Mathematical Physics, Analysis and Geometry - Volume 8

381 Pages·2005·4.73 MB·English

by  A. Boutet de Monvel, V. A. Marchenko, H. McKean (Editors)

Mathematical Physics, Analysis and Geometry (2005) 8: 1–39 © Springer 2005
DOI: 10.1007/s11040-004-1670-2
Symplectic Structures for the Cubic Schrödinger
⋆
Equation in the Periodic and Scattering Case
K. L. VANINSKY
Department of Mathematics, Michigan State University, MI 48824, East Lansing, U.S.A.
e-mail:
2 K. L. VANINSKY
and
2
λ
2 ′
V3(x, t) = iσ3 − λY0 + |ψ| iσ3 − iσ3Y
0.
2
The corresponding auxiliary linear problem
( )
f1
(∂x − V )f = 0, f =
f2
can be written in the form of an eigenvalue problem for the Dirac operator
[( ) ( )]
1 0 0 −iψ λ
Df = i∂x + f = f .
0 −1 iψ 0 2
Another important feature of soliton systems is the Hamiltonian formulation.
Here we assume that the potential ψ(x, t) is 2l-periodic: ψ(x + 2l, t) = ψ(x, t).
For instance, the NLS flow can be written as
•
ψ = {ψ, H3},
∫
1 l ′ 2 4
with Hamiltonian H3 =
2 −l |ψ | + |ψ| dx = energy and bracket
∫
l
∂A ∂B ∂A ∂B
{A, B} = 2i − dx.
−l ∂ψ(x) ∂ψ(x) ∂ψ(x) ∂ψ(x)
The bracket is nondegenerate. The corresponding symplectic form (up to a scalar)
is:
∫
l
1
ω0 = 2i⟨δψ ∧ δψ⟩, ⟨•⟩ = dx.
2l −l
A priori it is not clear why the dynamical system, which arises as a compatibility
condition has a Hamiltonian formulation. To put it differently, is it possible to
obtain Hamiltonian formalism from the spectral problem?
Here we would like to make some historical remarks. Originally, the Hamil-
tonian formulation of basic integrable models was found as an experimental fact.
For the KdV equation the computation of symplectic structure in terms of the
scattering data was performed by Faddeev and Zakharov [5]. It involved some
nontrivial identities for the products of solutions. Later Kulish and Reiman [14]
noted that all higher symplectic structures also can be written in terms of the scat-
tering data. Again, they used the scheme of [5] and explicit calculations. Finally,
we note that Zakharov and Manakov [28] for the NLS equation adopted a different
approach. Instead of the symplectic structure they worked with the corresponding
Poisson bracket. Again, using explicit formulas for the product of solutions they
computed the Poisson bracket between the coefficients of the scattering matrix. An
appearance of explicit formulas that are the moving force of all these computations
SYMPLECTIC STRUCTURES FOR THE CUBIC SCHRÖDINGER EQUATION 3
seems to be quite mysterious. This was already discussed in the literature [4], and
described as a “computational miracle”.
The standard assumption needed to carry out spectral analysis is that the po-
tential either is periodic or has rapid decay at infinity. We refer to the latter case
as scattering. Recently, in connection with the Seiberg–Witten theory [23, 24],
Krichever and Phong [13] developed a new approach for the construction of sym-
plectic formalism. The latest exposition of their results can be found in [3]. The
main idea of the Krichever–Phong approach is to introduce in a universal way the
two-form on the space of auxiliary linear operators. This form is written in terms
of the operator itself and its eigenfunctions. The goal of this paper is to review the
Krichever–Phong approach in the case of 1D periodic NLS and to extend it to the
scattering case. Within the unified approach, we reduce the number of formulas
and eliminate unnecessary explicit computations. For instance, computation of the
symplectic form in terms of the spectral data (both in the periodic and the scattering
case) becomes an application of the Cauchy residue theorem.
1.2. THE PERIODIC CASE
We assume that the potential is periodic with the period 2l: ψ(x +2l, t) = ψ(x, t).
The Krichever–Phong formula, in the NLS context, takes the form
∑
∗
ω0 = res⟨e J δV ∧ δe⟩ dλ.
P±
This formula defines a closed 2-form ω0 on the space of operators ∂x − V (x, λ)
⋆
with 2l periodic potential. The set-up for this formula is broadly as follows.
The eigenvalue problem
( )
0 1
[J ∂x − JV (x, λ)]e(x, λ) = 0, J = iσ2 =
−1 0
has special solutions, so-called Floquet solutions determined by the property e(x +
2l, λ) = w(λ)e(x, λ). The complex constant w(λ) is called a Floquet multiplier.
For each value of the spectral parameter λ there are two linearly independent
Floquet solutions and two distinct Floquet multipliers. These solutions and cor-
respondingly multipliers become single-valued functions of a point on the two-
sheeted covering of the plane of spectral parameter λ. The simple points of the pe-
riodic/antiperiodic spectrum of the eigenvalue problem constitute branching points
of the cover. We assume that there is a finite number of simple points (so-called
finite gap potential).
This two sheeted covering constitutes a hyperelliptic Riemann surface Ŵ with
two infinities P+ and P− (Figure 1). Each point Q = (λ, ±) of Ŵ is specified by
the value of spectral parameter λ and the sheet “+” or “−” which determines the
⋆
We refer to Section 2 for detailed discussion.
4 K. L. VANINSKY
Figure 1. Smooth Riemann surface Ŵ.
Floquet multiplier w(Q) corresponding to this λ. At every point of the curve we
also have a Floquet solution e(x, Q) which becomes a function of the point Q and
satisfies the identity e(x + 2l, Q) = w(Q)e(x, Q). The Floquet solution e(x, Q)
has an exponential singularity at infinities and plays the role of so-called Baker–
Akhiezer function for the curve Ŵ.
∗
At every point of the curve Ŵ we can define another solution e (x, Q). This is
the Floquet solution which is brought from a point on the different sheet but with
the same value of the spectral parameter λ. It is transposed and suitably normalized.
∗
The operator J ∂x − JV (x, λ) acts on the solution e (x, Q) as an adjoint, i.e. on
the right:
∗
e (x, Q)[J ∂x − JV ] = 0.
It is assumed that the phase space consists of smooth 2l-periodic functions ψ(x)
or equivalently operators ∂x − V (x, λ) with 2l-periodic potential. The NLS flow
acts on this space as it acts on the space of functions ψ(x). All notions of differen-
tial geometry with obvious conventions can be applied to this space of operators.
On the space of potentials we have a variation δψ(x). Thus for a fixed value of
the spectral parameter λ we have well defined variation δV (x, λ). The variations
∗
δe(x, Q), δe (x, Q) are defined correctly when λ = λ(Q) is fixed. Therefore,
at each point Q of the surface Ŵ we have well defined meromorphic in Q the
two-form
∗
⟨e J δV ∧ δe⟩ dλ.
It takes values in the space of skew-symmetric two-forms on the space of operators
∂x − V . The result of Krichever and Phong states that the sum of residues of this
form at infinities P± is nothing but the symplectic form ω0.
The formula has a lot of good properties. First, it produces all higher symplectic
n
structures by introducing the weight λ under the residue
∑
n ∗
ωn = res λ ⟨e J δV ∧ δe⟩ dλ, n = 1, 2, . . . .
P±
SYMPLECTIC STRUCTURES FOR THE CUBIC SCHRÖDINGER EQUATION 5
Second, it easily leads to the Darboux coordinates, or in physics terminology the
separation variables, see Sklyanin [25]. These are local coordinates where the
symplectic form ω0 takes the simple canonical form
∑
2
ω0 = δp(γk) ∧ δλ(γk).
i
k
This merits special explanation. It is well known since the work of Flashka and
McLaughlin [6], that the poles γk of Floquet solutions lead to the Darboux co-
⋆
ordinates for symplectic forms. Recently, a lot of work was performed [15] to
construct such variables for the Ruijsenaars–Sneider and the Moser–Calogero sys-
tems. This required formidable technical machinery and extensive computations.
At the same time, as it was demonstrated by Krichever [11], the formula leads to
the same result only by applying the Cauchy residue theorem.
1.3. THE SCATTERING CASE
The main goal of the present paper is to show that suitably interpreted the new
approach can be adopted for soliton systems with rapidly decaying initial data on
⋆⋆
the entire line. This is the so-called scattering case.
For such potentials one can define so-called Jost solutions J±(x, λ). These are
′
matrix solutions of the auxiliary linear problem J
± = V J± with the asymptotics
′ λ (1) (2)
J
±(x, λ) = exp (−i 2xσ3) + o(1), as x → ±∞. Their columns J± = [j± , j± ]
are analytic in the corresponding upper/lower half-plane.
Our construction of the associated Riemann surface Ŵ∞ is a geometrical inter-
pretation of what is called the Riemann–Hilbert approach to the scattering problem,
see [4]. A singular curve Ŵ∞ is obtained by taking two copies of the complex plane
and gluing them to each other along the real line (Figure 2). The curve Ŵ∞ has two
infinities P+ and P− and continuum set of singular points above the real line. The
standard Jost solutions are lifted on Ŵ∞ and become the single valued function of a
point on the curve. Different branches of BA function are connected along the real
line by the scattering matrix S:
[ ]
1 1 b
S(λ) = .
a −b 1
The Jost solution has exponential singularity at infinities and plays the role of the
Baker–Akhiezer function for the curve Ŵ∞. This construction is explained in detail
in Section 3.
The formula of Theorem 3.4 looks similar to the periodic case
1 ∗ ∗
ω0 = trace res
2 [⟨H+JδV ∧ δH+⟩ + ⟨H−JδV ∧ δH−⟩] dλ.
⋆
See also Novikov and Veselov [22], for general discussion.
⋆⋆
We refer to Section 3 for detailed definitions.
6 K. L. VANINSKY
Figure 2. Singular Riemann surface Ŵ∞.
The only difference now is that we work with the matrix solutions
[ ] [ ]
(1) (2) (1) (2)
H+(λ) = j
− (λ), j+ (λ) and H−(λ) = j+ (λ), j− (λ)
+ T + T
and H
+ (λ) = σ1H+ , H− (λ) = σ1H− , with
( )
0 1
σ1 = .
1 0
The averaging now corresponds to the integration on the entire line
∫
+∞
⟨•⟩ = dx.
−∞
The residue can be computed explicitly ω0 = 2i⟨δψ ∧ δψ⟩. Theorem 3.6 states
that the symplectic structure can be put in the Darboux form
∫
+∞ ¯
1 δb(λ) ∧ δb(λ)
ω0 = dλ,
2
πi −∞ |a(λ)|
where a and b are coefficients of the scattering matrix S. Again identically to
the periodic case this result is obtained by applying the Cauchy residue theorem.
Only now the sum of the residues in the affine part of the curve is replaced by its
continuous analog. This is the integral which stays in the right hand side of the
formula.
The unified approach to construction of symplectic forms produces an interest-
ing problem. As we see, the symplectic form constructed in the periodic case has
two systems of Darboux coordinates. One system is associated with poles of the
Floquet solution. It is the divisor-quasimomentum Darboux coordinates. Another
system of Darboux coordinates is the action-angle variables. At the same time in
the scattering case we know only one system of Darboux coordinates. These are
action-angle variables. What is the correct analog of the divisor-quasimomentum
in the scattering case? This is a subject of future publication [27].
SYMPLECTIC STRUCTURES FOR THE CUBIC SCHRÖDINGER EQUATION 7
2. The Periodic Case
2.1. THE DIRECT SPECTRAL PROBLEM
We provide here information needed in the next section for construction of sym-
plectic forms. We refer to classical books [20, 21] for standard facts of spectral
theory and algebraic-geometrical approach to solitons.
The NLS equation
• ′′ 2
iψ = −ψ + 2|ψ| ψ, (2.1)
where ψ(x, t) is a smooth complex function 2l-periodic in x, is a Hamiltonian
system
•
ψ = {ψ, H},
∫
1 l ′ 2 4
with the Hamiltonian H = |ψ | + |ψ| dx = energy and the bracket
2 −l
∫
l
∂A ∂B ∂A ∂B
{A, B} = 2i − dx.
−l ∂ψ(x) ∂ψ(x) ∂ψ(x) ∂ψ(x)
The NLS Hamiltonian H = H3 is one in the infinite series of conserved integrals
of motion.
∫
l
1
2
H1 = |ψ| dx,
2 −l
∫
l
1
′
H2 = ψψ¯ dx,
2i −l
∫
l
1
′ 2 4
H3 = |ψ | + |ψ| dx, etc.
2 −l
tXm
These Hamiltonians produce an infinite hierarchy of flows e , m = 1, 2, . . . .
tX1
The first in the hierarchy is the phase flow e generated by the vector field
•
X1: ψ = {ψ, H1} = −iψ.
The phase flow is a compatibility condition for
[∂t − V1, ∂x − V2] = 0, (2.2)
⋆ i
with V1 = σ3 and
2
( ) ( )
iλ
iλ − 0 0 ψ
2
V2 = − σ3 + Y0 =
iλ + .
2 0 ψ 0
2
⋆
Here and below σ denotes the Pauli matrices
( ) ( ) ( )
0 1 0 −i 1 0
σ1 = , σ2 = , σ3 = .
1 0 i 0 0 −1
8 K. L. VANINSKY
tX2
We often omit the subscript V = V2. The second, translation flow e generated
by
• ′
X2: ψ = {ψ, H2} = ψ
is equivalent to (2.2) with V1 replaced by V2. Finally, the third, original NLS
flow (2.1) is a compatibility condition for (2.2) with V1 replaced by
2
λ
2 ′
V3 = iσ3 − λY0 + |ψ| iσ3 − iσ3Y
0.
2
tXm
All flows of infinite hierarhy e , m = 1, 2, . . . commute with each other
[∂τ
m − Vm, ∂τn − Vn] = 0.
The first times τ1, τ2 and τ3 correspond to the first three flows.
We introduce a 2 × 2 transition matrix M(x, y, λ), x ⩾ y; that satisfies
′
M (x, y, λ) = V (x, λ)M(x, y, λ), M(y, y, λ) = I.
The solution is given by the formula
∫
x
M(x, y, λ) = exp V (ξ, λ) dξ.
y
The matrix M(x, y, λ) is unimodular because V is traceless.
The symmetry
¯
σ1V (x, λ)σ1 = V (x, λ)
produces the same relation for the transition matrix
σ1M(x, y, λ)σ1 = M(x, y, λ¯ ). (2.3)
Another symmetry
T
V (x, λ)J = −JV (x, λ),
where J = iσ2, implies
T −1
M (x, y, λ) J = JM(x, y, λ). (2.4)
1
The quantity (λ) = trace M(l, −l, λ) is called a discriminant. The for-
2
mula (2.3) implies (λ) = ( λ¯ ) and (λ) is real for real λ. The eigenvalues
of the monodromy matrix have a name of Floquet multipliers and they are roots of
the quadratic equation
2
w − 2w + 1 = 0. (2.5)
SYMPLECTIC STRUCTURES FOR THE CUBIC SCHRÖDINGER EQUATION 9
√
2
The Floquet multipliers are given by the formula w = ± − 1. The values
of λ : w(λ) = ±1 constitute the points of the periodic/antiperiodic spectrum. The
corresponding auxiliary linear problem
T
(∂x − V )f = 0, f = (f1, f2);
can be written in the form of an eigenvalue problem for the self-adjoint Dirac
operator
[( ) ( )]
1 0 0 −iψ λ
Df = i∂x + f = f .
0 −1 iψ 0 2
The self-adjointness implies that points of the spectra are real.
EXAMPLE. Let ψ ≡ 0. The corresponding monodromy matrix can be easily com-
λ
puted M(x, y, λ) = e−i 2 σ3(x−y). We have (λ) = cos λl and double eigenvalues
± πn ±
at the points λ = . If n is even/odd, then the corresponding λ belongs to the
n l n
periodic/anti-periodic spectrum.
±
For a generic potential the double points λ of the periodic/anti-periodic spec-
n
trum split, but they always stay real. The size of the spectral gap is determined,
roughly speaking, by the corresponding Fourier coefficients of the potential. In our
considerations we assume that there is a finite number of g + 1 open gaps in the
spectrum
− + − + − + −
· · · < λ = λ < λ < λ < · · · < λ < λ < λ
n−1 n−1 n n n+g n+g n+g+1
+
= λ < · · ·
n+g+1
These are so-called finite gap potentials which are dense among all potentials.
The Floquet multipliers become single-valued on the Riemann surface:
2
Ŵ = {Q = (λ, w) ∈ C : R(λ, w) = det[M(l, −l, λ) − wI] = 0}.
The Riemann surface consists of two sheets covering the plane of the spectral
parameter λ.
EXAMPLE. Let ψ ≡ 0. We have (λ) = cos λl and quadratic equation (2.5) has
±ilλ
the solutions w(λ) = e . The Riemann surface Ŵ = Ŵ+ + Ŵ− is reducible and
consists of two copies of the complex plane C that intersect each other at the points
±
of the double spectrum λ
n . Each part Ŵ+ or Ŵ− contains the corresponding infinity
P+ or P−. The Floquet multipliers are single valued on Ŵ:
+iλl
w(Q) = e , Q ∈ Ŵ+;
−iλl
w(Q) = e , Q ∈ Ŵ−.
For a finite gap potential the Riemann surface Ŵ is irreducible. There are three
types of important points on Ŵ. These are the singular points, the points above
λ = ∞ and the branch points which we discuss now in detail.
10 K. L. VANINSKY
• The singular points are determined by the condition
∂λR(λ, w) = ∂wR(λ, w) = 0.
±
These are the points (λ , ±1) of the double spectrum. At these points two
sheets of the curve intersect.
⋆
• There are two nonsingular points P+ and P− above λ = ∞. At these points
+iλl
w(Q) = e (1 + O(1/λ)), Q ∈ (P+); (2.6)
−iλl
w(Q) = e (1 + O(1/λ)), Q ∈ (P−). (2.7)
• The branch points are specified by the condition
∂wR(λ, w) = 0.
They are different from the singular points and correspond to the simple pe-
± ± k
riodic/antiperiodic spectrum. We denote these points by s = (λ , (−1) ),
k k
k = n, . . . , n+g. There are 2(g+1) of them, each has a ramification index 2.
The desingularized curve Ŵ is biholomorphicaly equivalent to a hyperelliptic curve
with branch points at the points of the simple spectrum. We also denote the hyper-
ellitic curve by Ŵ. The Riemann–Hurwitz formula for the genus of Ŵ implies
R
genus = − n + 1,
2
where R is a total ramification index and n is the number of sheets. Each branch
point has a ramification index 1 and therefore R = 2(g + 1) and n = 2. Therefore,
the genus of Ŵ is g, one off the number of open gaps in the spectrum.
Let ϵ± be a holomorphic involution on the curve Ŵ permuting sheets
ϵ±: (λ, w) −→ (λ, 1/w).
The fixed points of ϵ± are the branch points of Ŵ. The involution ϵ± permutes
infinities ϵ± : P− → P+. Let us also define on Ŵ an antiholomorphic involution
ϵa: (λ, w) −→ (λ¯ , w¯ ).
The involution ϵa also permutes infinities and commutes with ϵ±. Points of the
− +
curve above gaps [λ
n , λn ] where |(λ) | ⩾ 1 form g + 1 fixed “real” ovals of ϵa.
We call them a-periods.
The quasimomentum p(Q) is a multivalued function on the curve Ŵ. It is intro-
ip(Q)2l πn
duced by the formula w(Q) = e . Evidently, it is defined up to , where n is
l
an integer. The asymptotic expansion for p(Q) at infinities can be easily computed
λ
± p1 p2
±p(λ) = − p
0 − − 2 . . . , Q ∈ (P±), λ = λ(Q),
2 λ λ
± πk±
where p
0 = l , k± is an integer and
1 1 1
p1 = H1, p2 = H2, p3 = H3, etc.
l l l
⋆
The notation Q ∈ (P ) means that the point Q is in the vicinity of the point P .