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| ===Schrodinger maps===
| | #REDIRECT [[Schrodinger maps]] |
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| [Many thanks to Andrea Nahmod for help with this section - Ed.]
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| Schrodinger maps are to the Schrodinger equation as [wave:wm wave maps] are to the wave equation; they are the natural Schrodinger equation when the target space is a complex manifold (such as the sphere S<sup>2</sup> or hyperbolic space H<sup>2</sup>). They have the form
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| <center>iu<sub>t</sub> + <font face="Symbol">D</font> u = Gamma(u)( Du, Du )</center>
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| where Gamma(u) is the second fundamental form. This is the same as the harmonic map heat flow but with an additional "i" in front of the u<sub>t</sub>. When the target is S<sup>2</sup>, this equation arises naturally from the Landau-Lifschitz equation for a macroscopic ferromagnetic continuum, see e.g. [[Bibliography#SucSupBds1986|SucSupBds1986]]; in this case the equation has the alternate form u<sub>t</sub> = u x <font face="Symbol">D</font> u, where x is the cross product, and is sometimes known as the Heisenberg model; similar models exist when the target is generalized from a sphere S<sup>2</sup> to a Hermitian symmetric space (see e.g. [TeUh-p]). The Schrodinger map equation is also related to the Ishimori equation [[Bibliography#Im1984|Im1984]] (see [[Bibliography#KnPoVe2000|KnPoVe2000]] for some recent results on this equation)
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| In one dimension local well posedness is known for smooth data by the [#d-nls general theory of derivative nonlinear Schrodinger equations], however this is not yet established in higher dimensions. Assuming this regularity result, there is a gauge transformation (obtained by differentiating the equation, and placing the resulting connection structure in the Coulomb gauge) which creates a null structure in the non-linearity. Roughly speaking, the equation now looks like
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| <center>iv<sub>t</sub> + <font face="Symbol">D</font> v = Dv D<sup>-1</sup>(v v) + D<sup>-1</sup>(v v) D<sup>-1</sup>(v v) v + v<sup>3</sup></center>
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| where v := Du. The cubic term Dv D<sup>-1</sup>(v v) has a null structure so that orthogonal interactions (which normally cause the most trouble with derivative <br /> Schrodinger problems) are suppressed.
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| For certain special targets (e.g. complex Grassmannians) and with n=1, the Schrodinger flow is a completely integrable bi-Hamiltonian system [TeUh-p].In the case of n=1 when the target is the sphere S<sup>2</sup>, the equation is equivalent to the [#Cubic_NLS_on_R cubic NLS] [[Bibliography#ZkTkh1979|ZkTkh1979]], [[Bibliography#Di1999|Di1999]].
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| As with [wave:wm wave maps], the scaling regularity is H^{n/2}.
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| * In one dimension one has global existence in the energy norm [[Bibliography#CgSaUh2000|CgSaUh2000]] when the target is a compact Riemann surface; it is conjectured that this is also true for general compact Kahler manifolds.
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| ** When the target is a complex compact Grassmannian, this is in [TeUh-p].
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| ** In the periodic case one has local existence and uniqueness of smooth solutions, with global existence if the target is compact with constant sectional curvature [[Bibliography#DiWgy1998|DiWgy1998]]. The constant curvature assumption was relaxed to non-positive curvature (or Hermitian locally symmetric) in [[Bibliography#PaWghWgy2000|PaWghWgy2000]]. It is conjectured that one should have a global flow whenever the target is compact Kahler [[Bibliography#Di2002|Di2002]].
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| *** When the target is S<sup>2</sup> this is in [[Bibliography#ZhGouTan1991|ZhGouTan1991]]
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| * In two dimensions there are results in both the radial/equivariant and general cases.
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| ** With radial or equivariant data one has global existence in the energy norm for small energy [[Bibliography#CgSaUh2000|CgSaUh2000]], assuming high regularity LWP as mentioned above.
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| *** The large energy case may be settled in [CkGr-p], although the status of this paper is currently unclear (as of Feb 2003).
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| ** In the general case one has LWP in H<sup>s</sup> for s > 2 [[Bibliography#NdStvUh2003|NdStvUh2003]] (plus later errata), at least when the target manifold is the sphere S<sup>2</sup>. It would be interesting to extend this to lower regularities, and eventually to the critical H<sup>1</sup> case. (Here regularity is stated in terms of u rather than the derivatives v).
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| ** When the target is S<sup>2</sup> there are global weak solutions [[Bibliography#KnPoVe1993c|KnPoVe1993c]], [HaHr-p], and local existence for smooth solutions [[Bibliography#SucSupBds1986|SucSupBds1986]].
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| ** When the target is H^2 one can have blowup in finite time [Di-p].Similarly for higher dimensions.
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| * In general dimensions one has LWP in H<sup>s</sup> for s > n/2+1 [[Bibliography#DiWgy2001|DiWgy2001]]
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| ** When the target is is S<sup>2</sup> this is in [[Bibliography#SucSupBds1986|SucSupBds1986]].
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| Some further discussion on this equation can be found in the survey [[Bibliography#Di2002|Di2002]].
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| [[Category:Equations]]
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