Table Of ContentEFFECTIVE INTEGRATION OF THE NONLINEAR
VECTOR SCHRO¨DINGER EQUATION
J N ELGIN, V Z ENOLSKI, AND A R ITS
5
0
0 Abstract. Acomprehensivealgebro-geometricintegrationofthe
2
two component Nonlinear Vector Schr¨odinger equation (Manakov
n system) is developed. The allied spectral variety is a trigonal Rie-
a mannsurface,whichisdescribedexplicitlyandthesolutionsofthe
J
equationsaregivenintermsofθ-functionsofthesurface. Thefinal
0
formulaeareeffectiveinthatsensethatallentriesliketranscenden-
3
talconstantsinexponentials,windingvectorsetc. areexpressedin
1 termsofprime-formofthecurveandwellalgorithmizedoperations
v on them. That made the result available for direct calculations in
2 applied problems implementing the Manakov system. The sim-
7
plest solutionsin Jacobianϑ-functions aregivenas particularcase
0
of general formulae and discussed in details.
1
0
5
0
/
h Contents
p
- 1. Introduction 1
h
t 2. Zero-curvature representation 3
a
m 3. The spectral curve 5
: 4. Differentials and Integrals 9
v
4.1. Holomorphic differentials and integrals 9
i
X
4.2. Meromorphic differentials and integrals 11
r
a 4.3. θ-function and prime-form 15
4.4. θ-functional construction of meromorphic integrals 18
5. Algebro-geometric solutions of the Manakov system 23
6. Example: solution in elliptic functions 31
7. Summary: computational algorithm 34
8. Appendix 36
Acknowledgements 39
References 40
1. Introduction
The Vector Nonlinear Schr¨odinger equation (VNSE) usefully models
the propagationof a polarized optical beam along an optical fiber. The
1
2 J N ELGIN, V Z ENOLSKI,AND A R ITS
vector nature of the dependent variable models the polarization state
of the beam. It is intended in this article to derive and investigate a
general class of periodic and quasi-periodic solutions of this equation.
As the spectral curve for this system is trigonal – rather then hyper-
elliptic as for the scalar case – existing formulae for these solutions
are rather formal and not tractable for applications. Here, we use a
method first devised by Krichever [Kri77] to effect an explicit integra-
tionof the VNSE. This approach permits us to investigate some special
caseswheretheformalsolutionsthusobtainedreduced tosimpler types
expressible in terms of hyperelliptic or elliptic functions.
In this paper we shall consider the integrable 2 dimensional focusing
Vector Nonlinear Schr¨odinger equation (VNSE)
∂q ∂2q
(1.1) i 1 + 1 +2 q 2 + q 2 q = 0,
∂t ∂x2 | 1| | 2| 1
∂q ∂2q (cid:0) (cid:1)
(1.2) i 2 + 2 +2 q 2 + q 2 q = 0.
∂t ∂x2 | 1| | 2| 2
It was proven by Manakov [Man(cid:0)74] that thi(cid:1)s system is completely in-
tegrable and, in consequence, (1.1,1.2) are now known as the Manakov
system.
Manakov’s method is based on the Lax representation
(1.3) φ = Mφ,
x
(1.4) φ = Bφ,
t
where
iz q q
1 2
−
(1.5) M(z) = q¯ iz 0 .
1
−
q¯ 0 iz
2
−
and
(1.6)
2z2 q 2 q 2 2iq z q 2iq z q
1 2 1 1x 2 2x
−| | −| | − −
B(z) = i 2iq¯ z q¯ 2z2 + q 2 q¯ q ,
1 1x 1 1 2
− − − − | |
2iq¯ z q¯ q q¯ 2z2 + q 2
2 2x 1 2 2
− − − | |
where bar denotes complex conjugation. The Manakov system can be
represented in the form
(1.7) B M = [M,B].
x t
−
The simplest solution Manakov’s soliton has the form
(1.8) q (x,t) = 2ηsech(2η(x+4ξt))
sol
exp 2iξx 4i(ξ2 η2)t c,
× {− − − }
EFFECTIVE INTEGRATION OF THE VNSE 3
where c = (c ,c )T is a unit vector, c 2 + c 2 = 1, independent of
1 2 1 2
| | | |
both x and t, and ξ and η are real constants.
Periodic and quasi-periodic solutions expressed in terms of explicit
θ-functional formulae have been quoted by several authors. The one
component case, i.e. standard nonlinear Schr¨odinger equation was de-
veloped in [Its76], [IK76], and [Pre85] (see also monograph [BBE+94]).
The multi-component case was studied in [Kri77, AHH90]. while the
special case of reduction to a dynamical system with two degree of
freedom was studied in [CEEK00]. In recent years attention has been
directed to modulation instabilities of the multi-component equation
and searching for homoclinic orbits [FSW00], [FMMW00], [WF00]. Al-
though we are not touching this interesting and important subject we
believe that effective θ-functional formulae could shed some new light
on it. Indeed, we believe that they will be as useful for studying homo-
clinic orbits of the Manakov model as the one component θ-functional
formulae are for studying the homoclinic orbits of the standard nonlin-
ear Schr¨odinger equation (see Sections 4.4 and 4.5 of [BBE+94]).
The article is organized as listed in Contents. The work is a mixture
of analysis and computer algebra implementations using the Maple
code described in [DvH01].
2. Zero-curvature representation
Denote by t = x,t = t,...,t ,... a set of “times” and introduce
1 2 n
the set of 3 3 matrices L (z),L (z),...,L (z),... satisfying the zero
1 2 n
×
curvature representation,
∂ ∂
(2.1) L (z) L (z) = [L (z),L (z)],
j i i j
∂t − ∂t
i j
where the matrices L and L are chosen to satisfy the Lax representa-
1 2
tion (1.3)- (1.6). More generally, L (z) is expanded as the n-th degree
n
polynomial
(2.2) L (z) = (2z)nL +(2z)n−1L +...+(2z)L +L , n = 1,...,
n 0 1 n−1 n
having the property
(2.3) L†(z¯) = L (z),
n − n
where dagger denotes conjugate transpose: c† = c¯T.
†
Here
1 i 0T 0 qT
L = −2 , L = ,
0 0 1 i1 1 q¯ 0
(cid:18) 2 2 (cid:19) (cid:18) − 2 (cid:19)
4 J N ELGIN, V Z ENOLSKI,AND A R ITS
qTq¯ qT
L = i x ,
2 q¯ q¯qT
(cid:18) x − (cid:19)
where
q = (q ,q )T,
1 2
while, for k > 2 introduce the following ansatz
α βT
(2.4) L = k k−1 +L(0).
k γ A k
(cid:18) k−1 k (cid:19)
In equation (2.4) A denotes a 2 2-matrix and L(0) denotes a constant
× k
matrix of the form
ck 0 0
1,1
(2.5) L(0) = 0 ck ck
k 2,2 2,3
0 ck ck
3,2 3,3
with arbitrary entries ck . In this article we set L(0) = 0.
p,q k
The following theorem is valid
Theorem 2.1. The entries to the matrix (2.4) in the zero-curvature
representation are defined as follows
the vectors β and γ are given by the equations
• n n
β = (iD)nq, γ = β¯ ,
n n n
−
where D acts on a vector f(x) as
x
∂
(2.6) Df(x) = f(x)+ q(x′)†,f(x′) dx′q(x),
∂x A
Z
. (cid:8) (cid:9)
where , denotes the matrix which is the anti-hermitian part
A
{· ·}
of anticommutator, so that
(2.7) a†,b = a†b b†a 1 +ba† ab†.
A 2
{ } − −
Therefore, the flows(cid:0)are defined(cid:1) as
∂
(2.8) q q = i (iD)nq.
n ≡ ∂t
n
The (1,1) element of the matrix L is defined recursively as
k+2
•
follows
k k−2
(2.9) α = i γT β i α α ,
k+2 − k−j j − k−j j+2
j=0 j=0
X X
EFFECTIVE INTEGRATION OF THE VNSE 5
with
i
α = , α = 0, α = iqTq¯,
0 1 2
−2
while the associated right lower 2 2 minor A is given re-
k+2
×
cursively as
k k−2
(2.10) A = i γ βT + i α A
k+2 k−j j k−j j+2
j=0 j=0
X X
with
i
A = 1 , A = 0, A = iq¯qT.
0 2 1 2
2 −
In each case, contributions from the second sum appear only for k 2.
≥
Proof. Theproofofthese results followsfromthesubstitution ofansatz
(2.2), (2.4) into (2.1) with i = 1 and j = n and solving the equation
recursively. Also, one has to take into account that ∂ Tr(L2) = 0. (cid:3)
∂ti n
3. The spectral curve
The spectral curve is fixed by defining the stationary flow as fol-
lows: let the system depend only on times t ,...,t . Then the zero
1 n−1
curvature representation (2.1) written for L and L has the form
1 n
∂
(3.1) L (z)[L (z),L (z)].
n 1 n
∂x
This relation suggests we consider the polynomial equation
(3.2) f(z,w) = 0, f(z,w) = det(L (z) w1 ).
n 3
−
We shall call the polynomial equation
(3.3) X := (z,w) f(z,w) = 0 .
{ | }
the spectral curve. Evidently coefficients of monomials zkwl of the
polynomial f(z,w) are constants of motion. In what follows we shall
consider the Riemann surface of the curve X, which we shall denote
by the same letter.
To proceed we recall that any rational function of its arguments,
φ(z,w) is called a function on the curve f(z,w) = 0. The order of
the function φ(z,w) on the curve X is the number N of common zeros
(z ,w ),...,(z ,w ) of equations f(z,w) = 0 and φ(z,w) = 0. The
1 1 N N
curve is hyperelliptic if it admits a function of the second order, it is
trigonal if it admits a function of third order etc.
In the case considered, the spectral curve can be written in the ex-
plicit form as
X = (z,w) f(z,w) = 0 ,
{ | }
6 J N ELGIN, V Z ENOLSKI,AND A R ITS
i i
f(z,w) = (w+ (2z)n)(w (2z)n)2
2 − 2
2n−1 n−2
i
(3.4) +(w (2z)n) λ (2z)2n−j−1 + µ (2z)j,
j n−2−j
− 2
j=n j=0
X X
where 2n 1 parameters λ i = n,...,n 1 and µ , j = 0,...,n
i j
− − −
2 are constants of motion and can be taken arbitrary, but satisfying
conditions given below in (3.9). The coordinate z of the curve is a
function of the third order and therefore the curve is trigonal.
The parameters λ of the curve X can be computed in terms of q as
j
follows:
n−1
1
λ = γTβ
j −2 k j−k−1
k=j−n
X
n−2
1
(3.5) α α , j = n,...,2n 1.
k+2 j−k−1
− 2 −
k=j−n−1
X
In particular,
(3.6) λ = α α ,
n 0 n+1
1 1
(3.7) λ = α α + γTβ + γTβ .
n+1 0 n+2 2 n 0 2 0 n
The structure of the second term in (3.4) has been obtained analyti-
cally, including the stated expressions for λ . By contrast,information
j
concerning the final term has been obtained using Maple, which gives
the polynomial structure of degree n 2 indicated.
−
It follows from (2.3) that the curve X admits the anti-involution
property
(3.8) σ : X X, where σ : (z,w) (z¯, w¯)
−→ → −
That implies in accordance with explicit formula for λ
i
¯
λ = λ , i = n,...,2n 1,
i i
(3.9) −
µ¯ = µ , j = 0,...,n 2.
j j
− −
Therefore we have
(3.10) σ f(z,w) = f(z¯, w¯) = f(z,w),
◦ − −
what means that the curve X has required anti-involution property.
Let us clarify now the question on the genus g of the curve X.
(0)
Lemma 3.1. Let L = 0 and the curve X is given by the equation
n
(3.4) with parameters λ ,µ in general position. Then the genus of X
i j
EFFECTIVE INTEGRATION OF THE VNSE 7
is given by the formula
(3.11) g = 2n 3.
−
Proof. Write equation (3.4) in the form
i i
f(z,w) = (w+ (2z)n)(w (2z)n)2
2 − 2
i
(3.12) +(w (2z)n)P (z)+P (z) = 0,
n−1 n−2
− 2
where P (z) and P (z) are polynomials of degrees n 1 and n 2
n−1 n−2
− −
correspondingly. The discriminant of (3.12) be of the form
∂
Discriminant(X) = Resultant f(z,w), f(z,w),w
∂w
(cid:18) (cid:19)
= 256iP (z)z3n +16P (z)2z2n +27P (z)2
n−2 n−1 n−2
+72iP (z)P (z)zn +16P (z)2z2n +4P (z)3.
n−2 n−1 n−1 n−1
The degree in z of the Discriminant(X) be 4n 2 because coefficient
−
of the leading power be
(3.13) λ2 4iµ = 0, n = 2,3,...
n − 0 6
for λ ,µ in general position. Moreover for general values of parameters
i j
the Discriminant(X) has no multiple roots and all zeros are simple
branch points of the curve X. Beside of that we remark that the curve
X has no branch points at infinities, , , Therefore the curve
1 2 3
∞ ∞ ∞
X has 4n 2 simple branch points altogether, which we will denote e ,
1
−
e , ..., e . The application of the Riemann-Hurwitz formula
2 4n−2
B
(3.14) g = N +1,
2 −
where B is total branch number, being equal in the case 4n 2 and N
−
is the number of sheets of the cover over Riemann sphere, which is 3
(cid:3)
in the case, completes the proof.
We remark that our formula for genus (3.11) is addressed to the
concrete curve which isfixed forour analysis. The inclusion ofconstant
(0)
matrices L can increase the genus. The discrepancy of our formulae
k
with results of [AHH90] and [Wri99] is due to the fact that in there an
estimate of upper bound for genus was given for more general curve
then our be.
Introduce further the Riemann surface of the curve X. To do that
we define local coordinate ξ(P) of a point P = (x,y) X in vicinity
∈
8 J N ELGIN, V Z ENOLSKI,AND A R ITS
of another point P = (z,w) X as follows
∈
z +ξ if P = (z,w) is regular point,
(3.15) x = a+ξ2 if P = (a,w(a)) is branch point,
1 if P = ( , ) is regular point at infinity.
ξ ∞ ∞
To comment this definition we remark that for general values of param-
eters λ and µ the curve has only simple branch points with ramifica-
i i
tion number one, what leads to the structure of the second line of the
definition. The curve X has 3-sheeted structure with regular points at
infinities, , and where the coordinate of the curve behave as
1 2 3
∞ ∞ ∞
follows
(3.16)
1 2n−1i i
z = , w = λ ξ +O(ξ2) on the first sheet,
ξ − ξn − 2 n
1 2n−1i i
z = , w = + (λ + λ2 4iµ )ξ +O(ξ2) on the second sheet,
ξ ξn 4 n n − 0
1 2n−1i i p
z = , w = + (λ λ2 4iµ )ξ +O(ξ2) on the third sheet.
ξ ξn 4 n − n − 0
p
We shall also assume that the branch points are all complex, form
the conjugated pairs, i.e.
(3.17) e¯ = e , Ime < 0,
2k−1 2k 2k−1
and Re e < Re e .
2k−1 2k+1
We are in position now to introduce a suitable homology basis on
the Riemann surface of the curve (3.4). A canonical basis of cycles
a and b respecting intersection property a a = 0, b b = 0,
i i i j i j
a b = b a = δ which also respect the i◦nvolution pro◦perty
i j i j ij
◦ − ◦
(3.18) σ(a ) = a ,
j j
−
σ(b ) = b 2a a .
j j j k
− −
k6=j
X
The homology basis for the case g = 3 is shown in figure 1; here, the
solid, dashed and dash-dotted lines connecting points e to e etc. are
1 2
cuts connecting the first to second, second to third and third to first
sheets respectively. See caption for further comments. The homology
basis for higher genera can be plotted analogously.
EFFECTIVE INTEGRATION OF THE VNSE 9
e
4 e
e 6
2
e e
8 10
b
3
a
3
b
b 2
1
e e
a 7 9
e 2
5
e e a
1 3 1
Figure 1. Basisofcycles ofthecurveX ofgenus3. The
solid, dashed and dash-dotted lines denote paths on the
first, second and third sheets respectively. Correspond-
ingly the solid to dashed line, dashed to dot-dashed line,
and dot-dashed to solid lines illustrate trajectories pass-
ing through these cuts. The cuts are similarly encoded
for clarity.
12 23 31
Figure 2. Contours passing from sheet 1 to sheet 2,
from sheet 2 to sheet 3 and from sheet 3 to sheet 1
4. Differentials and Integrals
4.1. Holomorphic differentials and integrals. Let X be algebraic
curve (3.4) of genus g and let du(Q) = (du (Q),...,du (Q))T be the
1 g
set of canonical holomorphic differentials, which are given at n > 2
explicitly as
10 J N ELGIN, V Z ENOLSKI,AND A R ITS
izj−1
du (Q) = dz, j = 1,...,n 2,
j ∂ f(z,w) −
∂w
(4.1)
z2n−3−j w i(2z)n
du (Q) = − 2 dz, j = n 1,...,2n 3,
j ∂ f(z,w) − −
∂w(cid:0) (cid:1)
where f(z,w) be the polynomial defining the curve (3.4). At n = 2 the
curve is elliptic; this case is studied in detail in Section 6.
If du = φ(z,w)dz is an Abelian differential, then the action of the
involution σ, σ∗ say, is defined by the relation
σ∗du = φ(z¯, w¯)dz.
−
For the differentials du we have that
j
(4.2) σ∗du = du .
j j
−
Introduce the matrix of a-periods,
A = du .
k
aI
j j,k=1,...,g
From the properties (3.18) and (4.2) of σ∗ it follows that all elements
of the matrix A are real. Normalized form of the above differential is
introduced as
g
(4.3) dv = C du ,
j jl l
l=1
X
where the matrix C = A−1. Evidently,
(4.4) dv = σ∗dv .
j j
−
Now introduce τ-matrix, as a matrix of b-periods of normalized dif-
ferentials,
(4.5) τ = dv .
k
bI
j j,k=1,...,g
Then using (3.18) and (4.4) we find
τ = dv = σ∗dv = dv
jk k k k
− −
bIj bIj σ(Ibj)
= τ +2δ + δ .
jk jk lk
−
l6=j
X
