Semilinear Schrodinger equation: Difference between revisions

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Dimension d
Dimension <math>d</math>
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Scattering power 1+2/d
Scattering power <math>1+2/d</math>
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L<sup>2</sup>-critical power 1+4/d
<math>L^2</math> -critical power <math>1+4/d</math>
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H<sup>1</sup>-critical power 1+4/(d-2)
<math>H^1</math>-critical power <math>1+4/(d-2)</math>
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3
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infinity
<math>\infty</math>
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The pseudoconformal transformation of the Hamiltonian gives that the time derivative of
The pseudoconformal transformation of the Hamiltonian gives that the time derivative of


<center><nowiki>|| (x + 2it </nowiki><font face="Symbol">Ñ</font>) u ||<sup>2</sup>_2 - 8 <font face="Symbol">l</font> t<sup>2</sup>/(p+1) || u ||_{p+1}^{p+1}</center>
<center><math>\|(x + 2it \tilde{N})u \|^2_2 - 81t^2/(p+1)\|U\|{P+1}^{P+1}</math></center>


is equal to
is equal to


<center>4dt<font face="Symbol">l</font>(p-(1+4/d))/(p+1) ||u||_{p+1}^{p+1}.</center>
<center><math>4dt\lambda(p-(1+4/d))/(p+1) \|u\|_{p+1}^{p+1}.</math></center>


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


The equation is invariant under Gallilean transformations
The equation is invariant under Gallilean transformations


<center>u(x,t) -> exp(i (v.x/2 - |v|<sup>2</sup> t)) u(x-vt, t).</center>
<center><math>u(x,t) \rightarrow e^{(i (vx/2 - |v|^{2}t)} u(x-vt, t).\,</math></center>


This can be used to show ill-posedness below L<sup>2</sup> 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>u</u><sup>2</sup>, then [#Quadratic_NLS one can go below L<sup>2</sup>]).
This can be used to show ill-posedness below <math>L^2\,</math> 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 <math>\underline{u^2}\,,</math> then [#Quadratic_NLS one can go below <math>L^2\,</math>]).


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


<center><font face="Symbol">ò</font> <font face="Symbol">ò</font> |u|^{p+1}/|x| dx dt</center>
<center> <math>\iint \frac{|u|^{p+1}}{|x|} dx dt</math></center>


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 defocussing case in terms of the <math>H^{1/2}\,</math> 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 <font face="Symbol">³</font> 0, s<sub>c</sub> [[Bibliography#CaWe1990|CaWe1990]]; see also [[Bibliography#Ts1987|Ts1987]]; for the case s=1, see [[Bibliography#GiVl1979|GiVl1979]]. In the L<sup>2</sup>-subcritical cases one has GWP for all s<font face="Symbol">³</font>0 by L<sup>2</sup> conservation; in all other cases one has GWP and scattering for small data in H<sup>s</sup>, s <font face="Symbol">³</font> s<sub>c</sub>. These results apply in both the focussing and defocussing cases. At the critical exponent one can prove Besov space refinements [[Bibliography#Pl2000|Pl2000]], [Pl-p4]. This can then be used to obtain self-similar solutions, see [[Bibliography#CaWe1998|CaWe1998]], [[Bibliography#CaWe1998b|CaWe1998b]], [[Bibliography#RiYou1998|RiYou1998]], [MiaZg-p1], [MiaZgZgx-p], [MiaZgZgx-p2], [[Bibliography#Fur2001|Fur2001]].
In the other direction, one has LWP for <math>s \ge 0, s_c\,</math>, [[Bibliography#CaWe1990|CaWe1990]]; see also [[Bibliography#Ts1987|Ts1987]]; for the case <math>s=1\,,</math> see [[Bibliography#GiVl1979|GiVl1979]]. In the <math>L^2\,</math>-subcritical cases one has GWP for all <math>s\ge 0\,</math> by <math>L^2\,</math> conservation; in all other cases one has GWP and scattering for small data in <math>H^s\,</math>, <math>s\, \ge s_c.\,</math> These results apply in both the focussing and defocussing cases. At the critical exponent one can prove Besov space refinements [[Bibliography#Pl2000|Pl2000]], [Pl-p4]. This can then be used to obtain self-similar solutions, see [[Bibliography#CaWe1998|CaWe1998]], [[Bibliography#CaWe1998b|CaWe1998b]], [[Bibliography#RiYou1998|RiYou1998]], [MiaZg-p1], [MiaZgZgx-p], [MiaZgZgx-p2], [[Bibliography#Fur2001|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:
Now suppose we remove the regularity assumption that <math>p\,</math> is either an odd integer or larger than <math>[s]+1\,.</math> 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 [[Bibliography#Ka1986|Ka1986]]
* In the <math>H^1\,</math> subcritical case one has GWP in <math>H^1\,,</math> assuming the nonlinearity is smooth near the origin [[Bibliography#Ka1986|Ka1986]]
** In R^6 one also has Lipschitz well-posedness [BuGdTz-p5]
** In <math>R^6\,</math> one also has Lipschitz well-posedness [BuGdTz-p5]


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


* For the defocussing case, one has GWP in the H<sup>1</sup>-subcritical case if the data is in H<sup>1</sup>. To improve GWP to scattering, it seems that needs p to be L<sup>2</sup> super-critical (i.e. p > 1 + 4/d). In this case one can obtain scattering if the data is in L<sup>2</sup>(|x|<sup>2</sup> dx) (since one can then use the pseudo-conformal conservation law).
* For the defocussing case, one has GWP in the <math>H^1\,</math>-subcritical case if the data is in <math>H^1\,.</math> To improve GWP to scattering, it seems that needs <math>p\,</math> to be <math>L^2\,</math> super-critical (i.e. <math>p > 1 + 4/d\,</math>). In this case one can obtain scattering if the data is in <math>L^{2}(|x|^2 dx)\,</math> (since one can then use the pseudo-conformal conservation law).
** In the d <font face="Symbol">³</font> 3 cases one can remove the L<sup>2</sup>(|x|<sup>2</sup> dx) assumption [[Bibliography#GiVl1985|GiVl1985]] (see also [[Bibliography#Bo1998b|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<sup>1</sup> norm to H<sup>s</sup> for some s<1 [[Bibliography#CoKeStTkTa-p7 |CoKeStTkTa-p7]].
** In the <math>d \ge 3\,</math> cases one can remove the <math>L^{2}(|x|^2 dx)\,</math> assumption [[Bibliography#GiVl1985|GiVl1985]] (see also [[Bibliography#Bo1998b|Bo1998b]]) by exploiting Morawetz identities, approximate finite speed of propagation, and strong decay estimates (the decay of <math>t^{-d/2}\,</math> is integrable). In this case one can even relax the <math>H^1\,</math> norm to <math>H^s\,</math> for some <math>s<1\,</math> [[Bibliography#CoKeStTkTa-p7 |CoKeStTkTa-p7]].
** For d=1,2 one can also remove the L<sup>2</sup>(|x|<sup>2</sup> dx) assumption [[Bibliography#Na1999c|Na1999c]] by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.
** For <math>d=1,2\,</math> one can also remove the <math>L^{2}(|x|^2 dx)\,</math> assumption [[Bibliography#Na1999c|Na1999c]] by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.


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


<center>d<sup>2</sup><sub>t</sub> <font face="Symbol">ò</font> x<sup>2</sup> |u|<sup>2</sup> dx ~ H(u);</center>
<center><math>\partial^2_t \int x^2 |u|^2 dx ~ H(u)</math>;</center>


see e.g. [[Bibliography#OgTs1991|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 <br /> is not known, but the H^s norm can still get arbitrarily large arbitrarily quickly for s < s_c [CtCoTa-p2]
see e.g. [[Bibliography#OgTs1991|OgTs1991]]. By scaling this implies that we have instantaneous blowup in <math>H^s\,</math> for <math>s < s_c\,</math> in the focusing case. In the defocusing case blowup <br /> is not known, but the <math>H^s\,</math> norm can still get arbitrarily large arbitrarily quickly for <math>s < s_c\,</math> [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<sup>1</sup>-stable [[Bibliography#Ws1985|Ws1985]], [[Bibliography#Ws1986|Ws1986]]. Below the H^1 norm, this is not known, but polynomial upper bounds on the instability are in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]].Multisolitons are also asymptotically stable under smooth decaying perturbations [[Bibliography#Ya1980|Ya1980]], [[Bibliography#Grf1990|Grf1990]], [[Bibliography#Zi1997|Zi1997]], [RoScgSf-p], [RoScgSf-p2], provided that p is betweeen 1+2/d and 1+4/d.
Suppose we are in the <math>L^2\,</math> subcritical case <math>p < 1 + 2/d\,</math>, with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy <math>E\,</math>. By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is <math>H^1\,</math>-stable [[Bibliography#Ws1985|Ws1985]], [[Bibliography#Ws1986|Ws1986]]. Below the <math>H^1\,</math> norm, this is not known, but polynomial upper bounds on the instability are in [[Bibliography#CoKeStTkTa2003b|CoKeStTkTa2003b]].Multisolitons are also asymptotically stable under smooth decaying perturbations [[Bibliography#Ya1980|Ya1980]], [[Bibliography#Grf1990|Grf1990]], [[Bibliography#Zi1997|Zi1997]], [RoScgSf-p], [RoScgSf-p2], provided that <math>p\,</math> is betweeen <math>1+2/d\,</math> and <math>1+4/d\,.</math>


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


<center>1 + 8 / (sqrt(d<sup>2</sup> + 12d + 4) + d - 2);</center>
<center><math>1 + 8 / (sqrt(d^2 + 12d + 4) + d - 2)\,</math>;</center>


see [[Bibliography#NaOz2002|NaOz2002]] for further discussion.
see [[Bibliography#NaOz2002|NaOz2002]] for further discussion.


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


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

Revision as of 18:33, 31 July 2006

[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

is an odd integer, or

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

Scattering power

-critical power

-critical power

1

3

5

N/A

2

2

3

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

is equal to

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

The equation is invariant under Gallilean transformations

This can be used to show ill-posedness below 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 then [#Quadratic_NLS one can go below ]).

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

in the defocussing case in terms of the 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 , CaWe1990; see also Ts1987; for the case see GiVl1979. In the -subcritical cases one has GWP for all by conservation; in all other cases one has GWP and scattering for small data in , 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 is either an odd integer or larger than Then some of the above results are still known to hold:

  • In the subcritical case one has GWP in assuming the nonlinearity is smooth near the origin Ka1986
    • In 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 if , with the endpoint being attained when Bo1993. In particular one has GWP in when or when and the data is small norm.For one also has GWP for random data whose Fourier coefficients decay like (times a Gaussian random variable) Bo1995c. (For one needs to impose a smallness condition on the norm or assume defocusing; for one needs to assume defocusing).

  • For the defocussing case, one has GWP in the -subcritical case if the data is in To improve GWP to scattering, it seems that needs to be super-critical (i.e. ). In this case one can obtain scattering if the data is in (since one can then use the pseudo-conformal conservation law).
    • In the cases one can remove the assumption GiVl1985 (see also Bo1998b) by exploiting Morawetz identities, approximate finite speed of propagation, and strong decay estimates (the decay of is integrable). In this case one can even relax the norm to for some CoKeStTkTa-p7.
    • For one can also remove the assumption Na1999c by finding a variant of the Morawetz identity for low dimensions, together with Bourgain's induction on energy argument.


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

;

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

Suppose we are in the subcritical case , with focusing non-linearity. Then there is a unique positive radial ground state (or soliton) for each energy . By translation and phase shift one thus obtains a four-dimensional manifold of ground states for each energy. This manifold is -stable Ws1985, Ws1986. Below the 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 is betweeen and

One can go beyond scattering and ask for asymptotic completeness and existence of the wave operators. When this is not possible due to the poor decay in time in the non-linear term Bb1984, Gs1977b, Sr1989, however at 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 and critical powers the wave operators are well-defined in the energy space LnSr1978, GiVl1985, Na1999c. At the critical exponent one can define wave operators assuming that we impose an 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 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 wave operators were also constructed in Na2001. However in order to construct wave operators in spaces such as (the space of functions with finite pseudoconformal charge) it is necessary that is larger than or equal to the rather unusual power

;

see NaOz2002 for further discussion.

Many of the global results for also hold true for . 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.