Semilinear Schrodinger equation: Difference between revisions

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[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.]
{{equation
| name = NLS
| equation = <math>iu_t + \Delta u = \pm |u|^{p-1} u</math>
| fields = <math>u: \R \times \R^d \to \mathbb{C}</math>
| data = <math>u(0) \in H^s(\R^d)</math>
| hamiltonian = [[Hamiltonian]]
| linear = [[free Schrodinger equation|Schrodinger]]
| nonlinear = [[semilinear]]
| critical = <math>\dot H^{\frac{d}{2} - \frac{2}{p-1}}(\R^d)</math>
| criticality = varies
| covariance = [[Galilean]]
| lwp = <math>H^s(\R)</math> for <math>s \geq \max(s_c, 0)</math> (*)
| gwp = varies
| parent = [[Schrodinger equations|Nonlinear Schrodinger equations]]
| special = [[Quadratic NLS|quadratic]], [[cubic NLS|cubic]], [[quartic NLS|quartic]], [[quintic NLS|quintic]],<br> [[mass critical NLS|mass critical]] NLS
| related = NLS [[NLS with potential|with potential]], [[NLS on manifolds and obstacles|on manifolds]]
}}


The '''semilinear Schrodinger equation''' (NLS) is


<center><math>i \partial_t u + \Delta u = \pm |u|^{p-1} u </math></center>
The '''semilinear Schrodinger equation''' (NLS) is displayed on the right-hand box, where the exponent ''p'' is fixed
and is larger than one.  One can also replace the nonlinearity <math>\pm |u|^{p-1} u</math> by a more general nonlinearity
<math>F(u)</math> of [[power type]].  The <math>+</math> sign choice is the [[defocusing]] case; <math>-</math> is [[focusing]]. 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.


for p>1.  There are many [[Schrodinger:specific equations|specific cases]] of this equation which are of interest, but in this page we shall focus on the general theory. The <math>+</math> sign choice is the ''defocusing'' case; <math>-</math> 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.
== Theory ==


In order to consider this problem in <math>H^s</math> one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume
The NLS has been extensively studied, and there is now a substantial theory on many topics concerning solutions to this equation.


<center><math>p</math> is an odd integer, or <math>p > [s]+1.</math></center>
* [[Algebraic structure of NLS|Algebraic structure]] (Symmetries, conservation laws, transformations, Hamiltonian structure)
* [[NLS wellposedness|Well-posedness]] (both local and global)
* [[NLS scattering|Scattering]] (as well as asymptotic completeness and existence of wave operators)
* [[NLS stability|Stability of solitons]] (orbital and asymptotic)
* [[NLS blowup|Blowup]]
* [[Unique continuation]]


This is a Hamiltonian flow with the Hamiltonian
== Specific semilinear Schrodinger equations ==


<center><math>H(u) = \int_{R^d} |\nabla u |^2  \pm  |u|^{p+1}/(p+1) dx</math></center>
There are many special cases of NLS which are of interest:


and symplectic form
* [[Quadratic NLS]]
 
* [[Cubic NLS]] ([[cubic NLS on R|on R]], [[cubic NLS on R2|on R^2]], [[cubic NLS on R3|on R^3]], etc.)
<center><math>\{u, v\} = Im \int_{R^d} u \overline{v} dx.</math></center>
* [[Quartic NLS]]
 
* [[Quintic NLS]] ([[Quintic NLS on R|on R]], [[Quintic NLS on T|on T]], [[Quintic NLS on R2|on R^2]], and [[Quintic NLS on R3|on R^3]])
From the phase invariance <math>u \to e^{iq}u</math> one also has conservation of the <math>L^2_x</math>norm.
* [[Septic NLS]]
 
* [[Mass critical NLS]]
The scaling regularity is <math>s_c = d/2 - 2/(p-1)</math>. The most interesting values of p are the <math>L^2_x</math>-critical or [[Pseudoconformal Transformation|pseudoconformal]] power <math>p=1+4/d</math> and the <math>H^1_x</math>-critical power <math>p=1+4/(d-2)</math> for <math>d>2</math> (for <math>d=1,2</math> there is no <math>H^1</math> conformal power). The power <math>p = 1 + 2/d</math> is also a key exponent for the scattering theory (as this is when the non-linearity <math>|u|^{p-1}u</math> has about equal strength with the decay <math>t^{-d/2}</math>). The cases <math>p=3,5</math> 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. <br /><br />
* [[Energy critical NLS]]
 
{| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"
|- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"
| style="padding: .75pt .75pt .75pt .75pt" |
Dimension <math>d</math>
| style="padding: .75pt .75pt .75pt .75pt" |
Scattering power <math>1+2/d</math>
| style="padding: .75pt .75pt .75pt .75pt" |
<math>L^2</math> -critical power <math>1+4/d</math>
| style="padding: .75pt .75pt .75pt .75pt" |
<math>H^1</math>-critical power <math>1+4/(d-2)</math>
|- style="mso-yfti-irow: 1"
| style="padding: .75pt .75pt .75pt .75pt" |
1
| style="padding: .75pt .75pt .75pt .75pt" |
3
| style="padding: .75pt .75pt .75pt .75pt" |
5
| style="padding: .75pt .75pt .75pt .75pt" |
N/A
|- style="mso-yfti-irow: 2"
| style="padding: .75pt .75pt .75pt .75pt" |
2
| style="padding: .75pt .75pt .75pt .75pt" |
2
| style="padding: .75pt .75pt .75pt .75pt" |
3
| style="padding: .75pt .75pt .75pt .75pt" |
<math>\infty</math>
|- style="mso-yfti-irow: 3"
| style="padding: .75pt .75pt .75pt .75pt" |
3
| style="padding: .75pt .75pt .75pt .75pt" |
5/3
| style="padding: .75pt .75pt .75pt .75pt" |
7/3
| style="padding: .75pt .75pt .75pt .75pt" |
5
|- style="mso-yfti-irow: 4"
| style="padding: .75pt .75pt .75pt .75pt" |
4
| style="padding: .75pt .75pt .75pt .75pt" |
3/2
| style="padding: .75pt .75pt .75pt .75pt" |
2
| style="padding: .75pt .75pt .75pt .75pt" |
3
|- style="mso-yfti-irow: 5"
| style="padding: .75pt .75pt .75pt .75pt" |
5
| style="padding: .75pt .75pt .75pt .75pt" |
7/5
| style="padding: .75pt .75pt .75pt .75pt" |
9/5
| style="padding: .75pt .75pt .75pt .75pt" |
7/3
|- style="mso-yfti-irow: 6; mso-yfti-lastrow: yes"
| style="padding: .75pt .75pt .75pt .75pt" |
6
| style="padding: .75pt .75pt .75pt .75pt" |
4/3
| style="padding: .75pt .75pt .75pt .75pt" |
5/3
| style="padding: .75pt .75pt .75pt .75pt" |
2
|}
 
The pseudoconformal transformation of the Hamiltonian gives that the time derivative of
 
<center><math>\|(x + 2it \tilde{N})u \|^2_2 - 81t^2/(p+1)\|U\|{P+1}^{P+1}</math></center>
 
is equal to
 
<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. <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
 
<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 <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
 
<center> <math>\iint \frac{|u|^{p+1}}{|x|} dx dt</math></center>
 
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 <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 <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 <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 <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 <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 <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 <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 <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 <math>L^2\,</math>-supercritical focussing case one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality
 
<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 <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 <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 <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><math>1 + 8 / (\sqrt{d^2 + 12d + 4} + d - 2)\,</math>;</center>
 
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.


[[Category:Equations]]
[[Category:Equations]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]

Latest revision as of 05:44, 21 July 2007

NLS
Description
Equation
Fields
Data class
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity
Criticality varies
Covariance Galilean
Theoretical results
LWP for (*)
GWP varies
Related equations
Parent class Nonlinear Schrodinger equations
Special cases quadratic, cubic, quartic, quintic,
mass critical NLS
Other related NLS with potential, on manifolds


The semilinear Schrodinger equation (NLS) is displayed on the right-hand box, where the exponent p is fixed and is larger than one. One can also replace the nonlinearity by a more general nonlinearity of power type. The sign choice is the defocusing case; is focusing. 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.

Theory

The NLS has been extensively studied, and there is now a substantial theory on many topics concerning solutions to this equation.

Specific semilinear Schrodinger equations

There are many special cases of NLS which are of interest: