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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. | | 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. |
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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
| | == Theory == |
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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|Local and global well-posedness theory]] |
| This is a Hamiltonian flow with the Hamiltonian
| | * [[NLS scattering|Scattering theory]] |
| | | * [[NLS blowup|Blowup]] |
| <center><math>H(u) = \int_{R^d} |\nabla u |^2 \pm |u|^{p+1}/(p+1) dx</math></center>
| | * [[Unique continuation]] |
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| and symplectic form
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| <center><math>\{u, v\} = Im \int_{R^d} u \overline{v} dx.</math></center>
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| From the phase invariance <math>u \to e^{iq}u</math> one also has conservation of the <math>L^2_x</math>norm.
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| 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 />
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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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| 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 Galilean 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>, [[CaWe1990]]; see also [[Ts1987]]; for the case <math>s=1\,,</math> see [[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 [[Pl2000]], [[Pl-p4]]. This can then be used to obtain self-similar solutions, see [[CaWe1998]], [[CaWe1998b]], [[RiYou1998]], [[MiaZg-p1]], [[MiaZgZgx-p]], [[MiaZgZgx-p2]], [[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 [[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> [[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) [[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 [[GiVl1985]] (see also [[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> [[CoKeStTkTa-p7]].
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| ** For <math>d=1,2\,</math> one can also remove the <math>L^{2}(|x|^2 dx)\,</math> assumption [[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. [[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 [[Ws1985]], [[Ws1986]]. Below the <math>H^1\,</math> 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 <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 [[Bb1984]], [[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 [[Oz1991]], [[GiOz1993]], [[HaNm1998]], [[ShiTon2004]], [[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]], [[GiVl1985]], [[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 [[GiVl1979]], [[GiVl1985]], [[Bo1998]] (see also [[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 [[CaWe1992]], [[GiOz1993]], [[GiOzVl1994]], [[Oz1991]]; see also [[GiVl1979]], [[Ts1985]]. For <math>H^s\,</math> wave operators were also constructed in [[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 [[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 [[CaWe1992]], [[GiOzVl1994]], [[Ka1995]], [[NkrOz1997]], [[NkrOz-p]]. See [[NaOz2002]] for further discussion.
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| Some semilinear Schrodinger equations are known to enjoy a [[unique continuation]] property.
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| == Specific semilinear Schrodinger equations == | | == Specific semilinear Schrodinger equations == |
