1. Introduction
Integrable discretizations of soliton equations have attracted much attention during the past few years. The celebrated Landau-Lifshits equation [2]
st=[s,sxx+Js], s∈ℝ3, J=diag(J1,J2,J3), J1+J2+J3=0,(1.1)
has two well-known integrable discretizations. One of them is the Sklyanin lattice which is determined by the Poisson brackets algebra (see (13) in [
7] or [
1])
{sα(n),s0(n)}=Jβγsβ(n)sγ(n), {sα(n),sβ(n)}=−s0(n)sγ(n),(1.2)
and the discrete variant of the Landau-Lifshits Hamiltonian (see (23) in [
7])
H(0)=∑nln(s0(n+1)s0(n)+∑α=13(K1K0−Jα)sα(n+1)sα(n)).(1.3)
Another discretization of the Landau-Lifshits equation (1.1) is the so-called Shabat-Yamilov lattice [5,6]
ut=2hu+−v+hv, vt=2hu−v−−hu,(1.4)
where
u =
u(
n,
t),
v =
v(
n,
t),
ut=du(n,t)dt,
vt=dv(n,t)dt,
u+ =
E+u =
u(
n + 1,
t),
v− =
E−b =
b(
n − 1,
t) (
E± are the shift operators and (
n,
t) ∈ × ℝ),
h =
h(
u,
v) is a symmetric polynomial of
u,
v degree no higher than two with respect to each of the variables and
huuu = 0. By introducing the complexified stereographic projection
S=S(u,v)=1u−v(1−uv,i+iuv,u+v), i=−1,(1.5)
one may relate the Shabat-Yamilov lattice
(1.4) to the Sklyanin lattice by a special transformation of variables [
1]. The linear combination of the flows of
(1.4) and its shift
ut=2hu−−v+hv, vt=2hu−v+−hu,(1.6)
give rise to [
5]
ut=ρ1(2hu+−v+hv)+ρ2(2hu−−v+hv),vt=ρ1(2hu−v−−hu)+ρ2(2hu−v+−hu),(1.7)
where
ρ1,
ρ2 are real constants. By specifying
ρ1 = 1,
ρ2 = 0,
h=12(u−v)2, the lattice
(1.7) reduces to the so-called Heisenberg ferromagnet (HF) lattice
{ut=(u−v)(u−u+)((u+−v)−1,vt=(u−v)(v−−v)(u−v−)−1.(1.8)
In the spin variables S (see (1.5)), the system (1.7) reads [1]
St=ρ1〈S,KS〉,([S,S+]1+[S,S+]+[S,S−]1+[S,S−])−2ρ1[S,KS]+ρ2〈S,KS〉(S+S+1+[S,S+]+S+S−1+[S,S−]),(1.9)
where
K = diag(
K1,
K2,
K3) and |
S| = 1. It is well-known that the lattice
(1.9) is integrable with a zero-curvature representation and can be reduced to the well-known Heisenberg lattice (see [
1] and references therein).
In this paper, we study quasi-periodic solutions of the whole Heisenberg ferromagnet hierarchy by using algebrogeometric method. In section 2, we construct the stationary and time-dependent HF hierarchy from its zero-curvature representation. Then spectral curves and an auxiliary function φ are introduced in section 3. In section 4, based on analytic and asymptotic properties of φ, we derive theta function representations for the entire HF hierarchy.
2. Heisenberg ferromagnet hierarchy
In this section, we construct the Heisenberg ferromagnet hierarchy by developing zero-curvature formulism of the HF lattice. For later use we denote by E± the shift operators acting on ψ={ψ(n,t)}n=−∞+∞∈ℂ according to (E± ψ)(n, t) = ψ(n±1, t), or E± ψ = ψ± for convenience. These notations are followed by [4].
It is well-known that the HF lattice has symplectic operator, Hamiltonian structure, recursion operator, nontrivial generalised symmetry and Lax representation [5]. To construct the HF hierarchy we start from the following two 2 × 2 Lax matrices
U=U(λ,u,v)=(λ−2u(u−v)−1−2(u−v)−12uv(u−v)−1λ+2v(u−v)−1),(2.1)
V=V(λ,u,v)=λ−1(u−v−)−1(u+v−2−2uv−−(u+v−)),(2.2)
where
λ ∈ ℂ is a spectral parameter and
u,
v are functions of the lattice variable
n and the time variable
t.
In the following we temporarily view u, v as functions of only n and define the sequence {aℓ, bℓ, cℓ}ℓ∈ℕ0
2δ(aℓ+aℓ+−ubℓ−vbℓ+)=bℓ−1+−bℓ−1, ℓ∈ℕ0,(2.3)
2uvδ(aℓ+aℓ++cℓ/u+cℓ+/v)=cℓ−1+−cℓ−1, ℓ∈ℕ0,(2.4)
2vδ(aℓ−aℓ+)−2uvδbℓ−2δcℓ+=−aℓ−1+aℓ−1+, ℓ∈ℕ0,(2.5)
where
aℓ,
bℓ,
cℓ are polynomials of
u,
v and their shifts, and
δ = (
u −
v)
−1. In matrix form, the relations
(2.3)–
(2.5) can be written as
(I+E+−u−vE+0I+E+0u−1+v−1E+I−E+−uv−1E+)(aℓbℓcℓ)=(0(2δ)−1(E+−I)000(2uvδ)−1(E+−I)E+−I00)(aℓ−1bℓ−1cℓ−1),(2.6)
where
I is the identity operator from ℂ
to ℂ
. Apparently, once the initial value (
a0,
b0,
c0) is given, {
aℓ,
bℓ,
cℓ}
ℓ∈ℕ can be recursively determined by the relations
(2.3)–
(2.5) or
(2.6). However, the calculations involved are rather big and therefore it is uneasy to obtain explicit form of {
aℓ,
bℓ,
cℓ}
ℓ∈ℕ. In spite of difficulties, we can still get the following result.
Theorem 2.1.
Solutions of the system (2.3)–(2.5) are explicitly given by the following recursion relations
aℓ=u+v−u−v−(E+−I)−1S1,ℓ−1, ℓ∈ℕ0,(2.7)
bℓ=S2,ℓ−1+2u−v−(E+−I)−1S1,ℓ−1, ℓ∈ℕ0,(2.8)
cℓ=S3,ℓ−1−2uv−u−v−(E+−I)−1S1,ℓ−1, ℓ∈ℕ0, (2.9)
where S1,ℓ−1,
S2,ℓ−1,
S3,ℓ−1 are defined by
S1,ℓ−1=uu2−(v−)2(v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ)−−v(u+)2−v2(u(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ)+−uu2−(v−)2u(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ+v(u+)2−v2v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ+bℓ−1+−bℓ−12δ,
S2,ℓ−1=1u2−(v−)2((v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ)−−u(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ),
S3,ℓ−1=uv−u2−(v−)2(uv−(v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ)−−v−uv(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ).
Proof.
First, from (2.3)–(2.5) it follows
2uδ(aℓ−aℓ+)=−2uvδbℓ+−2δcℓ+aℓ−1−aℓ−1+.(2.10)
Eliminating bℓ+ from (2.3) and (2.10), we have
2uδ(2aℓ−ubℓ+cℓu)=−u(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+.(2.11)
Then insertion of (2.3) into (2.5) yields
4vδaℓ+−2v2δbℓ++2δcℓ+=v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+,(2.12)
or equivalently,
2aℓ−v−bℓ+cℓv−=(v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ)−.(2.13)
Combining (2.11) with (2.13), we obtain
bℓ+1uv−cℓ=(v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2vδ)−−u(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2uδ(2.14)
bℓ−2aℓu+v−=1u2−(v−)2(v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2vδ)−−1u2−(v−)2×u(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2δ.(2.15)
Thus, inserting (2.15) into (2.3), we conclude (2.7) and (2.8) hold. Finally, the formula (2.9) can be derived from the following two equalities
2uaℓ−uv−bℓ+uv−cℓ=u(v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2vδ)−,(2.16)
2v−aℓ−uv−bℓ+v−ucℓ=v−(v(bℓ−1+−bℓ−1)+aℓ−1−aℓ−1+2uδ).(2.17)
by eliminating
bℓ.
Next we compute the stationary HF hierarchy. To this end we start from the stationary zero-curvature equation
0=UW−(EW)U, W=(ABC−A),(2.18)
where
A,
B,
C,
D are polynomials of
λ:
A=∑ℓ=1N+1aN+1−ℓλ−ℓ, B=∑ℓ=1N+1bN+1−ℓλ−ℓ, C=∑ℓ=1N+1cN+1−ℓλ−ℓ, N∈ℕ.(2.19)
Using (2.1), one may rewrite Eq. (2.18) as
(λ−2uδ)A−2δC=(λ−2uδ)A++2uvδB+,(2.20)
(λ−2uδ)B+2δA=−2δA++(λ+2vδ)B+,(2.21)
2uvδA+(λ+2vδ)C=(λ−2uδ)C+−2uvδA+,(2.22)
2uvδB−(λ+2vδ)A=−2δC+−(λ+2uv)δA+.(2.23)
Inserting (2.19) into (2.20)–(2.23) and taking into account (2.3)–(2.5), one derives the stationary HF hierarchy
aN−aN+=0,bN−bN+=0,cN−cN+=0.(2.24)
In the case N = 0, the system (2.24) is reduced to the stationary HF lattice
(u−v)(u−u+)(u+−v)−1=0,(u−v)(v−−v)(u−v−)−1=0.(2.25)
The time-dependent HF hierarchy can be derived as follows. First, aℓ, bℓ, cℓ are considered as functions of both n and t. To distinguish different higher order flows that are related to different time variables, we replace t by tr. Then inserting (2.1), (2.3)–(2.5) and (2.19) into the time-dependent zero-curvature equation
0=Utr+UW−(E+W)U,(2.26)
we obtain the
r-th equation in the HF hierarchy
(2uδ)tr=−ar+ar+,(2.27)
(2δ)tr=−br+br+,(2.28)
(2uvδ)tr=cr−cr+,(2.29)
(2vδ)tr=−ar+ar+.(2.30)
The system (2.27)–(2.30) is overdetermined and can be simplified into the r-th HF lattice.
Theorem 2.2.
The r-th HF lattice has the form of
utr=u−v2(u(br−br+)−ar+ar+),(2.31)
vtr=u−v2(v(br−br+)−ar+ar+), r∈ℕ0.(2.32)
Proof.
First, using (2.27), we obtain
2utrδ+2uδtr=−ar+ar+.(2.33)
Then the relations (2.28) and (2.33) give rise to
2utrδ=u(br−br+)−ar+ar+,(2.34)
and hence
(2.31) holds. Using
(2.28) and
(2.30), we arrive at
(2.32). To complete the proof, it remains to show that
(2.28),
(2.29) are compatible with
(2.31),
(2.32). Actually, using
(2.31) and
(2.32), we obtain
(2δ)tr=(2u−v)tr=−2δ2(utr−vtr)=−δ(u(br−br+)−ar+ar+)+δ(v(br−br+)−ar+ar+)=−br+br+.(2.35)
Then by (2.31), (2.32) and (2.35), it follows
2utrvδ=uv(br−br+)−v(ar−ar+),2uvtrδ=uv(br−br+)−u(ar−ar+),2uvδtr=−uv(br−br+),
and consequently,
(2uvδ)tr=2utrvδ+2uvtrδ+2uvδtr=uv(br−br+)−(u+v)(ar−ar+)=cr−cr+.(2.36)
Here we use the recursion relations
2vδ(aℓ−aℓ+)−2uvδbℓ−2δcℓ+=−aℓ−1+aℓ−1+, ℓ∈ℕ0,2uδ(aℓ−aℓ+)+2uvδbℓ+2δcℓ+=aℓ−1−aℓ−1+, ℓ∈ℕ0
in the last equality of
(2.36).
3. Spectral curves and a basic meromorphic function
In this section, we first introduce the spectral curves associated with the HF hierarchy. Then we introduce a basic meromorphic function φ and study its analytic and asymptotic properties.
Let (ψ1, ψ2)T and (ϕ1, ϕ2)T be two fundamental solutions of auxiliary linear problem
E+Φ(λ,n,tr)=U(λ,n,tr)Φ(λ,n,tr),Φtr(λ,n,tr)=W(λ,n,tr)Φ(λ,n,tr),(3.1)
where Φ(
λ,
n,
tr) = (Φ
1(
λ,
n,
tr),Φ
2(
λ,
n,
tr))
T . Moreover, we define
f(λ,n,tr)=−(λ2−2λ)−nψ1(λ,n,tr)φ1(λ,n,tr),(3.2)
h(λ,n,tr)=(λ2−2λ)−nψ2(λ,n,tr)φ2(λ,n,tr),(3.3)
g(λ,n,tr)=2−1(λ2−2λ)−n(ψ1(λ,n,tr)φ2(λ,n,tr)+φ1(λ,n,tr)ψ2(λ,n,tr)).(3.4)
Theorem 3.1.
The functions f, g, h defined in (3.2)–(3.4) satisfy
(gfh−g)+=U(gfh−g)U−1,(3.5)
(gfh−g)tr=[(ABC−A),(gfh−g)],(3.6)
where [·,·]
is the Lie bracket of two matrices P,
Q
[P,Q]=PQ−QP
and g2 +
fh is independent of x and tr.
Proof.
The relation (3.5) can be derived from (3.1), (3.2), (3.3) and (3.4). Indeed, we have
g+=2−1(λ2−2λ)−(n+1)ψ1+φ2++φ1+ψ2+=2−1(λ2−2λ)−(n+1)(((λ−2uδ)ψ1−2δψ2)(2uvδφ1+(λ+2vδ)φ2)+((λ−2uδ)φ1−2δφ2)(2uvδψ1+(λ+2vδ)ψ2))=−(λ2−2λ)−1(2uvδ(λ−2uδ)f−(λ−2uδ)(λ+2vδ)g+4uvδ2g+2δ(λ+2vδ)h,
f+=−(λ2−2λ)−(n+1)ψ1+φ1+=−(λ2−2λ)−(n+1)((λ−2uδ)ψ1−2δψ2)((λ−2uδ)φ1−2δφ2)=(λ2−2λ)−1((λ−2uδ)(λ−2uδ)f+2δ(λ−2uδ)2g−(−2δ)2h),
h+=(λ2−2λ)−(n+1)ψ2+φ2+,=(λ2−2λ)−(n+1)(2uvδψ1+(λ+2vδ)ψ2)(2uvδφ1+(λ+2vδ)φ2)=(λ2−2λ)−1(−(2uvδ)2f+2uvδ(λ+2vδ)2g+(λ+2vδ)2h).
Similarly, according to the definition of f, g, h, ϕ1, ϕ2, ψ1, ψ1, it follows that
gtr=2−1(λ2−2λ)−n(ψ1,trφ2+ψ1φ2,tr+φ1,trψ2+φ1ψ2,tN)=2−1(λ2−2λ)−n((Aψ1+Bψ2)φ2+ψ1(Cφ1−Aφ2)+(Aφ1+Bφ2)ψ2+φ1(Cψ1−Aψ2))+Bh−Cf,
ftr=−(λ2−2λ)−n(ψ1φ1)tr=−(λ2−2λ)−n(ψ1,trφ1+ψ1φ1,tr)=−(λ2−2λ)−n((Aψ1+Bψ2)φ1+ψ1(Aφ1+Bφ2))=2Af−2Bg
htr=(λ2−2λ)−n(ψ2φ2)tr=(λ2−2λ)−n(ψ2,trφ2+ψ2φ2,tr)=(λ2−2λ)−n((Cψ1−Aψ2)φ2+ψ2(Cφ1−Aφ2))=2Cg−2Ah,
which indicates
(3.6) holds. Finally, using
(det(gfh−g))tr=det(gfh−g)tr((gfh−g)tr(gfh−g)−1),(3.7)
det(gfh−g)=−g2−fh,(3.8)
we conclude that
g2 +
fh is independent of
n and
tr.
In what follows it is convenient to introduce
gM=∑ℓ=1M+1aM+1−λλ−ℓ, fM=∑ℓ=1M+1bM+1−ℓλ−ℓ, hM=∑ℓ=1M+1cM+1−ℓλ−ℓ, M∈ℕ0.(3.9)
Then one may obtain the following result.
Theorem 3.2.
Assume u, v are solutions of the r-th HF lattice and the M-th stationary HF lattice (2.24). Then for fixed M ∈ ℕ0,
g=gM, f=fM, h=hM,(3.10)
solve the system (3.6).
Proof.
First we show that (3.10) satisfy the first equation in (3.6). To this end we make the ansatz
g=∑ℓ=1M+1a⌣M+1−λλ−ℓ, f=∑ℓ=1M+1b⌣M+1−ℓλ−ℓ, h=∑ℓ=1M+1c⌣M+1−ℓλ−ℓ,(3.11)
where
a⌣j,
b⌣j,
c⌣j,
j = 1,...,
M +1 are polynomials with respect to
u,
v and their shift. Inserting
(3.11) into
(3.6), one derives
a⌣j=aj, b⌣j=bj, c⌣j=cj, j=1,…,M+1.(3.12)
and hence
(3.10) and
(3.9) hold. To complete the proof, one has to prove that
g,
f,
h also satisfy
(3.6). Since
u,
v are both solutions of the
M-th time-independent lattice, there exists a common eigenfunction
χ = (
χ1,
χ2) for the following two linear problems
χ+=Uχ, VMχ=λ−(M+1)yχ,(3.13)
where the 2 × 2 matrix
VM=(gMfMhM−gM).(3.14)
The relation (3.13) implies
λM+1(gM2+fMhM)1/2=y.
Next we introduce a new function
ϕ′=χ2χ1=λ−(M+1)y−gMfM=hMλ−(M+1)y+gM.(3.15)
A direct computation shows that φ′ satisfies
(λ−2uδ)ϕ′+−2δϕ′ϕ′=2uvδ+(λ+2vδ)ϕ′.(3.16)
Differentiating (3.16) with respect to tN then yields
((λ−2uδ−2δϕ′)E+−2δϕ′+−(λ+2vδ))ϕtr′=(2uvδ)tr+(2vδ)trϕ′+(λ+2vδ)ϕtr′.(3.17)
On the other hand, we have
((λ−2uδ−2δϕ′)E+−2δϕ′+−(λ+2vδ))(C−2Aϕ′−B(ϕ′)2)=(λ−2uδ−2δϕ′)C+−2δϕ′+C−(λ+2vδ)C+(2uvδ+(λ+2vδ)ϕ′ϕ′+E++2uvδ−(λ−2uδ)ϕ′+ϕ′)(−2Aϕ′−B(ϕ′+)2)=(2uvδ)tr+(2vδ)trϕ′+(λ+2vδ)ϕtr′.(3.18)
Combining (3.17) with (3.18), one obtains
(λ−2uδ−2δϕ′)E+−2δϕ′+−(λ+2vδ))(ϕtr′−C+2Aϕ′−B(ϕ′)2)=0,(3.19)
and consequently,
ϕtr′=C−2Aϕ′−B(ϕ′)2.(3.20)
On the other hand, from the definition of ψi, ϕi, one infers that the function φ (see (3.32)) satisfies
(λ−2uδ)ϕ+−2δϕ+ϕ−2uvδ−(λ+2vδ)ϕ=0.(3.21)
Then by (3.1), we have
(ψ2ψ1)tr=ψ2,tNψ1−ψ2ψ1,tNψ12=Cψ1−Aψ2ψ1−ψ2ψ1Aψ1+Bψ2ψ1=C−2Aψ2ψ1−B(ψ2ψ1)2.(3.22)
Similarly one may derive
(φ2φ1)tr=C−2Aφ2φ1−B(φ2φ1)2.(3.23)
Thus one arrives at
ϕtr=C−2Aϕ−Bϕ2.(3.24)
A comparison of (3.17), (3.20), (3.21) and (3.24) yields
ϕ=F(λ)ϕ′,(3.25)
where
F(
λ) is an arbitrary function of
λ. Without loss of generality, one can take
F(
λ) ≡ 1. Therefore, one infers from
(3.25) that there exists common eigenfunctions of the following linear system
χ+=Uχ,(3.26)
χtr=Vχ,(3.27)
VMχ=λ−(M+1)yχ.(3.28)
From the compatibility condition of (3.27) and (3.28), we finally obtain
VM,tr=[V,VM],(3.29)
which completes the proof.
Next let us define
Δ±(λ,n,tr)=±2−1(λ2−2λ)−n(ψ1(λ,n,tr)φ2(λ,n,tr)−φ1(λ,n,tr)ψ2(λ,n,tr)).(3.30)
It is not difficult to check that Δ± satisfy
Δ±2(λ,n,tr)=g2(λ,n,tr)+f(λ,n,tr)h(λ,n,tr),Δ+(λ,n,tr)+g(λ,n,tr)=−Δ−(λ,n,tr)+g(λ,n,tr)=(λ2−2λ)−nψ1(λ,n,tr)φ2(λ,n,tr),φ2(λ,n,tr)/φ1(λ,n,tr)=(Δ−(λ,n,tr)−g(λ,n,tr))/f(λ,n,tr),ψ2/ψ1(λ,n,tr)=h(λ,n,tr)/(Δ+(λ,n,tr)+g(λ,n,tr))=(Δ+(λ,n,tr)−g(λ,n,tr))/f(λ,n,tr).
For fixed M ∈ ℕ, the spectral curve 𝒦M−1 of genus M − 1 can be introduced as follows
𝒦M−1:y2=λ2M+2(f2+gh)=∏j=02M−1(λ−Ej), Ej∈ℂ.
Throughout this paper, one assumes that
Ej≠Eℓ, j≠ℓ, j=0,1,…,2M−1.(3.31)
The next step is crucial; we lift the functions ϕ2/ϕ1 and ψ2/ψ1 on the Riemann surface 𝒦M−1 by
ϕ(P,x,tr)={ψ2(λ,n,tr)/ψ1(λ,n,tr),P∈𝒦M−1+φ2(λ,n,tr)/φ1(λ,n,tr),P∈𝒦M−1−,(3.32)
where
𝒦M−1± are two sheets of
𝒦M−1. The spectral curve
𝒦M−1 is compactified by the point at infinity
P∞. A point on
𝒦M−1 is denoted by
P = (
λ,
y(
P)), where
λ ∈ ℂ and
y(
P) is a holomorphic function defined on the two sheets
𝒦M−1± of 𝒦
M−1:
y(P)={∏j=02M−1(λ−Ej),P∈𝒦M−1+−∏j=02M−1(λ−Ej),P∈𝒦M−1−.(3.33)
In particular, one introduces P0±=(0,±(∏j=02M−1Ej)1/2).
Based on above preparations, we turn to study the analytic and asymptotic properties of φ.
Lemma 3.3.
The function φ defined in (3.32) is meromorphic. Its divisor is given by a
(ϕ(⋅,n,tr))=𝒟ν^_(n,tr)ν^M(n,tr)−𝒟μ^_(n,tr)μ^M(n,tr).(3.34)
Here the notations ν^_(n,tr) and ν^_(n,tr) represent two vectors
ν^_(n,tr)=(ν^1(n,tr),…,ν^M−1(n,tr)), μ^_(n,tr)=(μ^1(n,tr),…,μ^M−1(n,tr)).
Lemma 3.4.
Near the points P0±, the function φ has the following expansions
ϕ(⋅,n,tr)=c∞++O(ζ), P→P∞±, ζ=λ−1(3.35)
and
ϕ(⋅,n,tr)={−v−−E−1(v−v−)(u−v)2(u−v−)ζ+O(ζ2),P→P0+, ζ=λ,−u+(u+−u)(u−v)2(u+−v)ζ+O(ζ2).P→P0−, ζ=λ,(3.36)
where c∞± are lattice constants.
Proof.
First one observes that the expansion (3.35) can be easily derived from the definition of φ. Thus it suffices to compute the expansions of φ near P0±. Using the relation
Δ±2=g2+fh=λ−2(M+1)((∑i=0Ngiλi)2−(∑i=0Nfiλi)(∑i=0Nhiλi))=λ−2(M+1)∑i=0N(∑k=0i(gkgi−k−fkhi−k))λi,
one infers
Δ±=ζ→0±ζ−(M+1)g02+f0h0(1+122g0g1+f0h1+f1h0g02+f0h0ζ+O(ζ2)),(3.37)
as
P →
P0±. Then from the relation
g02+f0h0=1 and
(3.32), it follows
ϕ(⋅,n,tr)={1−u+v−u−v−2u−v−+O(ζ),P→P0+, λ=ζ,−1−u+v−u−v−2u−v−+O(ζ),P→P0−, λ=ζ,={−v−+O(ζ),P→P0+, λ=ζ,−u+O(ζ),P→P0−, λ=ζ.(3.38)
To get the coefficient of ζ in (3.36), we use the equation (see (3.16))
(λ−2uδ)ϕ+−2δϕ+ϕ=2uvδ+(λ+2vδ)ϕ.(3.39)
Inserting the ansatz
ϕ=ϕ0±+ϕ1±ζ+ϕ2±ζ2+O(ζ3), as P→P0±(3.40)
into
(3.39) and comparing the coefficients of
ζ lead to
ϕ0±+−2uδϕ1±+−2δϕ0±+ϕ1±−2δϕ0±ϕ1±+=ϕ0±+2vδϕ1±.(3.41)
Explicitly, Eq. (3.41) can be equivalently written as
−v−2uδϕ1±++2vδϕ1++2v−δϕ1++=−v−+2vδϕ1+,(3.42)
−u+−2uδϕ1−++2u+δϕ1−+2uδϕ1−+=−u+2vδϕ1−,(3.43)
which implies
ϕ1+=E−1(v−v−)(u−v)2(v−−u), ϕ1−=(u+−u)(u−v)2(u+−v).(3.44)
Using similar method we can prove (3.35).
4. Quasi-periodic solutions
In this section, we obtain theta function representations for the meromorphic function φ and quasiperiodic solutions u, v.
First, we choose a convenient base point Q0 ∈ KM−1\{P∞±, P0±}. The Abel maps A_Q0(⋅), α_Q0(⋅) are defined by
A_Q0:X→J(X)=ℂM−1/LM−1,
P↦A_Q0(P)=(AQ0,1(P),…,QQ0,M−1(P))=(∫Q0Pω1,…,∫Q0PωM−1)(modLM−1),
and
α_Q0:Div(X)→J(X), 𝒟↦α_Q0(𝒟)=∑P∈𝒦M−1𝒟(P)A_Q0(P),
where
LM−1={z_∈ℂM−1|z_=N_+ΓM_,N_,M_∈M−1}, and Γ,
Ξ_Q0 are the Riemann matrix and the vector of Riemann constants, respectively. Moreover, we choose a homology basis
{aj,bj}j=1M−1 on
X in such a way that the intersection matrix of the cycles satisfies
aj∘bk=δj,k, aj∘ak=0, bj∘bk=0, j,k=1,…,n.(4.1)
For brevity, we introduce
z_(P,Q_)=Ξ_Q0−A_Q0(P)+α_Q0(𝒟Q_),P∈𝒦M−1, Q_=(Q1,…,QM−1)∈σM−1𝒦M−1,(4.2)
where
z_(⋅,Q_) is independent of the choice of
Q0. The Riemann theta function
θ(z_) associated with
X and the homology homology basis
{aj,bj}j=1M−1 is defined by
θ(z_)=∑n_∈exp(2πi〈n_,z_〉+πi〈n_,n_Γ〉), z_∈ℂM−1,
where
〈B_,C_〉=∑j=1M−1B¯jCj is the scalar product in ℂ
M−1.
Next, we introduce the differential of the third kind with simple zero and pole respectively at ν^M(n,tr) and μ^M(n,tr) by
ων^M(n,tr)μ^M(n,tr)(3)(P)=(y+y(ν^M(n,tr))λ−νM(n,tr)−y+y(μ^M(n,tr))λ−μM(n,tr))dλ2y+λ0y∏i=1M−2(λ−λj)dλ,
where
λj,
j = 0,
...,
M − 2, are constants that are uniquely determined by the requirement of vanishing
a-period, i.e.
∫ajων^M(n,tr)μ^M(n,tr)(3)(P)=0, j=1,…,M−1.(4.3)
A simple computation shows
∫Q0Pων^M(n,tr)μ^M(n,tr)(3)(P)={lnζ+d0(n,tr)+O(ζ),P→ν^M(n,tr),−lnζ+d1(n,tr)+O(ζ),P→μ^M(n,tr),dP0±+O(ζ)P→P0±,dP∞±+d2(n,tr)ζ+O(ζ2),P→P∞±.
Here d0(n,tr), d1(n,tr), d2(n,tr) are functions of variables n, tr and dP∞±, dP0± are integration constants.
Theta function representations for quasiperiodic solutions of the rth HF lattice can be obtained as follows.
Theorem 4.1.
The function φ defined in (3.32) is meromorphic on 𝒦M−1 and has the following theta function representation
ϕ(P,n,tr)=cP∞+θ(λ_(P∞+,μ^_(n,tr)))θ(λ_(P∞+,ν^_(n,tr)))θ(λ_(P,ν^_(n,tr)))θ(λ_(P,μ^_(n,tr)))exp(∫Q0Pων^M(n,tr)μ^M(n,tr)(3)−dP∞+).(4.4)
Proof.
To derive the formula (4.4), one has to use Riemann vanishing theorem and Riemann-Roch theorem in standard literature [3]. From Lemma 2.3, one infers that the function φ(P,n,tN) take the form
ϕ(P,n,tr)=C(n,tr)θ(λ_(P,ν^_(n,tr)))θ(λ_(P,μ^_(n,tr)))exp(∫Q0Pων^M(n,tr)μ^M(n,tr)(3)),(4.5)
where
C(
n,
tr) is an undetermined function. Then taking the limit
P →
P∞+ on both sides of
(4.5), one derives
cP∞+=C(n,tN)θ(λ_(P∞+,ν^_(n,tr)))θ(λ_(P∞+,μ^_(n,tr)))exp(dP∞+).(4.6)
Inserting(4.6) into (4.5) and eliminating C(n,tr), one finally obtains (4.4).
Next, we obtain theta function representations for solutions u,v with the help of (4.4).
Theorem 4.2.
Quasiperiodic solutions of the r-th HF lattice have the following Riemann theta function representation
dP0+-dP∞+,(4.7)
v(n,tr)=-cP∞+θ(λ_(P∞+,μ^_+(n,tr)))θ(λ_(P∞+,ν^_+(n,tr)))θ(λ_(P0+,ν^_+(n,tr)))θ(λ_(P0+,μ^_+(n,tr)))exp(dP0+-dP∞+).(4.8)
Proof.
Using the coordinate λ = ζ near P0±, we have
ϕ(n,x,tr)=cP∞+θ(λ_(P∞+,μ^_(n,tr)))θ(λ_(P∞+,ν^_(n,tr)))θ(λ_(P,ν^_(x,tr)))θ(λ_(P,μ^_(x,tr)))×exp(∫Q0Pων^M(x,tr)μ^M(x,tr)(3)−dP∞+)=ζ→0cP∞+θ(λ_(P∞+,μ^_(n,tr)))θ(λ_(P∞+,ν^_(n,tr)))θ(λ_(P0±,ν^_(x,tr)))θ(λ_(P0±,μ^_(x,tr)))×exp(dP0±−dP∞+)(1+O(ζ)).(4.9)
Comparing the leading coefficients of (3.36) and (4.9) naturally gives rise to (4.7) and (4.8).
