# 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** (NLS) is

for p>1. There are many specific cases of this equation which are of interest, but in this page we shall focus on the general theory. The sign choice is the *defocusing* case; is *focussing*. There are also several variants of NLS, such as NLS with potential or NLS on manifolds and obstacles; see the general page on Schrodinger equations for more discussion.

In order to consider this problem in one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume

This is a Hamiltonian flow with the Hamiltonian

and symplectic form

From the phase invariance one also has conservation of the norm.

The scaling regularity is . The most interesting values of p are the -critical or pseudoconformal power and the -critical power for (for there is no conformal power). The power is also a key exponent for the scattering theory (as this is when the non-linearity has about equal strength with the decay ). The cases 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 |
L |
H |

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

^{2}_2 - 8 l t

^{2}/(p+1) || u ||_{p+1}^{p+1}

is equal to

This law is useful for obtaining a priori spacetime estimates on the solution given sufficient decay in space (e.g. xu(0) in L^{2}), especially in the L^{2}-critical case p=1+4/d (when the above derivative is zero). The L^{2} norm of xu(0) is sometimes known as the *pseudoconformal charge*.

The equation is invariant under Gallilean transformations

^{2}t)) u(x-vt, t).

This can be used to show ill-posedness below L^{2} 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 __u__^{2}, then [#Quadratic_NLS one can go below L^{2}]).

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

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 L^{2}-subcritical cases one has GWP for all s³0 by L^{2} 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 L^{2} norm or assume defocusing; for p>6 one needs to assume defocusing).

- For the defocussing case, one has GWP in the H
^{1}-subcritical case if the data is in H^{1}. To improve GWP to scattering, it seems that needs p to be L^{2}super-critical (i.e. p > 1 + 4/d). In this case one can obtain scattering if the data is in L^{2}(|x|^{2}dx) (since one can then use the pseudo-conformal conservation law).- In the d ³ 3 cases one can remove the L
^{2}(|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 H^{1}norm to H^{s}for some s<1 CoKeStTkTa-p7. - For d=1,2 one can also remove the L
^{2}(|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 d ³ 3 cases one can remove the L

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

^{2}

_{t}ò x

^{2}|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 H^{1}-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 L^{2} and H^{1} critical powers the wave operators are well-defined in the energy space LnSr1978, GiVl1985, Na1999c. At the L^{2} 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 L^{2} 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 L^{2}(|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

^{2}+ 12d + 4) + d - 2);

see NaOz2002 for further discussion.

Many of the global results for H^{s} also hold true for L^{2}(|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.

Some semilinear Schrodinger equations are known to enjoy a unique continuation property.