b. See also [Ts1994].

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 [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 [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 [GiOz1993])

For n>1 and n > 1 this is in [NwOz1992]

In the defocusing case, all solutions in suitable spaces have asymptotic states in L², and one has asymptotic completeness when n > 4/3 [HaTs1987].

For n < 1, n ³3, and 1 - n/2 < s < 1 this is in [Nak-p4]

Many earlier results in [HaKakNm1998], [HaKaiNm1998], [HaNm2001], [HaNm1998b]

In the Gevrey and real analytic categories there are some large data results in [GiVl2000], [GiVl2000b], [GiVl2001], covering the cases n< 1 and n > 1.
For small decaying data one has some invertibility of the wave operators [HaNm1998]

Maxwell-Schrodinger system in R³

This system is a partially non-relativistic analogue of the 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 D₀ u - D_j u D^j u

giving rise to the system of PDE

i u_t = D_j u D^j u/2 + A₀ u
d^a F_ab = J_b

where the current density J_b is given by

J₀ = |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 [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 [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 [Ts1993]

In one dimension, GWP in the energy class is known [Ts1995]
In two dimensions, GWP for smooth solutions is known [TsNk1985]

Quasilinear NLS (QNLS)

These are general equations of the form

u_t = i a(x,t,u,Du) D^2 u + b_1(x,t,u,Du) Du + b_2(x,t,u,Du) Du + first order terms,

where a, b_1, b_2, and the lower order terms are smooth functions of all variables. These general systems arise in certain physical models (see e.g. [BdHaSau1997]). Also under certain conditions they can be derived from fully nonlinear Schrodinger equations by differentiating the equation.

In order to qualify as a quasilinear NLS, we require that the quadratic form a is real and elliptic. It is also natural to assume that the metric structure induced by a obeys a non-trapping condition (all geodesics eventually reach spatial infinity), as this is what is necessary for the optimal local smoothing estimate to occur. For a similar reason it is useful to assume that the magnetic field b_1 (or more precisely, the imaginary part of this field) is uniformly integrable along lines in space in the time independent case (for the time dependent case the criterion involves the bicharacteristic flow and is more complicated, see [Ic1984]); without this condition even the linear equation can be ill-posed.

A model example of QNLS is the equation

u_t = i (Delta – V(x)) u – 2iu h’(|u|^2) Delta h(|u|^2) + i u g(|u|^2)

for smooth functions h,g.

· When V=0 local existence for small data is known in H^6(R^n) for n=1,2,3 [BdHaSau1997]

o Under certain conditions on the initial data the LWP can be extended to GWP for n=2,3 [BdHaSau1997].

o For large data, LWP is known in H^s(R^n) for any n and any sufficiently large s > s(n) [Col2002]

· For suitable choices of V LWP is also known for H^infty(R^n) for any n [Pop2001]; this uses the Nash-Moser iteration method.

In one dimension, the fully nonlinear Schrodinger equation has LWP in H^infty(R^n) assuming a cubic nonlinearity [Pop2001b]. Other LWP results for the one-dimensional QNLS have been obtained by [LimPo-p] using gauge transform arguments.

In general dimension, LWP for data in H^{s,2} for sufficiently large s has been obtained in [KnPoVe-p] assuming non-trapping, and asymptotic flatness of the metric a and of the magnetic field Im b_1 (both decaying like 1/|x|^2 or better up to derivatives of second order).