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

$i\partial _{t}u+\Delta u=\pm |u|^{p-1}u$

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 $H^{s}$ 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

$H(u)=\int _{R^{d}}|\nabla u|^{2}\pm |u|^{p+1}/(p+1)dx$

and symplectic form

$\{u,v\}=Im\int _{R^{d}}u{\overline {v}}dx.$

From the phase invariance $u\to e^{iq}u$ one also has conservation of the $L_{x}^{2}$norm.

The scaling regularity is $s_{c}=d/2-2/(p-1)$. The most interesting values of p are the $L_{x}^{2}$-critical or pseudoconformal power $p=1+4/d$ and the $H_{x}^{1}$-critical power $p=1+4/(d-2)$ for $d>2$ (for $d=1,2$ there is no $H^{1}$ 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 $1+2/d$	$L^{2}$ -critical power $1+4/d$	$H^{1}$-critical power $1+4/(d-2)$
1	3	5	N/A
2	2	3	$\infty$
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

$\|(x+2it{\tilde {N}})u\|_{2}^{2}-81t^{2}/(p+1)\|U\|{P+1}^{P+1}$

is equal to

$4dt\lambda (p-(1+4/d))/(p+1)\|u\|_{p+1}^{p+1}.$

This law is useful for obtaining a priori spacetime estimates on the solution given sufficient decay in space (e.g. $xu(0)\,$ in $L^{2}\,$), 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 Galilean transformations

$u(x,t)\rightarrow e^{(i(vx/2-|v|^{2}t)}u(x-vt,t).\,$

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 ${\underline {u^{2}}}\,,$ then one can go below $L^{2}\,$).

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

$\iint {\frac {|u|^{p+1}}{|x|}}dxdt$

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 $s\geq 0,s_{c}\,$, CaWe1990; see also Ts1987; for the case $s=1\,,$ see GiVl1979. In the $L^{2}\,$-subcritical cases one has GWP for all $s\geq 0\,$ by $L^{2}\,$ conservation; in all other cases one has GWP and scattering for small data in $H^{s}\,$, $s\,\geq s_{c}.\,$ 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 $1\leq p\leq 4\,$ Bo1993. In particular one has GWP in $L^{2}\,$ when $p<4\,,$ or when $p=4\,$ and the data is small norm.For $6>p\geq 4\,$ 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 $d\geq 3\,$ 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 $L^{2}\,$-supercritical focussing case one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality

$\partial _{t}^{2}\int x^{2}|u|^{2}dx~H(u)$;

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 $p\leq 1+2/d\,$ 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 $L_{x,t}^{p}\,$ 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

$1+8/({\sqrt {d^{2}+12d+4}}+d-2)\,$;

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.

Specific semilinear Schrodinger equations

• Quadratic NLS
• Cubic NLS (on R, on R^2, on R^3, etc.)
• Quartic NLS
• Quintic NLS
• Septic NLS
• Mass critical NLS