Schrodinger:Schrodinger maps

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

iu_t + 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 u_t. When the target is S², this equation arises naturally from the Landau-Lifschitz equation for a macroscopic ferromagnetic continuum, see e.g. references:SucSupBds1986 SucSupBds1986; in this case the equation has the alternate form u_t = 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 references:Im1984 Im1984 (see references:KnPoVe2000 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

iv_t + 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] references:ZkTkh1979 ZkTkh1979, references:Di1999 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 references:CgSaUh2000 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 references:DiWgy1998 DiWgy1998. The constant curvature assumption was relaxed to non-positive curvature (or Hermitian locally symmetric) in references:PaWghWgy2000 PaWghWgy2000. It is conjectured that one should have a global flow whenever the target is compact Kahler references:Di2002 Di2002.
    • When the target is S² this is in references:ZhGouTan1991 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 references:CgSaUh2000 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 references:NdStvUh2003 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 references:KnPoVe1993c KnPoVe1993c, [HaHr-p], and local existence for smooth solutions references:SucSupBds1986 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 references:DiWgy2001 DiWgy2001
  • When the target is is S² this is in references:SucSupBds1986 SucSupBds1986.

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

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