Semilinear Schrodinger equation

[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 [#Quadratic_NLS 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 references:CoKeStTkTa-p7 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.