CHAPTER 1

Introduction

In this article we consider the Schr¨ odinger map equation in R2+1 with values

into S2,

(1.1) ut = u × Δu, u(0) = u0

This equation admits a conserved energy,

E(u) =

1

2

R2

|∇u|2dx

and is invariant with respect to the dimensionless scaling

u(t, x) →

u(λ2t,

λx).

The energy is invariant with respect to the above scaling, therefore the Schr¨odinger

map equation in

R2+1

is energy critical.

Local solutions for regular large initial data have been constructed in [25] and

[18]. Low regularity small data Schr¨ odinger maps were studied in several works,

see [1], [2], [3], [10], [11], [13], [14], [15], [20], [21], [22]. The definitive result for

the small data problem was obtained by the authors and collaborators in [4]. There

global well-posedness and scattering are proved for initial data which is small in

the energy space

˙

H

1.

However, such a result cannot hold for large data. In particular there exists a

collection of families

Qm

of finite energy stationary solutions, indexed by integers

m ≥ 1. To describe these families we begin with the maps

Qm

defined in polar

coordinates by

Qm(r,

θ) =

emθR

¯m(r),

Q

¯m(r)

Q =

⎛

⎝

h1

m(r)

0

h3

m(r)

⎞

⎠

, m ∈ Z \ {0}

with

h1

m(r)

=

2rm

r2m + 1

, h3

m(r)

=

r2m

− 1

r2m + 1

.

Here R is the generator of horizontal rotations, which can be interpreted as a matrix

or, equivalently, as the operator below

R =

⎛

⎝

0 −1 0

1 0 0

0 0 0

⎞

⎠

, Ru =

− →

k × u

The families

Qm

are constructed from

Qm

via the symmetries of the problem,

namely scaling and isometries of the base space

R2

and of the target space

S2.

Q−m

generates the same family

Qm.

The elements of

Qm

are harmonic maps

from

R2

into

S2,

and admit a variational characterization as the unique energy

1