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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]] |
| | }} |
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| The '''semilinear Schrodinger equation''' (NLS) is
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| <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. |
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| 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 == |
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| 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. |
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| <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]] |
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| This is a Hamiltonian flow with the Hamiltonian
| | == Specific semilinear Schrodinger equations == |
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| <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: |
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| 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]] |
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| Dimension <math>d</math>
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| Scattering power <math>1+2/d</math>
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| <math>L^2</math> -critical power <math>1+4/d</math>
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| <math>H^1</math>-critical power <math>1+4/(d-2)</math>
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| <math>\infty</math>
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| The pseudoconformal transformation of the Hamiltonian gives that the time derivative of
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| <center><math>\|(x + 2it \tilde{N})u \|^2_2 - 81t^2/(p+1)\|U\|{P+1}^{P+1}</math></center>
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| is equal to
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| <center><math>4dt\lambda(p-(1+4/d))/(p+1) \|u\|_{p+1}^{p+1}.</math></center>
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| 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''.
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| The equation is invariant under Gallilean transformations
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| <center><math>u(x,t) \rightarrow e^{(i (vx/2 - |v|^{2}t)} u(x-vt, t).\,</math></center>
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| 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>]).
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| From scaling invariance one can obtain Morawetz inequalities, which usually estimate quantities of the form
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| <center> <math>\iint \frac{|u|^{p+1}}{|x|} dx dt</math></center>
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| 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.
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| 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]].
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| 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:
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| * 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]]
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| ** In <math>R^6\,</math> one also has Lipschitz well-posedness [BuGdTz-p5]
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| <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 />
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| * 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).
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| ** 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. | |
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| <br /> In the <math>L^2\,</math>-supercritical focussing case one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality
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| <center><math>\partial^2_t \int x^2 |u|^2 dx ~ H(u)</math>;</center>
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| 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]
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| 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>
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| 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
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| <center><math>1 + 8 / (\sqrt{d^2 + 12d + 4} + d - 2)\,</math>;</center>
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| see [[Bibliography#NaOz2002|NaOz2002]] for further discussion.
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| 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.
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| Some semilinear Schrodinger equations are known to enjoy a [[unique continuation]] property.
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| [[Category:Equations]] | | [[Category:Equations]] |
| [[Category:Schrodinger]] | | [[Category:Schrodinger]] |
