8 2. AN OUTLINE OF THE PAPER
which we will call the reduced field. The relevance of the variable ψ comes from the
following reinterpretation. If W is defined as the vector
W = ∂ru
m
r
u × Ru
Tu(S2)
then ψ is the representation of W with respect to the frame (v, w). On the other
hand, a direct computation, see for instance [8], leads to
E(u) = π

0
|∂r
¯|2
u +
m2
r2
u ×
R¯|2
u rdr = π
¯
WL2(rdr)
2
+ 4πm
where we recall that u(r, θ) = emθR ¯(r). u Therefore ψ = 0 is a complete character-
ization of u being a harmonic map. Moreover the mass of ψ is directly related to
the energy of u via
(2.2) ψ
2
L2
=
¯
WL2
2
=
E(u) 4πm
π
.
A second goal of the next section is to derive an equation for the time evolution
of ψ. This is governed by a cubic NLS type equation,
(2.3) (i∂t + Δ
2
r2
= A0 2
A2
r2

1
r
(ψ2
¯)
ψ ψ.
In addition, we show that ψ is connected back to (ψ2, A2) via the ODE system
(2.4) ∂rA2 =
¯
ψ
2
) +
1
r
|ψ2|2,
∂rψ2 = iA2ψ
1
r
A2ψ2
with the conservation law A2+|ψ2|2 2 = 1. However, this does not uniquely determine
(ψ2, A2) and, by extension, the Schr¨ odinger map u as we are missing a suitable
boundary condition.
2.3. Linearizations and the operators H,
˜
H
This is the point in our work where we specialize in the case m = 1 and, for
convenience, drop the upper-script m from all elements involved, i.e. use h1,h3
instead of h1,h3,
1 1
etc.
A key role in our analysis is played by the linearization of the Schr¨ odinger Map
equation around the soliton Q. A solution to the linearized flow is a function
ulin :
R2+1

TQS2.
The Coulomb frame associated to Q has the form
vQ(θ, r) =
eθR¯Q(r),
v wQ(θ, r) =
eθR
¯Q(r) w
with
¯Q(r) v =


h3(r)
0
−h1(r)


, ¯Q(r) w =


0
1
0


.
Expressing ulin in this frame,
φlin = ulin,vQ + i ulin,wQ
one obtains the Schr¨ odinger type equation
(2.5) (i∂t H)φlin = 0
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