Schrodinger maps

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Schrodinger maps are to the free Schrodinger equation as wave maps are to the free wave equation; they are the natural Schrodinger equation when the target space is a complex manifold (such as the sphere $S^2\,$ or hyperbolic space $H^2\,$). They have the form

$i\partial_tu + Du = \Gamma(u)( Du, Du )\,$

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 $u_t\,$. When the target is $S^2\,$, this equation arises naturally from the Landau-Lifschitz equation for a macroscopic ferromagnetic continuum, see e.g. SucSupBds1986; in this case the equation has the alternate form $u_t = u x Du\,,$ 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 $S^2\,$ to a Hermitian symmetric space (see e.g. TeUh-p). The Schrodinger map equation is also related to the Ishimori equation Im1984 (see KnPoVe2000 for some recent results on this equation)

In one dimension local well posedness is known for smooth data by the 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

$i\partial_tv + Dv = Dv D^{-1}(v v) + D^{-1}(v v) D^{-1}(v v) v + v^3$

where $v := Du\,$. The cubic term Dv $D^{-1}(v v)\,$ has a null structure so that orthogonal interactions (which normally cause the most trouble with derivative Schrodinger equations) are suppressed.

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 $S^2\,$, the equation is equivalent to the cubic NLS ZkTkh1979, Di1999.

As with wave maps, the scaling regularity is $H^{n/2}\,.$

• In one dimension one has global existence in the energy norm CgSaUh2000 when the target is a compact Riemann surface; it is conjectured that this is also true for general compact Kahler manifolds.
• When the target is a complex compact Grassmannian, this is in TeUh-p.
• 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 DiWgy1998. The constant curvature assumption was relaxed to non-positive curvature (or Hermitian locally symmetric) in PaWghWgy2000. It is conjectured that one should have a global flow whenever the target is compact Kahler Di2002.
• When the target is $S^2\,$ this is in ZhGouTan1991
• In two dimensions there are results in both the radial/equivariant and general cases.
• With radial or equivariant data one has global existence in the energy norm for small energy CgSaUh2000, assuming high regularity LWP as mentioned above.
• The large energy case may be settled in CkGr-p, although the status of this paper is currently unclear (as of Feb 2003).
• In the general case one has LWP in $H^s\,$ for $s > 2\,$ NdStvUh2003 (plus later errata), at least when the target manifold is the sphere $S^2\,$. It would be interesting to extend this to lower regularities, and eventually to the critical $H^1\,$ case. (Here regularity is stated in terms of $u\,$ rather than the derivatives $v\,$).
• When the target is $S^2\,$ there are global weak solutions KnPoVe1993c, HaHr-p, and local existence for smooth solutions SucSupBds1986.
• When the target is $H^2\,$ one can have blowup in finite time Di-p.Similarly for higher dimensions.
• In general dimensions one has LWP in $H^s\,$ for $s > n/2+1\,$ DiWgy2001
• When the target is is $S^2\,$ this is in SucSupBds1986.

Some further discussion on this equation can be found in the survey Di2002.