Schrodinger maps

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Schrodinger maps

[Many thanks to Andrea Nahmod for help with this section - Ed.]

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 S2 or hyperbolic space H2). They have the form

iut + D u = 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 ut. When the target is S2, 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 ut = u x D 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 S2 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 [#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

ivt + D v = Dv D-1(v v) + D-1(v v) D-1(v v) v + v3

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 problems) 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 S2, the equation is equivalent to the [#Cubic_NLS_on_R cubic NLS] ZkTkh1979, Di1999.

As with [wave:wm 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.
  • 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 Hs for s > 2 NdStvUh2003 (plus later errata), at least when the target manifold is the sphere S2. It would be interesting to extend this to lower regularities, and eventually to the critical H1 case. (Here regularity is stated in terms of u rather than the derivatives v).
    • When the target is S2 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 Hs for s > n/2+1 DiWgy2001

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