Non-linear Schrodinger equations


Overview

The free Schrodinger equation

i ut + 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

ut - 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 Lqt Lrx or Lrx Lqt, 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:

On T:

On T^d:

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

|| uv ||1/2+, 0 <~ || u ||1/2+, 1/2+  || v ||0+, 1/2+

|| 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+

 

|| 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 ||_{Lt H1/3x} <~ || 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.
 

 

|| 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

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


Multilinear estimates

|| 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 ut + 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 Hs 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) = | u|2/2 - l |u|^{p+1}/(p+1) dx

and symplectic form

{u, v} = Im 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 sc = 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 ) 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

|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 0, sc [CaWe1990]; see also [Ts1987]; for the case s=1, see [GiVl1979].  In the L2-subcritical cases one has GWP for all s0 by L2 conservation; in all other cases one has GWP and scattering for small data in Hs, s sc.  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 periodic setting these results are much more difficult to obtain.  On the one-dimensional torus T one has LWP for s > 0, sc 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 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).
 


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

d2t 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 Lpx,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 Hs 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 Hs 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 Hs 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 d3 and p < 5 [BuGdTz-p3].

        For the cubic equation on two-dimensional surfaces one has LWP in H^s for s > [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.

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

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 ut + 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.

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.

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?


Quadratic NLS on R


Quadratic NLS on T


Quadratic NLS on R2


Quadratic NLS on T2


Quadratic NLS on R3


Quadratic NLS on T3


Cubic NLS on R


Cubic NLS on T


Cubic NLS on R2


Cubic NLS on RxT and T2


Cubic NLS on R3


Cubic NLS on T3


Cubic NLS on R4


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


Cubic NLS on S6


Quartic NLS on R


Quartic NLS on T


Quartic NLS on R^2


Quintic NLS on R


Quintic NLS on T

 


Quintic NLS on R2


Quintic NLS on R3


Septic NLS on R


Septic NLS on R^2


Septic NLS on R^3



L^2 critical NLS on R^d

The L^2 critical situation sc = 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

ut - 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 Ds 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' 0.


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 > sc = d/2 + 1 - 1/(k-1), s1, and k 2 is an integer; similarly when f = D(u^k) one has LWP when s > sc = d/2 - 1/(k-1), s 0, and k 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} |ux|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

iut + 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 ut.  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 ut = 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

ivt + 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}.

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:

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 ux u).

For non-linearities of the form f = a (u u)x u + b (u u)x ux 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 ut + 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:
 


Maxwell-Schrodinger system in R3

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

\int Fab Fab + 2 Im u D0 u - Dj u Dj u

giving rise to the system of PDE

i ut =   Dj u Dj u/2 + A0 u
da Fab = Jb

where the current density Jb is given by

J0 = |u|^2;  Jj = - Im u Dj 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.


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).