# Algebro-geometric solutions to the lattice potential modified
Kadomtsev–Petviashvili equation

Xiaoxue Xu1, Cewen Cao1, Da-jun Zhang2
1School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan
450001, P.R. China
2Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Algebro-geometric solutions of the lattice potential modified
Kadomtsev–Petviashvili (lpmKP) equation are constructed. A Darboux
transformation of the Kaup–Newell spectral problem is employed to generate a
Lax triad for the lpmKP equation, as well as to define commutative integrable
symplectic maps which generate discrete flows of eigenfunctions. These maps
share the same integrals with the finite-dimensional Hamiltonian system
associated to the Kaup–Newell spectral problem. We investigate asymptotic
behaviors of the Baker–Akhiezer functions and obtain their expression in terms
of Riemann theta function. Finally, algebro-geometric solutions for the lpmKP
equation are reconstructed from these Baker–Akhiezer functions.

Key words: lattice potential modified Kadomtsev–Petviashvili equation,
algebro-geometric solution, Kaup–Newell spectral problem, Baker–Akhiezer
function

Mathematics Subject Classification (2000): 35Q51, 37K60, 39A36

##  1 Introduction

In recent two decades discrete integrable systems have undergone a true
development (see [31] and the references therein). One of remarkable results
is the Adler–Bobenko–Suris (ABS) classification of quadrilateral equations
that are consistent around a cube (CAC) [52, 48, 5, 1] with certain extra
restrictions (affine linear, D4 symmetry and tetrahedron property) [1]. The
ABS list contains only 9 equations, named H1, H2, H${}_{3}(\delta)$,
A${}_{1}(\delta)$, A2, Q${}_{1}(\delta)$, Q2, Q${}_{3}(\delta)$ and Q4. The
ABS equations serve as main objects in the study of discrete integrable
systems, although many equations in the list are already found before. During
the study, some methods have been developed or elaborated for discrete
integrable systems, such as the Cauchy matrix approach [49, 63], bilinear
method [32], inverse scattering transform [7] and algebraic geometry approach
[9].

The finite-gap integration method was created for solving the Korteweg–de
Vries (KdV) equation with periodic initial value problem by Novikov, Maveev
and their collaborators Dubrovin, Its and Krichever in 1970s [20, 21, 34, 35,
42, 54]. The obtained periodic solutions are called finite-gap solutions or
algebro-geometry solutions. After the original work, the theory has undergone
a true development (e.g.[4, 28, 29]) and had a strong impact on the evolution
of modern mathematics and theoretical physics (see [45] and the references
therein for the historical review of finite-gap integration method).

For fully discrete equations, such as the ABS equations, algebro-geometry
solutions have been derived from several equations in a series of papers [9,
10, 12, 13, 60, 61, 62] by Cao, Xu and their collaborators, which establish an
algebro-geometric approach to periodic solutions for discrete equations that
are multi-dimensionally consistent. This approach can be sketched as the
following three steps. For a given quadrilateral equation which admits a Lax
pair, in the first step we find an associated continuous spectral problem that
is compatible with the discrete Lax pair. In step 2, consider the Hamiltonian
system that arises from the associated continuous spectral problem and derive
independent integrals of the Hamiltonian system. Then, view the discrete Lax
pair as maps by which discrete flows of eigenfunctions can be generated. Using
the compatibility between the maps and the associated continuous spectral
problem, one can prove that the maps are symplectic and integrable in
Liouville sense, sharing the same integrals with the Hamiltonian system.
Finally, in step 3, using algebro-geometric technique, introduce
Baker–Akhiezer functions and Abel–Jacobi variables, and express the
Baker–Akhiezer functions in terms of the Riemann theta function according to
their divisors, and recover the discrete potential functions which are
expressed in terms of the Riemann theta function. The approach was also
extended to the lattice potential Kadomtsev–Petviashvili (lpKP) equation [10]
which is 4D consistent [2].

The first step of the approach is a starting point but it is highly
nontrivial. In fact, since the Lax pair are compatible with the associated
continuous spectral problem, the Lax pair can be considered as Darboux
transformations of the continuous spectral problem (cf.[44]). With regard to
an ABS equation, its spectral problem is thought to be a Darboux
transformation of some continuous spectral problem, but such associated
continuous spectral problems are still unknown for H2, H${}_{3}(\delta)$, Q2,
Q${}_{3}(\delta)$ and Q4. In this step, one needs to secure a continuous
spectral problem and simultaneously to find a “suitable” Darboux
transformation which not only can act as a spectral problem of the target
discrete equation, but also can be used to recover the discrete potential
functions in the follow-up steps. And moreover, the Darboux transformation is
usually different from the discrete spectral problem obtained from
multidimensional consistency. Of course, the step 2 and 3 are also different
for different equations. Let us list out the associated continuous spectral
problems that have been used in this approach. The Schrödinger spectral
problem (in matrix form)

$\phi_{x}=\left(\begin{array}[]{cc}0&-\lambda+u\\\ 1&0\end{array}\right)\phi$

was used in [9] for solving H1. To solve Hirota’s discrete sine-Gordon
equation, spectral problems

$\phi_{x}=\left(\begin{array}[]{cc}u&\lambda\\\
\lambda&-u\end{array}\right)\phi$

was employed [12]. Ref.[61] made use of

$\phi_{x}=\left(\begin{array}[]{cc}u&\lambda\\\ 0&-u\end{array}\right)\phi$

to solve Q${}_{1}(0)$. Ref.[62] solves H1, H${}_{3}(0)$ and Q${}_{1}(0)$ and
the three associated continuous spectral problems are, respectively,

$\phi_{x}=\left(\\!\begin{array}[]{cc}v&-\lambda+u\\\
1&-v\end{array}\\!\right)\phi,~{}~{}\phi_{x}=\left(\begin{array}[]{cc}\frac{\lambda^{2}}{2}&\lambda
u\\\ \lambda
v&-\frac{\lambda^{2}}{2}\end{array}\\!\right)\phi,~{}~{}\phi_{x}=\left(\\!\\!\begin{array}[]{cc}-\frac{\lambda^{2}}{2}+u+v&\lambda
u\\\ -\lambda&\frac{\lambda^{2}}{2}-u-v\end{array}\\!\\!\right)\phi,$

where the second one is known as the Kaup–Newell spectral problem. In [60], to
solve Q${}_{1}(\delta)$, spectral problem

$\phi_{x}=\frac{1}{\lambda
u_{x}}\left(\begin{array}[]{cc}0&\lambda^{2}\delta^{2}+u_{x}^{2}\\\
1&0\end{array}\right)\phi$

has been used. For the coupled lattice nonlinear Schrödinger equations, a
nonsymmetric 3D lattice, and the lpKP equation, the
Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur (ZS–AKNS) spectral problem

$\phi_{x}=\left(\begin{array}[]{cc}-\lambda&u\\\
v&\lambda\end{array}\right)\phi$ (1.1)

was employed respectively in [13] and [10]. Note that one discrete equation
may have more than one associated continuous spectral problems, e.g. H1 and
Q${}_{1}(0)$.

This paper is devoted to finding algebro-geometric solutions to the lattice
potential modified KP (lpmKP) equation,

$\Xi^{(0,3)}\equiv\beta_{1}^{2}(e^{-\overline{\widetilde{W}}+\overline{W}}-e^{-\widehat{\widetilde{W}}+\widehat{W}})+\beta_{2}^{2}(e^{-\widehat{\overline{W}}+\widehat{W}}-e^{-\widetilde{\overline{W}}+\widetilde{W}})+\beta_{3}^{2}(e^{-\widetilde{\widehat{W}}+\widetilde{W}}-e^{-\overline{\widehat{W}}+\overline{W}})=0,$
(1.2)

where $j,k$ in the notation $\Xi^{(j,k)}$ respectively stand for the numbers
of continuous and discrete independent variables in the equation,
$\beta_{j}(j=1,2,3)$ are the lattice parameters associated with three
directions $m_{j}(j=1,2,3)$, and $\widetilde{W},\overline{W},\widehat{W}$ are
conventional notations denoting shifts in different directions, i.e.

$\widetilde{W}=W(m_{1}+1,m_{2},m_{3}),~{}~{}\overline{W}=W(m_{1},m_{2}+1,m_{3}),~{}~{}\widehat{W}=W(m_{1},m_{2},m_{3}+1).$

This equation is first given in [50] (Eq.(4.16)), derived in a framework of
direct linearisation for 3D lattice equations developed in [51]. It is one of
the five octahedron-type equations that are 4D consistent [2]. In continuous
limit the equation goes to the potential mKP equation [50]

$\Xi^{(3,0)}\equiv\frac{1}{4}(W_{xxx}-2W_{x}^{3})_{x}-\frac{3}{2}W_{xx}W_{y}+\frac{3}{4}W_{yy}-W_{xt}=0.$
(1.3)

The associated continuous spectral problem we will employ to solve the lpmKP
equation is the KN spectral problem [36, 37]

$\partial_{x}\chi=U_{1}\chi,\quad
U_{1}(\lambda;u,v)=\left(\begin{array}[]{cc}\lambda^{2}/2&\lambda u\\\ \lambda
v&-\lambda^{2}/2\end{array}\right).$ (1.4)

The paper is organized as follows. In Section 2 we present a Lax triad for the
lpmKP equation, explain connections between the Lax triad and the associated
KN spectral problem. In Section 3, a nonlinear integrable symplectic map is
obtained based on a finite-dimensional Hamiltonian system arising from the KN
spectral problem. In Section 4, finite-gap solutions to the lpmKP equation
(1.2) are constructed by introducing the Baker-Akhiezer functions and using a
discrete analogue of the Liouville-Arnold theory. After that, we present an
example of the obtained solutions for the case of genus one in Section 5. The
concluding remarks are given in Section 6. We will also obtain a hierarchy of
equations that are related to the lpmKP equation, in terms of the number of
discrete independent variables. Together with some links between
$(1+1)$-dimensional and $(2+1)$-dimensional integrable systems, these will be
given in Appendix A. In addition, Appendix B provides some complex algebraic
geometry preliminaries that will be used in our approach.

##  2 The lpmKP equation arising from compatibility

The idea of introducing discretisation by transform originated in [43, 44]. A
Darboux transformation of a continuous spectral problem can serve as a
discrete spectral problem to generate semi-discrete and fully discrete
integrable systems, e.g. [14, 38, 41, 53, 64]. In this section, we will
explain how the lpmKP equation (1.2) is connected with the KN spectral problem
(1.4). These connections are important for us to reconstruct the discrete
potential function $W$ in the follow-up sections.

Consider a Darboux transformation of the KN spectral problem (1.4): (cf.[64])

$T_{1}\chi\equiv\widetilde{\chi}=D^{(\beta)}\chi,\quad
D^{(\beta)}=\left(\begin{array}[]{cc}\lambda^{2}(ab+1)-\beta^{2}&\lambda a\\\
\lambda b&1\end{array}\right),$ (2.1)

where $\beta$ serves as a soliton parameter, which transforms (1.4) to
$\partial_{x}\widetilde{\chi}=\widetilde{U}_{1}\widetilde{\chi}=U_{1}(\lambda,\widetilde{u},\widetilde{v})\widetilde{\chi}$.
This requires that111Considering (2.1) as a discrete spectral problem,
equation (2.2) is the semi-discrete zero curvature equation from the
compatibility between (2.1) and (1.4)

$\begin{split}0&=D^{(\beta)}_{x}-\widetilde{U}_{1}D^{(\beta)}+D^{(\beta)}U_{1}\\\
&=\left(\begin{array}[]{cc}\lambda^{2}(Z_{x}-\tilde{u}b+va)&\lambda^{3}(-a+Zu)+\lambda(a_{x}-\tilde{u}-\beta^{2}u)\\\
\lambda^{3}(b-Z\tilde{v})+\lambda(b_{x}+v+\beta^{2}\tilde{v})&\lambda^{2}(ub-\tilde{v}a)\end{array}\right),\end{split}$
(2.2)

where we have taken

$Z=ab+1.$ (2.3a) It follows that $\displaystyle
a_{x}-\widetilde{u}-\beta^{2}u=0,\quad b_{x}+v+\beta^{2}\widetilde{v}=0,$
(2.3b) $\displaystyle a=Zu,\quad b=Z\widetilde{v},$ (2.3c) $\displaystyle
Z_{x}=(\widetilde{u}\widetilde{v}-uv)Z,$ (2.3d)

which allows a special formulation222Note that this formulation is different
from the Case $T_{22}$ in [64] and therefore the spectral problem (2.1) (with
formulation (2.4)) is different from the spectral problem introduced in [59].

$\displaystyle Z=2/(1+\sqrt{1-4u\tilde{v}}),$ (2.4a) $\displaystyle
a=2u/(1+\sqrt{1-4u\tilde{v}}),\quad b=2\tilde{v}/(1+\sqrt{1-4u\tilde{v}}),$
(2.4b)

As a byproduct, equation (2.3b) (in terms of the variables $(u,v)$) provides a
Bäcklund transformation for the KN potentials in (1.4),

$\displaystyle\Xi_{1}^{(1,1)}\equiv
u_{x}+(\tilde{u}\tilde{v}-uv)u-\frac{1}{2}(1+\sqrt{1-4u\tilde{v}})(\tilde{u}+\beta^{2}u)=0,$
(2.5a)
$\displaystyle\Xi_{2}^{(1,1)}\equiv\tilde{v}_{x}+(\tilde{u}\tilde{v}-uv)\tilde{v}+\frac{1}{2}(1+\sqrt{1-4u\tilde{v}})(v+\beta^{2}\tilde{v})=0,$
(2.5b)

which can also be regarded as a semi-discrete derivative nonlinear Schrödinger
(dNLS) equation as it continuum limit yields the dNLS equation (A.4) (see
Appendix A).

######  Remark 2.1.

Note that $Z$ given in (2.4a) is one of roots of the quadratic equation
$Z=Z^{2}u\widetilde{v}+1$, which results from the relation (2.3). If we take
another root, $Z=2/(1-\sqrt{1-4u\tilde{v}})$, we will have an analogue of
(2.5):

$\displaystyle\Xi_{1}^{{}^{\prime}(1,1)}\equiv
u_{x}+(\tilde{u}\tilde{v}-uv)u-\frac{1}{2}(1-\sqrt{1-4u\tilde{v}})(\tilde{u}+\beta^{2}u)=0,$
$\displaystyle\Xi_{2}^{{}^{\prime}(1,1)}\equiv\tilde{v}_{x}+(\tilde{u}\tilde{v}-uv)\tilde{v}+\frac{1}{2}(1-\sqrt{1-4u\tilde{v}})(v+\beta^{2}\tilde{v})=0.$

However, it cannot recover the continuous dNLS equation (A.4) in continuum
limit due to the fatal minors sign in front of the square roots. In this
context, in the paper we will only consider the consequential results of
(2.4a).

Next, in order to derive the lpmKP equation (1.2), we consider discrete
spectral problems, which are three replicas of equation (2.1) with the
formulation (2.4),

$\displaystyle
T_{1}\chi=D^{(\beta_{1})}\chi=\left(\begin{array}[]{cc}\lambda^{2}Z^{(1)}-\beta_{1}^{2}&\lambda
Z^{(1)}u\\\ \lambda Z^{(1)}\widetilde{v}&1\end{array}\right)\chi,$ (2.6c)
$\displaystyle
T_{2}\chi=D^{(\beta_{2})}\chi=\left(\begin{array}[]{cc}\lambda^{2}Z^{(2)}-\beta_{2}^{2}&\lambda
Z^{(2)}u\\\ \lambda Z^{(2)}\overline{v}&1\end{array}\right)\chi,$ (2.6f)
$\displaystyle
T_{3}\chi=D^{(\beta_{3})}\chi=\left(\begin{array}[]{cc}\lambda^{2}Z^{(3)}-\beta_{3}^{2}&\lambda
Z^{(3)}u\\\ \lambda Z^{(3)}\widehat{v}&1\end{array}\right)\chi,$ (2.6i)

where $T_{i}$ stands for the shift operator in $m_{i}$-direction, i.e.
$T_{1}f=\widetilde{f},T_{2}f=\overline{f},T_{3}f=\widehat{f}$, $\beta_{i}$
serves as the spacing parameter of the $m_{i}$-direction, and in light of
(2.4a),

$Z^{(1)}=\frac{2}{1+\sqrt{1-4u\widetilde{v}}},\quad
Z^{(2)}=\frac{2}{1+\sqrt{1-4u\overline{v}}},\quad
Z^{(3)}=\frac{2}{1+\sqrt{1-4u\widehat{v}}}.$ (2.7)

One can quickly check that the compatibility between (2.6c) and (2.6f)
requires that

$\overline{Z}^{(1)}Z^{(2)}=\widetilde{Z}^{(2)}Z^{(1)}.$ (2.8)

This relation, together with (2.3d), leads us to introducing

$W_{x}=uv$ (2.9)

and

$\displaystyle\widetilde{W}-W=\ln Z^{(1)},$ (2.10a)
$\displaystyle\overline{W}-W=\ln Z^{(2)},$ (2.10b)
$\displaystyle\widehat{W}-W=\ln Z^{(3)}.$ (2.10c)

Later in Section 4 we will derive $Z^{(i)}$ in an explicit form, from which
$W$ can be “integrated” from (2.10). In this sense, $u$ and $v$ act as
auxiliary variables and $x$ is a dummy variable. Note that explicit $u$ and
$v$ can also be obtained from $Z^{(i)}$ (see Remark 4.1).

In order to show that $W$ defined by the triad (2.6) via (2.10) satisfies the
lpmKP equation (1.2), we first look at equations resulting from the
compatibility of the first two equations in (2.6).

######  Lemma 2.1.

If $u$ and $v$ are functions of $(m_{1},m_{2})$ such that $W(m_{1},m_{2})$ can
be well defined by equations (2.10a) and (2.10b), and equations (2.6c) and
(2.6f) are compatible for $\chi$, then the pair $(u,v)$ solves the lattice
dNLS (ldNLS) equation

$\displaystyle\Xi_{1}^{(0,2)}\equiv\widetilde{Z}^{(2)}\widetilde{u}-\overline{Z}^{(1)}\overline{u}+(\beta_{1}^{2}Z^{(2)}-\beta_{2}^{2}Z^{(1)})u=0,$
(2.11a) $\displaystyle\Xi_{2}^{(0,2)}\equiv
Z^{(1)}\widetilde{v}-Z^{(2)}\overline{v}-(\beta_{1}^{2}\widetilde{Z}^{(2)}-\beta_{2}^{2}\overline{Z}^{(1)})\widetilde{\overline{v}}=0,$
(2.11b)

where $Z^{(j)}$ are defined in (2.7). Moreover, the following relation

$Y_{12}\equiv\beta_{1}^{2}\Bigl{[}\Bigl{(}\overline{Z}^{(1)}\Bigr{)}^{-1}-\Bigl{(}Z^{(1)}\Bigr{)}^{-1}\Bigr{]}-\beta_{2}^{2}\Bigl{[}\Bigl{(}\widetilde{Z}^{(2)}\Bigr{)}^{-1}-\Bigl{(}Z^{(2)}\Bigr{)}^{-1}\Bigr{]}+(\widetilde{u}\widetilde{v}-\overline{u}\,\overline{v})=0$
(2.12)

holds.

###### Proof.

By making use of relation (2.8), the compatibility of equations (2.6c) and
(2.6f) gives rise to

$\mathbf{0}=\widetilde{D}^{(\beta_{2})}D^{(\beta_{1})}-\overline{D}^{(\beta_{1})}D^{(\beta_{2})}=\left(\begin{array}[]{cc}\lambda^{2}e^{\overline{\widetilde{W}}-W}Y_{12}&\lambda\Xi_{1}^{(0,2)}\\\
\lambda\Xi_{2}^{(0,2)}&0\end{array}\right),$

where $\Xi_{1}^{(0,2)}$ and $\Xi_{2}^{(0,2)}$ are given in (2.11). In
addition, noticing that

$Y_{12}=e^{W-\widetilde{\overline{W}}}\Bigl{(}Z^{(1)}\widetilde{v}\Xi_{1}^{(0,2)}+\overline{Z}^{(1)}\overline{u}\Xi_{2}^{(0,2)}\Bigr{)},$
(2.13)

we obtain equation (2.12) as a consequence of (2.11). The proof is then
completed.

∎

Note that in light of (2.7), equations (2.11a) and (2.11b) can be written as

$\displaystyle\frac{1}{2}(1+\sqrt{1-4u\widetilde{v}}\,)(\widetilde{u}+\beta_{1}^{2}u)-\frac{1}{2}(1+\sqrt{1-4u\overline{v}}\,)(\overline{u}+\beta_{2}^{2}u)-(\widetilde{u}\widetilde{v}-\overline{u}\,\overline{v})u=0,$
(2.14a)
$\displaystyle\frac{1}{2}(1+\sqrt{1-4\widetilde{u}\overline{\widetilde{v}}}\,)(\widetilde{v}+\beta_{2}^{2}\overline{\widetilde{v}})-\frac{1}{2}(1+\sqrt{1-4\overline{u}\overline{\widetilde{v}}}\,)(\overline{v}+\beta_{1}^{2}\overline{\widetilde{v}})-(\widetilde{u}\widetilde{v}-\overline{u}\,\overline{v})\overline{\widetilde{v}}=0.$
(2.14b)

Next, we can show that $W$ satisfies the lpmKP equation (1.2) after
consistently introducing evolution in the third direction.

######  Theorem 2.1.

If $u$ and $v$ are functions of $(m_{1},m_{2},m_{3})$ such that
$W(m_{1},m_{2},m_{3})$ can be well defined by (2.10) and equations in the
triad (2.6) are compatible for $\chi$, then $W$ satisfies the lpmKP equation
(1.2).

###### Proof.

In fact, in addition to the relation (2.13), compatibility between any two
equations in the triad (2.6) yields

$\displaystyle Y_{23}$
$\displaystyle\equiv\beta_{2}^{2}\Bigl{[}\Bigl{(}\widehat{Z}^{(2)}\Bigr{)}^{-1}-\Bigl{(}Z^{(2)}\Bigr{)}^{-1}\Bigr{]}-\beta_{3}^{2}\Bigl{[}\Bigl{(}\overline{Z}^{(3)}\Bigr{)}^{-1}-\Bigl{(}Z^{(3)}\Bigr{)}^{-1}\Bigr{]}+(\overline{u}\,\overline{v}-\widehat{u}\widehat{v})=0,$
$\displaystyle Y_{31}$
$\displaystyle\equiv\beta_{3}^{2}\Bigl{[}\Bigl{(}\widetilde{Z}^{(3)}\Bigr{)}^{-1}-\Bigl{(}Z^{(3)}\Bigr{)}^{-1}\Bigr{]}-\beta_{1}^{2}\Bigl{[}\Bigl{(}\widehat{Z}^{(1)}\Bigr{)}^{-1}-\Bigl{(}Z^{(1)}\Bigr{)}^{-1}\Bigr{]}+(\widehat{u}\,\widehat{v}-\widetilde{u}\widetilde{v})=0.$

Then, noticing that

$\Xi^{(0,3)}=Y_{12}+Y_{23}+Y_{31}$ (2.15)

and replacing $Z^{(i)}$ by $W$ using (2.10), we arrive at the lpmKP equation
(1.2). The proof is completed.

∎

Since the spectral problem (2.6c) arises from the Darboux transformation
(2.1), which commutes with the KN spectral problem and therefore may serve as
a Darboux transformation for the whole KN hierarchy, we can have more
equations in this frame, which have different number of discrete independent
variables and compose a hierarchy of the lpmKP family. Since we will focus on
the lpmKP equation (1.2), we will list out these semi-discrete equations in
Appendix A.

##  3  Nonlinear integrable map $S_{\beta}$

In this section, we will prove that the Darboux transformation (2.1) (with
(2.3)) is an integrable symplectic map.

For the sake of self-containedness of the paper, let us recall some results
given in [11] for the Hamiltonian system associated with the KN spectral
problem (1.4). Let $N$ be any positive integer,
$<\xi,\eta>=\Sigma_{j=1}^{N}\xi_{j}\eta_{j}$, and $A\doteq{\rm
diag}(\alpha_{1},\cdots,\alpha_{N})$ with distinct and non-zero
$\alpha_{1}^{2},\cdots,\alpha_{N}^{2}$. A Liouville integrable Hamiltonian
system $(\mathbb{R}^{2N},{\rm d}p\wedge{\rm d}q,H_{1})$ is constructed by $N$
copies of the KN spectral problem (1.4), by imposing a constraint (3.1c) on
$(u,v)$, as

$\displaystyle H_{1}=-\frac{1}{2}<A^{2}p,q>+\frac{1}{2}<Ap,p><Aq,q>,$ (3.1a)
$\displaystyle\partial_{x}{p_{j}\choose q_{j}}={-\partial H_{1}/\partial
q_{j}\choose\partial H_{1}/\partial p_{j}}=U_{1}(\alpha_{j};u,v){p_{j}\choose
q_{j}},\quad 1\leq j\leq N,$ (3.1b) $\displaystyle u=-<Ap,p>,\quad v=<Aq,q>,$
(3.1c)

where $p=(p_{1},p_{2},\cdots,p_{N})^{T}$ and
$q=(q_{1},q_{2},\cdots,q_{N})^{T}$. Note that equation (3.1c), coinciding with
squared eigenfunction symmetry constraint, converts the KN spectral problem
(1.4) to the Hamiltonian equation (3.1b), which is nonlinear with respect to
the eigenfunction $(p,q)$. Such a procedure is usually referred to as
nonlinearisation of a Lax pair (cf.[8]). The associated Lax equation
$L_{x}=[U_{1},L]$ has a solution, which is

$L(\lambda;p,q)=\left(\begin{array}[]{cc}L^{11}(\lambda)&L^{12}(\lambda)\\\
L^{21}(\lambda)&-L^{11}(\lambda)\end{array}\right)=\left(\begin{array}[]{cc}\frac{1}{2}+Q_{\lambda}(A^{2}p,q)&-\lambda
Q_{\lambda}(Ap,p)\\\ \lambda
Q_{\lambda}(Aq,q)&-\frac{1}{2}-Q_{\lambda}(A^{2}p,q)\end{array}\right),$ (3.2)

where $Q_{\lambda}(\xi,\eta)=<(\lambda^{2}-A^{2})^{-1}\xi,\eta>$. It satisfies
the $r$-matrix Ansatz [3, 22, 27]

$\\{L(\lambda)\underset{,}{\otimes}L(\mu)\\}=[r(\lambda,\mu),L(\lambda)\otimes
I]+[r^{\prime}(\lambda,\mu),I\otimes L(\lambda)],$ (3.3)

where

$\displaystyle
r(\lambda,\mu)=\frac{2\lambda}{\lambda^{2}-\mu^{2}}P_{\lambda\mu},\quad
r^{\prime}(\lambda,\mu)=\frac{2\mu}{\lambda^{2}-\mu^{2}}P_{\mu\lambda}=-r(\mu,\lambda),$
$\displaystyle P_{\lambda\mu}=\left(\begin{array}[]{cccc}\lambda&0&0&0\\\
0&0&\mu&0\\\ 0&\mu&0&0\\\ 0&0&0&\lambda\end{array}\right).$

This implies the Poisson commutativity $\\{F(\lambda),F(\mu)\\}=0$, where
$F(\lambda)=\det L(\lambda)$ (cf.[60]) and the Poisson bracket is defined as

$\\{A,B\\}=\sum^{N}_{k=1}\biggl{(}\frac{\partial A}{\partial
q_{k}}\frac{\partial B}{\partial p_{k}}-\frac{\partial A}{\partial
p_{k}}\frac{\partial B}{\partial q_{k}}\biggr{)}.$

The Hamiltonian $H_{1}$ can be determined by the expansion

$H(\lambda)=-\displaystyle\frac{\sqrt{-F(\lambda)}}{2}=-\displaystyle\frac{1}{4}+\sum_{j=1}^{\infty}H_{j}\zeta^{-j},\
\ \zeta=\lambda^{2},$ (3.4)

which implies that $\\{H_{1},F(\lambda)\\}=0$. The generating function
$F(\lambda)$ can be expanded as

$F(\lambda)=-\frac{1}{4}+\sum_{k=1}^{N}\frac{\alpha_{k}^{2}E_{k}}{\lambda^{2}-\alpha_{k}^{2}},$
(3.5)

which yields a complete set of integrals for the Hamiltonian system
$(\mathbb{R}^{2N},{\rm d}p\wedge{\rm d}q,H_{1})$,

$E_{k}=(2<p,q>-1)p_{k}q_{k}-p_{k}^{2}q_{k}^{2}+\frac{\alpha_{k}}{2}\sum_{\begin{subarray}{c}1\leq
j\leq N;\\\ j\neq
k\end{subarray}}\Big{(}\frac{(p_{j}q_{k}-p_{k}q_{j})^{2}}{\alpha_{k}-\alpha_{j}}-\frac{(p_{j}q_{k}+p_{k}q_{j})^{2}}{\alpha_{k}+\alpha_{j}}\Big{)},$
(3.6)

where $1\leq k\leq N$. Suppose that the roots of $F(\lambda)$ are
$\zeta_{j}=\lambda_{j}^{2},~{}j=1,\ldots,N$, then we have the factorization

$F(\lambda)=-\frac{\prod_{j=1}^{N}(\zeta-\lambda_{j}^{2})}{4\alpha(\zeta)}=-\frac{R(\zeta)}{4\alpha^{2}(\zeta)},$
(3.7)

where
$\alpha(\zeta)=\prod_{j=1}^{N}(\zeta-\alpha_{j}^{2}),R(\zeta)=\alpha(\zeta)\prod_{j=1}^{N}(\zeta-\lambda_{j}^{2})$.
Thus a hyperelliptic curve

$\mathcal{R}:~{}~{}\xi^{2}=R(\zeta)=\prod_{j=1}^{N}(\zeta-\alpha_{j}^{2})(\zeta-\lambda_{j}^{2}),$
(3.8)

with genus $g=N-1$, is defined. The Riemann surface where $\zeta$ is consists
of two sheets, and the curve $\mathcal{R}$ is of hyperelliptic involution in
the sense that $\tau:(\zeta,\xi)\to(\zeta,-\xi)$ maps $\mathcal{R}$ to itself.
For a non-branching point $\zeta$ on the Riemann surface, when necessary, we
distinguish the two corresponding points on $\mathcal{R}$ by

$\mathfrak{p}_{+}(\zeta)=\big{(}\zeta,\,\xi=\sqrt{R(\zeta)}\big{)},\quad~{}\mathfrak{p}_{-}(\zeta)=\big{(}\zeta,\,\xi=-\sqrt{R(\zeta)}\big{)};$
(3.9)

and in particular, for the infinity $\infty$ on the Riemann surface, we denote
the two corresponding points on $\mathcal{R}$ by $\infty_{+},\,\infty_{-}$.

Generically [23, 30, 47] based on the curve (3.8), one can introduce Abelian
differentials of the first kind, by which a Riemann theta function can be
defined. One can refer to Appendix B for details.

Next, let us introduce our integrable map. Using the Darboux transformation
(2.1) (with (2.3)), we define the following linear map:

$\displaystyle
S_{\beta}:\mathbb{R}^{2N}\rightarrow\mathbb{R}^{2N},\quad(p,q)\mapsto(\widetilde{p},\widetilde{q}),$
(3.10a)
$\displaystyle{\widetilde{p}_{j}\choose\widetilde{q}_{j}}=\frac{1}{\sqrt{\alpha_{j}^{2}-\beta^{2}}}D^{(\beta)}(\alpha_{j};a,b){p_{j}\choose
q_{j}},\quad 1\leq j\leq N.$ (3.10b)

The extra factor in front of $D^{(\beta)}$ is to make the determinant to be 1,
which is useful in proving $S_{\beta}$ to be symplectic. One can convert the
constraint (3.1c) on $(u,v)$ to be the following constraint on $(a,b)$:

$\displaystyle P_{1}(a)\equiv(<Ap,p>b+1)a+<Ap,p>=0,$ (3.11a) $\displaystyle
P(\beta b)\equiv(\beta b)^{2}L^{12}(\beta)-2(\beta
b)L^{11}(\beta)-L^{21}(\beta)=0.$ (3.11b)

In fact, (3.11a) is nothing but

$P_{1}(a)=Z(<Ap,p>+u),$

where $a=Zu=(ab+1)u$ in (2.3c) and $u=-<Ap,p>$ have been used. To achieve
(3.11b), one may start from the constraint $v=<Aq,q>$ and consider

$P(\beta b)=\beta(<A\widetilde{q},\widetilde{q}>-\widetilde{v}).$

Replacing $\widetilde{v}$ by $b=Z\widetilde{v}$ from (2.3c) and replacing
$(\widetilde{p},\widetilde{q})$ using the map (3.10b), yield (3.11b).

In the following, by the nonlinearized map $S_{\beta}$ we mean the map (3.10)
after imposing constraint (3.11) on $(a,b)$. In fact, (3.11) indicates that
$(a,b)$ can be explicitly expressed in terms of $(p,q)$, which is denoted as

$(a,b)=f_{\beta}(p,q)$ (3.12a) where
$a=\frac{-<Ap,p>}{1+<Ap,p>b},~{}~{}b=\frac{-1}{\beta^{2}Q_{\beta}(Ap,p)}\Big{(}\frac{1}{2}+Q_{\beta}(A^{2}p,q)\pm\mathcal{H}(\beta)\Big{)},$
(3.12b)

with
$\mathcal{H}(\beta)=-2H(\beta)=\sqrt{-F(\beta)}=\displaystyle\frac{\sqrt{R(\beta^{2})}}{2\alpha(\beta^{2})}$
where $R(\beta^{2})$ is defined by (3.7). Note that $b$ is single-valued in
light of the monodromy formulation (3.9). So is $a$. Thus, after replacing
$(a,b)$ in (3.10) using (3.12b), the map is nonlinear with respect to $(p,q)$.

######  Proposition 3.1.

The nonlinearized map $S_{\beta}$ is symplectic and Liouville integrable,
sharing the same integrals $E_{1},\cdots,E_{N}$, defined by equation (3.6),
with the Hamiltonian system $(\mathbb{R}^{2N},{\rm d}p\wedge{\rm d}q,H_{1})$.

###### Proof.

Direct calculation from equation (3.10b) yields

$\sum_{j=1}^{N}({\rm d}\widetilde{p}_{j}\wedge{\rm d}\widetilde{q}_{j}-{\rm
d}p_{j}\wedge{\rm d}q_{j})=\frac{1}{2}{\rm d}\frac{P_{1}(a)}{ab+1}\wedge{\rm
d}b+\frac{1}{2}{\rm d}a\wedge{\rm
d}\Big{(}\frac{b^{2}P_{1}(a)}{ab+1}+\beta^{-1}P(\beta b)\Big{)},$ (3.13)

which vanishes when constraint (3.11) makes sense. This means the map
$S_{\beta}$ is symplectic. In addition, consider the Lax matrix $L$ given in
equation (3.2) and the Darboux matrix $D^{(\beta)}(\lambda;a,b)$ given in
(2.1). In light of the constraint (3.11), it turns out that

$\begin{split}L(\lambda;&\widetilde{p},\widetilde{q})D^{(\beta)}(\lambda;a,b)-D^{(\beta)}(\lambda;a,b)L(\lambda;p,q)\\\
&=\left(\begin{array}[]{cc}\beta^{2}b&\lambda\\\ \lambda
b^{2}&b\end{array}\right)P_{1}(a)+\left(\begin{array}[]{cc}\beta^{2}a&0\\\
\lambda(ab+1)&a\end{array}\right)\beta^{-1}P(\beta b)=0.\end{split}$ (3.14)

Thus $\det L(\lambda;\widetilde{p},\widetilde{q})=\det L(\lambda;p,q)$, which
indicates

$F(\lambda;\widetilde{p},\widetilde{q})=F(\lambda;p,q).$ (3.15)

It then follows from (3.5) that
$E_{k}(\widetilde{p},\widetilde{q})=E_{k}(p,q)$, which are invariants of the
map $S_{\beta}$.

∎

##  4 Algebro-geomitric solutions to the lpmKP equation

In this section we proceed to derive algebro-geometric solutions to the lpmKP
equation (1.2).

First, using the integrable symplectic map (3.10), we define discrete phase
flow $\big{(}p(m),\,q(m)\big{)}=S_{\beta}^{m}(p(0),q(0))$ with initial point
$(p(0),q(0))\in\mathbb{R}^{2N}$, and then we use (3.12) to define finite genus
potential $(a,b)$ in (2.1), i.e.

$(a_{m},b_{m})=f_{\beta}\big{(}p(m),\,q(m)\big{)},$ (4.1)

which coincides with $u_{m}=-<Ap(m),p(m)>,\,v_{m}=<Aq(m),q(m)>$. In light of
(2.3), we also have

$\displaystyle a_{m}=Z_{m}u_{m},\quad b_{m}=Z_{m}v_{m+1},$ (4.2a)
$\displaystyle Z_{m}=a_{m}b_{m}+1=\frac{2}{1+\sqrt{1-4u_{m}v_{m+1}}}.$ (4.2b)

We will finally reconstruct $Z_{m}$ in terms of theta function (see equation
(4.25)) and by “integration” from (2.10) we recover the lpmKP solution $W$.

Next, consider the discrete KN spectral problem (2.1) and the discrete Lax
equation (3.14) along the $S_{\beta}^{m}$-flow, which are rewritten as

$h(m+1,\lambda)=D_{m}(\lambda)h(m,\lambda)$ (4.3)

and

$L_{m+1}(\lambda)D_{m}(\lambda)=D_{m}(\lambda)L_{m}(\lambda),$ (4.4)

where the Darboux matrix $D_{m}(\lambda)$ is

$D_{m}(\lambda)=D^{(\beta)}(\lambda;a_{m},b_{m})=\left(\begin{array}[]{cc}\lambda^{2}Z_{m}-\beta^{2}&\lambda
Z_{m}u_{m}\\\ \lambda Z_{m}v_{m+1}&1\end{array}\right)$ (4.5)

and $L_{m}(\lambda)=L\big{(}\lambda;p(m),q(m)\big{)}$ is defined as the form
(3.2). Let $M(m,\lambda)=\Bigl{(}\begin{smallmatrix}M^{11}&M^{12}\\\
M^{21}&M^{22}\end{smallmatrix}\Bigr{)}$ be a fundamental solution matrix of
(4.3) with $M(0,\lambda)$ being the unit matrix $I$. It turns out that

$\displaystyle M(m,\lambda)=D_{m-1}(\lambda)D_{m-2}(\lambda)\cdots
D_{0}(\lambda),$ (4.6a) $\displaystyle
L_{m}(\lambda)M(m,\lambda)=M(m,\lambda)L_{0}(\lambda),$ (4.6b)

and we then have $\det M(m,\lambda)=(\zeta-\beta^{2})^{m}$ due to $\det
D_{m}(\lambda)=\zeta-\beta^{2}$. For such an $M(m,\lambda)$ one can obtain its
asymptotic behaviors from (4.6a).

######  Lemma 4.1.

$M^{11}(m,\lambda),\lambda M^{12}(m,\lambda),\lambda M^{21}(m,\lambda)$ and
$M^{22}(m,\lambda)$ are polynomials of $\zeta=\lambda^{2}$. When
$\zeta\sim\infty$, for $m\geq 2$ we have

$\displaystyle M^{11}(m,\lambda)=Z_{0}Z_{1}\cdots
Z_{m-1}\zeta^{m}+O(\zeta^{m-1}),$ (4.7a) $\displaystyle\lambda
M^{12}(m,\lambda)=u_{0}Z_{0}Z_{1}\cdots Z_{m-1}\zeta^{m}+O(\zeta^{m-1}),$
(4.7b) $\displaystyle\lambda M^{21}(m,\lambda)=v_{m}Z_{0}Z_{1}\cdots
Z_{m-1}\zeta^{m}+O(\zeta^{m-1}),$ (4.7c) $\displaystyle
M^{22}(m,\lambda)=u_{0}v_{m}Z_{0}Z_{1}\cdots
Z_{m-1}\zeta^{m-1}+O(\zeta^{m-2}),$ (4.7d)

and for $m=1$ they are still valid except $M^{22}(1,\lambda)=1$. When
$\zeta\sim 0$, we have $(m\geq 1)$

$\begin{split}&M^{11}=(-\beta^{2})^{m}+O(\zeta),\quad\lambda
M^{12}=O(\zeta),\\\ &\lambda M^{21}=O(\zeta),\quad
M^{22}=1+O(\zeta).\end{split}$ (4.8)

Equation (4.4) indicates that the solution space of equation (4.3) is
invariant under the action of the linear map $L_{m}(\lambda)$. From (3.2), the
traceless $L_{m}(\lambda)$ allows two opposite eigenvalues, denoted as
$\pm\mathcal{H}(\lambda)=\pm\sqrt{-F(\lambda)}$, which are independent of the
discrete argument $m$ due to relation (3.15). Denoting the corresponding
eigenvectors by $h_{\pm}(m,\lambda)=(h_{\pm}^{(1)},h_{\pm}^{(2)})^{T}$, we
have

$L_{m}(\lambda)h_{\pm}(m,\lambda)=\pm\mathcal{H}(\lambda)h_{\pm}(m,\lambda),$
(4.9a) and $h_{\pm}(m+1,\lambda)=D_{m}(\lambda)h_{\pm}(m,\lambda),$ (4.9b)

simultaneously. Noting that the rank of
$L_{m}(\lambda)\mp\mathcal{H}(\lambda)$ is 1, which means in each case the
common eigenvector is uniquely determined up to a constant factor, we select
two eigenvectors $h_{\pm}(m,\lambda)$ defined through $M(m,\lambda)$, as the
following,

$h_{\pm}(m,\lambda)={h_{\pm}^{(1)}\choose
h_{\pm}^{(2)}}=M(m,\lambda){c_{\lambda}^{\pm}\choose 1},$ (4.10)

where the constants $c_{\lambda}^{\pm}$ are determined through (4.9a) and
(4.6b), by

$L_{0}(\lambda)\begin{pmatrix}c_{\lambda}^{+}&c_{\lambda}^{-}\\\
1&1\end{pmatrix}=\begin{pmatrix}c_{\lambda}^{+}&c_{\lambda}^{-}\\\
1&1\end{pmatrix}\begin{pmatrix}\mathcal{H}(\lambda)&0\\\
0&-\mathcal{H}(\lambda)\end{pmatrix},$

i.e. taking $m=0$ in equation (4.9a). It turns out that

$c_{\lambda}^{\pm}=\frac{L_{0}^{11}(\lambda)\pm\mathcal{H}(\lambda)}{L_{0}^{21}(\lambda)}=\frac{-L_{0}^{12}(\lambda)}{L_{0}^{11}(\lambda)\mp\mathcal{H}(\lambda)}.$
(4.11)

Next, we will investigate these eigenvectors $h_{\pm}(m,\lambda)$ using the
Baker–Akhiezer functions, which can be expressed by theta function on the
hyperelliptic Riemann surface corresponding to the spectral curve
$\mathcal{R}$. Let us introduce the Baker–Akhiezer functions, which are
meromorphic on $\mathcal{R}$, by

$\begin{split}&\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{+}(\lambda^{2})\big{)}=\lambda
h_{+}^{(1)}(m,\lambda),\quad\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{-}(\lambda^{2})\big{)}=\lambda
h_{-}^{(1)}(m,\lambda),\\\
&\mathfrak{h}^{(2)}\big{(}m,\mathfrak{p}_{+}(\lambda^{2})\big{)}=h_{+}^{(2)}(m,\lambda),\quad\mathfrak{h}^{(2)}\big{(}m,\mathfrak{p}_{-}(\lambda^{2})\big{)}=h_{-}^{(2)}(m,\lambda).\end{split}$
(4.12)

To associate them with the Riemann theta function (see (B.8)), we investigate
their analytic behaviors and divisors. To this end, introduce elliptic
variables $\mu_{j},\,\nu_{j}$ in $L^{12}$ and $L^{21}$ by

$\displaystyle\lambda^{-1}L_{m}^{12}(\lambda)=-Q_{\lambda}(Ap(m),p(m))=\frac{u_{m}}{\alpha(\zeta)}\prod_{j=1}^{N-1}\big{(}\zeta-\mu_{j}^{2}(m)\big{)},$
(4.13a)
$\displaystyle\lambda^{-1}L_{m}^{21}(\lambda)=Q_{\lambda}(Aq(m),q(m))=\frac{v_{m}}{\alpha(\zeta)}\prod_{j=1}^{N-1}\big{(}\zeta-\nu_{j}^{2}(m)\big{)},$
(4.13b)

which lead us, from equations (4.6a) and (4.6b), to

$\displaystyle\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{+}(\lambda^{2})\big{)}\cdot\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{-}(\lambda^{2})\big{)}=\zeta(\zeta-\beta^{2})^{m}\frac{-u_{m}}{v_{0}}\prod_{j=1}^{N-1}\frac{\zeta-\mu_{j}^{2}(m)}{\zeta-\nu_{j}^{2}(0)},$
(4.14a)
$\displaystyle\mathfrak{h}^{(2)}\big{(}m,\mathfrak{p}_{+}(\lambda^{2})\big{)}\cdot\mathfrak{h}^{(2)}\big{(}m,\mathfrak{p}_{-}(\lambda^{2})\big{)}=(\zeta-\beta^{2})^{m}\frac{v_{m}}{v_{0}}\prod_{j=1}^{N-1}\frac{\zeta-\nu_{j}^{2}(m)}{\zeta-\nu_{j}^{2}(0)}.$
(4.14b)

With regard to asymptotic behaviors of the Baker–Akhiezer functions, we have
the following lemmas.

######  Lemma 4.2.

The Baker–Akhiezer functions (4.12) have the following asymptotic behaviors as
$\zeta=\lambda^{2}\sim\infty$,

$\displaystyle\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{+}(\lambda^{2})\big{)}=\frac{1}{2v_{0}}Z_{0}Z_{1}\cdots
Z_{m-1}\zeta^{m+1}\big{(}1+O(\zeta^{-1})\big{)},$ (4.15a)
$\displaystyle\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{-}(\lambda^{2})\big{)}=\frac{-2u_{m}}{Z_{0}Z_{1}\cdots
Z_{m-1}}\big{(}1+O(\zeta^{-1})\big{)},$ (4.15b)
$\displaystyle\mathfrak{h}^{(2)}\big{(}m,\mathfrak{p}_{+}(\lambda^{2})\big{)}=\frac{v_{m}}{2v_{0}}Z_{0}Z_{1}\cdots
Z_{m-1}\zeta^{m}\big{(}1+O(\zeta^{-1})\big{)},$ (4.15c)
$\displaystyle\mathfrak{h}^{(2)}\big{(}m,\mathfrak{p}_{-}(\lambda^{2})\big{)}=\frac{2}{Z_{0}Z_{1}\cdots
Z_{m-1}}\big{(}1+O(\zeta^{-1})\big{)}.$ (4.15d)

###### Proof.

First, as $\lambda\sim\infty$ we find $\lambda
c_{\lambda}^{+}=(\zeta/2v_{0})\big{(}1+O(\zeta^{-1})\big{)}$ from equations
(4.11) and (3.2). Then, using (4.10) and the asymptotic results of
$M(m,\lambda)$ given in Lemma 4.1, one can obtain (4.15a) and (4.15c). The
other two in (4.15) follow from (4.14).

∎

######  Lemma 4.3.

When $\zeta=\lambda^{2}\sim 0$, the following asymptotic behaviors hold,

$\displaystyle\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{+}(\lambda^{2})\big{)}=(-\beta^{2})^{m}\frac{1-2<p(0),q(0)>}{-<A^{-1}q(0),q(0)>}\big{(}1+O(\zeta)\big{)},$
(4.16a)
$\displaystyle\mathfrak{h}^{(1)}\big{(}m,\mathfrak{p}_{-}(\lambda^{2})\big{)}=\zeta\frac{<A^{-1}q(0),q(0)>u_{m}}{(1-2<p(0),q(0)>)v_{0}}\left(\prod_{j=1}^{N-1}\frac{\mu_{j}^{2}(m)}{\nu_{j}^{2}(0)}\right)\big{(}1+O(\zeta)\big{)}.$
(4.16b)

###### Proof.

From (4.11) we have $\lambda
c_{\lambda}^{+}=-\frac{1-2<p(0),q(0)>}{<A^{-1}q(0),q(0)>}\big{(}1+O(\zeta)\big{)}$
as $\lambda\sim 0$. Equations (4.8) and (4.10) yield (4.16a), which further
gives rise to (4.16b) by using (4.14a).

∎

Now we are able to write down divisors of the Baker–Akhiezer functions
$\mathfrak{h}^{(1)}(m,\mathfrak{p}),\,\mathfrak{h}^{(2)}(m,\mathfrak{p})$ on
$\mathcal{R}$, which are, respectively, (cf.[23, 30, 47])

$\displaystyle\mathcal{D}(\mathfrak{h}^{(1)}(m,\mathfrak{p}))=\sum_{j=1}^{g}\Big{(}\mathfrak{p}\big{(}\mu_{j}^{2}(m)\big{)}-\mathfrak{p}\big{(}\nu_{j}^{2}(0)\big{)}\Big{)}+\\{\mathfrak{o}_{-}\\}+m\\{\mathfrak{p}(\beta^{2})\\}-(m+1)\\{\infty_{+}\\},$
(4.17a)
$\displaystyle\mathcal{D}(\mathfrak{h}^{(2)}(m,\mathfrak{p}))=\sum_{j=1}^{g}\Big{(}\mathfrak{p}\big{(}\nu_{j}^{2}(m)\big{)}-\mathfrak{p}\big{(}\nu_{j}^{2}(0)\big{)}\Big{)}+m\\{\mathfrak{p}(\beta^{2})\\}-m\\{\infty_{+}\\},$
(4.17b)

where $\mathfrak{o}_{-}=(\zeta=0,\,\xi=-\sqrt{R(0)})$, $g=N-1$.

Next, introduce the Abel–Jacobi variables

$\vec{\psi}(m)=\mathcal{A}\Big{(}\hbox{$\sum$}_{j=1}^{g}\mathfrak{p}\big{(}\mu_{j}^{2}(m)\big{)}\Big{)},\quad\vec{\phi}(m)=\mathcal{A}\Big{(}\hbox{$\sum$}_{j=1}^{g}\mathfrak{p}\big{(}\nu_{j}^{2}(m)\big{)}\Big{)},$
(4.18)

by using the Abel map $\mathcal{A}$ (see Appendix B). Employing Toda’s dipole
technique [56], from (4.18) and (4.17) we have

$\displaystyle\vec{\psi}(m)\equiv\vec{\phi}(0)+m\vec{\Omega}_{\beta}+\vec{\Omega}_{0},\quad({\rm
mod}\,\mathcal{T}),$ (4.19a)
$\displaystyle\vec{\phi}(m)\equiv\vec{\phi}(0)+m\vec{\Omega}_{\beta},\quad({\rm
mod}\,\mathcal{T}),$ (4.19b)
$\displaystyle\vec{\Omega}_{\beta}=\int_{\mathfrak{p}(\beta^{2})}^{\infty_{+}}\vec{\omega},\quad\vec{\Omega}_{0}=\int_{\mathfrak{o}_{-}}^{\infty_{+}}\vec{\omega}.$
(4.19c)

Then, as usual treatment (cf.[9, 23, 30, 47]), by comparing divisors we obtain
express the Baker-Akhiezer functions in terms of the Riemann theta function
(B.8):

$\displaystyle\mathfrak{h}^{(1)}(m,\mathfrak{p})=C_{m}^{(1)}\frac{\theta(-\mathcal{A}(\mathfrak{p})+\vec{\psi}(m)+\vec{K};B)}{\theta(-\mathcal{A}(\mathfrak{p})+\vec{\phi}(0)+\vec{K};B)}\exp\int_{\mathfrak{p}_{0}}^{\mathfrak{p}}(m\,\omega[\mathfrak{p}(\beta^{2}),\infty_{+}]+\omega[\mathfrak{o}_{-},\infty_{+}]),$
(4.20a)
$\displaystyle\mathfrak{h}^{(2)}(m,\mathfrak{p})=C_{m}^{(2)}\frac{\theta(-\mathcal{A}(\mathfrak{p})+\vec{\phi}(m)+\vec{K};B)}{\theta(-\mathcal{A}(\mathfrak{p})+\vec{\phi}(0)+\vec{K};B)}\exp\int_{\mathfrak{p}_{0}}^{\mathfrak{p}}m\,\omega[\mathfrak{p}(\beta^{2}),\infty_{+}],$
(4.20b)

where $C_{m}^{(1)}$ and $C_{m}^{(2)}$ are constant factors and the Riemann
constant vector $\vec{K}$ is defined in (B.9). Here,
$\omega[\mathfrak{p},\mathfrak{q}]$ is the dipole, a meromorphical
differential that has only simple poles at $\mathfrak{p}$ and $\mathfrak{q}$
with residues $+1$ and $-1$, respectively.

Our purpose is to derive explicit expression of $Z_{m}$ in terms of the
Riemann theta function. To achieve that, first, we take
$\mathfrak{p}\rightarrow\infty_{-}$ in equation (4.20b) and then compare the
result with the asymptotic formula (4.15d). This gives rise to

$C_{m}^{(2)}=\frac{2}{Z_{0}Z_{1}\cdots
Z_{m-1}}\frac{\theta[\vec{\phi}(0)+\vec{K}+\vec{\eta}_{\infty_{-}}]}{\theta[\vec{\phi}(m)+\vec{K}+\vec{\eta}_{\infty_{-}}]}\exp\int_{\infty_{-}}^{\mathfrak{p}_{0}}m\,\omega[\mathfrak{p}(\beta^{2}),\infty_{+}],$
(4.21)

where $\vec{\eta}_{\infty_{-}}=-\mathcal{A}(\infty_{-})$. Next, we consider
the second row in equation (4.9b), i.e.

$\mathfrak{h}^{(2)}(m+1,\mathfrak{p})=b_{m}\mathfrak{h}^{(1)}(m,\mathfrak{p})+\mathfrak{h}^{(2)}(m,\mathfrak{p}),$
(4.22)

which reads

$\mathfrak{h}^{(2)}(m+1,\mathfrak{o}_{-})=\mathfrak{h}^{(2)}(m,\mathfrak{o}_{-})$
(4.23)

at the point $\mathfrak{o}_{-}$ since
$\mathfrak{h}^{(1)}(m,\mathfrak{o}_{-})=0$. Substituting (4.20b) with
$\mathfrak{p}=\mathfrak{o}_{-}$ into (4.23) immediately yields

$\frac{C_{m}^{(2)}}{C_{m+1}^{(2)}}=\frac{\theta(\vec{\phi}(m+1)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}};B)}{\theta(\vec{\phi}(m)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}};B)}\exp\int_{\mathfrak{p}_{0}}^{\mathfrak{o}_{-}}\omega[\mathfrak{p}(\beta^{2}),\infty_{+}],$
(4.24)

where $\vec{\eta}_{\mathfrak{o}_{-}}=-\mathcal{A}(\mathfrak{o}_{-})$. Now,
substituting (4.21) into the above equation, we arrive at an explicit
expression of $Z_{m}$ in terms of theta function, i.e.

$Z_{m}=\frac{\theta(\vec{\phi}(m+1)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}};B)\cdot\theta(\vec{\phi}(m)+\vec{K}+\vec{\eta}_{\infty_{-}};B)}{\theta(\vec{\phi}(m+1)+\vec{K}+\vec{\eta}_{\infty_{-}};B)\cdot\theta(\vec{\phi}(m)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}};B)}\exp\int_{\infty_{-}}^{\mathfrak{o}_{-}}\omega[\mathfrak{p}(\beta^{2}),\infty_{+}].$
(4.25)

With $Z_{m}$ in hand, for a function $W_{m}$ that obeys equation
$W_{m+1}-W_{m}=\ln Z_{m}$ where $Z_{m}$ is given in (4.25), one can obtain an
explicit solution by “integration”,

$W_{m}=W_{0}+\ln\frac{\theta[\vec{\phi}(m)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}}]\cdot\theta[\vec{\phi}(0)+\vec{K}+\vec{\eta}_{\infty_{-}}]}{\theta[\vec{\phi}(m)+\vec{K}+\vec{\eta}_{\infty_{-}}]\cdot\theta[\vec{\phi}(0)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}}]}+m\int_{\infty_{-}}^{\mathfrak{o}_{-}}\omega[\mathfrak{p}(\beta^{2}),\infty_{+}].$
(4.26)

######  Remark 4.1.

Explicit expression for $u_{m}$ and $v_{m}$ can be computed with the help of
equations (4.15b), (4.15c), (4.20) and (4.25). This will lead to the finite-
gap solutions of the ldNLS equation (2.11), in a similar way to the following.

The above discussions and results are valid for $(m,\beta)=(m_{i},\beta_{i})$,
$i=1,2,3$. Thus, we have three integrable symplectic maps
$S_{\beta_{1}},\,S_{\beta_{2}}$ and $S_{\beta_{3}}$, which commute with each
other since they share the same Liouville integrals $E_{1},\cdots,E_{N}$
(cf.[6, 12, 13, 57, 58]). This enables us to define

$\big{(}p(m_{1},m_{2},m_{3}),\,q(m_{1},m_{2},m_{3})\big{)}=S_{\beta_{1}}^{m_{1}}S_{\beta_{2}}^{m_{2}}S_{\beta_{3}}^{m_{3}}(p(0,0,0),q(0,0,0)),$
(4.27)

and the components $(p_{j},q_{j})$ satisfy the equation (3.10b) for
$\beta=\beta_{1},\beta_{2},\beta_{3}$ simultaneously in the case of
$\lambda=\alpha_{j}$. This leads to a compatible solution

$\chi(m_{1},m_{2},m_{3})=(\alpha_{j}^{2}-\beta_{1}^{2})^{\frac{m_{1}}{2}}(\alpha_{j}^{2}-\beta_{2}^{2})^{\frac{m_{2}}{2}}(\alpha_{j}^{2}-\beta_{3}^{2})^{{\frac{m_{3}}{2}}}(p_{j},q_{j})$
(4.28)

that satisfies the three equations in (2.6) with spectral parameter
$\lambda=\alpha_{j}$. Thus, one can obtain $W(m_{1},m_{2},m_{3})$ by
“integrating” (2.6), and in light of Theorem 2.1, such $W(m_{1},m_{2},m_{3})$
provides solutions to the lpmKP equation (1.2).

######  Theorem 4.1.

The lpmKP equation (1.2) admits algebro-geometric solutions expressed in terms
of the Riemann theta function,

$\displaystyle W(m_{1},m_{2},m_{3})=$
$\displaystyle\ln\frac{\theta(\sum_{k=1}^{3}m_{k}\vec{\Omega}_{\beta_{k}}+\vec{\phi}(0,0,0)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}};B)\cdot\theta(\vec{\phi}(0,0,0)+\vec{K}+\vec{\eta}_{\infty_{-}};B)}{\theta(\sum_{k=1}^{3}m_{k}\vec{\Omega}_{\beta_{k}}+\vec{\phi}(0,0,0)+\vec{K}+\vec{\eta}_{\infty_{-}};B)\cdot\theta(\vec{\phi}(0,0,0)+\vec{K}+\vec{\eta}_{\mathfrak{o}_{-}};B)}$
$\displaystyle+\sum_{k=1}^{3}m_{k}\int_{\infty_{-}}^{\mathfrak{o}_{-}}\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]+W(0,0,0),$
(4.29)

where $\vec{\eta}_{\mathfrak{o}_{-}}=-\mathcal{A}(\mathfrak{o}_{-})$,
$\vec{\eta}_{\infty_{-}}=-\mathcal{A}(\infty_{-})$, $\vec{K}$ is the Riemann
constant vector (B.9),

$\vec{\Omega}_{\beta_{k}}=\int_{\mathfrak{p}(\beta^{2}_{k})}^{\infty_{+}}\vec{\omega},$
(4.30)

and the dipole differential $\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]$
is defined as

$\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]=\left(\zeta+\frac{\xi+\sqrt{R(\beta_{k}^{2})}}{\zeta-\beta_{k}^{2}}\right)\frac{\mathrm{d}\zeta}{2\sqrt{R(\zeta)}},$
(4.31)

with $\xi$ and $R(\zeta)$ given by (3.8).

##  5 An example: $g=1$ case

As an example of the solution (4.1), in the following we explore the simplest
case, where genus $g=1$. The elliptic curve $\mathcal{R}$ (3.8) reads

$\xi^{2}=R(\zeta)=(\zeta-\zeta_{1})(\zeta-\zeta_{2})(\zeta-\zeta_{3})(\zeta-\zeta_{4}),$
(5.1)

with
$\zeta_{1}=\lambda_{1}^{2},\zeta_{2}=\lambda_{2}^{2},\zeta_{3}=\alpha_{1}^{2},\zeta_{4}=\alpha_{2}^{2}$.

In our example, we assume all $\\{\zeta_{j}\\}$ are on real axis and
$\zeta_{1}<\zeta_{2}<\zeta_{3}<\zeta_{4}$ so that related branch cuts are
taken as $[\zeta_{1},\zeta_{2}]$ and $[\zeta_{3},\zeta_{4}]$. Thus, we have
the Abelian differential of the first kind

$\omega_{1}=\displaystyle\frac{C_{11}\mathrm{d}\zeta}{2\sqrt{(\zeta-\zeta_{1})(\zeta-\zeta_{2})(\zeta-\zeta_{3})(\zeta-\zeta_{4})}},$
(5.2)

where the normalization constant $C_{11}$ is

$C_{11}^{-1}=\int_{a_{1}}\displaystyle\frac{\mathrm{d}\zeta}{2\sqrt{(\zeta-\zeta_{1})(\zeta-\zeta_{2})(\zeta-\zeta_{3})(\zeta-\zeta_{4})}}$
(5.3)

along with an $a_{1}$-period. Then we have the two periods

$\displaystyle 1=$
$\displaystyle\int_{a_{1}}\omega_{1}=\int_{\zeta_{3}}^{\zeta_{4}}\displaystyle\frac{C_{11}\mathrm{d}\zeta}{\sqrt{(\zeta-\zeta_{1})(\zeta-\zeta_{2})(\zeta-\zeta_{3})(\zeta-\zeta_{4})}},$
(5.4a) $\displaystyle B_{11}=$
$\displaystyle\int_{b_{1}}\omega_{1}=\int_{\zeta_{2}}^{\zeta_{3}}\displaystyle\frac{C_{11}\mathrm{d}\zeta}{\sqrt{(\zeta-\zeta_{1})(\zeta-\zeta_{2})(\zeta-\zeta_{3})(\zeta-\zeta_{4})}}.$
(5.4b)

Note that by certain linear fractional transformation (see [56]) the above two
formulae can be converted to the elliptic integrals of the first kind:

$\displaystyle 1=$
$\displaystyle\int_{a_{1}}\omega_{1}=A_{0}\int_{1}^{\frac{1}{\kappa}}\displaystyle\frac{\mathrm{d}s}{\sqrt{(1-s^{2})(1-\kappa^{2}s^{2})}},$
(5.5a) $\displaystyle B_{11}=$
$\displaystyle\int_{b_{1}}\omega_{1}=A_{0}\int_{-1}^{1}\displaystyle\frac{\mathrm{d}s}{\sqrt{(1-s^{2})(1-\kappa^{2}s^{2})}},$
(5.5b)

where $\kappa\in(0,1)$ is a constant.

In this case the Riemann theta function reduces to the Jacobi theta function
$\vartheta_{3}$,

$\theta(z;B_{11})=\sum_{n=-\infty}^{+\infty}\exp[\pi\sqrt{-1}(n^{2}B_{11}+2nz)]=\vartheta_{3}(z\mid
B_{11}),\ \ z\in\mathbb{C}.$ (5.6)

Hence, the algebro-geometric solution (4.1) in the case of $g=1$ can be
expressed as

$\displaystyle W(m_{1},m_{2},m_{3})=$
$\displaystyle\ln\frac{\vartheta_{3}(\sum_{k=1}^{3}m_{k}\Omega_{\beta_{k}}+\phi(0,0,0)+K_{1}+\eta_{\mathfrak{o}_{-}}|B_{11})\cdot\vartheta_{3}(\phi(0,0,0)+K_{1}+\eta_{\infty_{-}}|B_{11})}{\vartheta_{3}(\sum_{k=1}^{3}m_{k}\Omega_{\beta_{k}}+\phi(0,0,0)+K_{1}+\eta_{\infty_{-}}|B_{11})\cdot\vartheta_{3}(\phi(0,0,0)+K_{1}+\eta_{\mathfrak{o}_{-}}|B_{11})}$
$\displaystyle+\sum_{k=1}^{3}m_{k}\int_{\infty_{-}}^{\mathfrak{o}_{-}}\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]+W(0,0,0),$
(5.7)

where

$\displaystyle
K_{1}=-\int_{a_{1}}\mathcal{A}\,\omega_{1}+\displaystyle\frac{B_{11}}{2},\ \
\mathcal{A}=\mathcal{A}(\mathfrak{p})=\int_{\mathfrak{p}_{0}}^{\mathfrak{p}}\omega_{1},$
(5.8a)
$\displaystyle\Omega_{\beta_{k}}=\int_{\mathfrak{p}(\beta_{k}^{2})}^{\infty_{+}}\omega_{1},\
\
\eta_{\mathfrak{o}_{-}}=-\int_{\mathfrak{p}_{0}}^{\mathfrak{o}_{-}}\omega_{1},\
\ \eta_{\infty_{-}}=-\int_{\mathfrak{p}_{0}}^{\infty_{-}}\omega_{1},$ (5.8b)
$\displaystyle\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]=\displaystyle\frac{1}{C_{11}}\left(\zeta+\displaystyle\frac{\xi+\sqrt{R(\beta_{k}^{2})}}{\zeta-\beta_{k}^{2}}\right)\omega_{1}.$
(5.8c)

Note that due to the arbitrariness of $\phi(0,0,0)$ we can always vanish
$\phi(0,0,0)+K_{1}$ and thus we come to

$\displaystyle
W(m_{1},m_{2},m_{3})=W_{2}(m_{1},m_{2},m_{3})+W_{1}(m_{1},m_{2},m_{3})$ (5.9a)
with $\displaystyle
W_{2}(m_{1},m_{2},m_{3})=\ln\frac{\vartheta_{3}(\sum_{k=1}^{3}m_{k}\Omega_{\beta_{k}}+\eta_{\mathfrak{o}_{-}}|B_{11})\cdot\vartheta_{3}(\eta_{\infty_{-}}|B_{11})}{\vartheta_{3}(\sum_{k=1}^{3}m_{k}\Omega_{\beta_{k}}+\eta_{\infty_{-}}|B_{11})\cdot\vartheta_{3}(\eta_{\mathfrak{o}_{-}}|B_{11})},$
(5.9b) $\displaystyle
W_{1}(m_{1},m_{2},m_{3})=\sum_{k=1}^{3}m_{k}\int_{\infty_{-}}^{\mathfrak{o}_{-}}\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]+W(0,0,0),$
(5.9c)

where $\Omega_{\beta_{k}},\eta_{\mathfrak{o}_{-}},\eta_{\infty_{-}}$ and
$\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]$ are computed from (5.8b), and
$W_{1}(m_{1},m_{2},m_{3})$ acts as a linear background of
$W(m_{1},m_{2},m_{3})$.

To illustrate the solution, we take

$\zeta_{1}=1,~{}\zeta_{2}=3,~{}\zeta_{3}=5,~{}\zeta_{4}=8,~{}\beta_{1}^{2}=10,~{}\beta_{2}^{2}=13,~{}\beta_{3}^{2}=14,$
(5.10)

and consequently,

$\mathfrak{p}(\beta_{1}^{2})=(10,25.0998),~{}~{}\mathfrak{p}(\beta_{2}^{2})=(13,69.282),~{}~{}\mathfrak{p}(\beta_{3}^{2})=(14,87.8749).$

It follows from (5.3), (5.4) and (5.8) that (the integrals are computed
numerically using Mathematica)

$\begin{split}&C_{11}=1.30467\,i,~{}~{}B_{11}=1.21091\,i,~{}~{}\eta_{\mathfrak{o}_{-}}=-0.391547\,i,~{}~{}\eta_{\infty_{-}}=-0.569795\,i,~{}~{}\\\
&\Omega_{\beta_{1}}=0.123456\,i,~{}~{}\Omega_{\beta_{2}}=0.0769913\,i,~{}~{}\Omega_{\beta_{3}}=0.068633\,i,~{}~{}\\\
&\int_{\infty_{-}}^{\mathfrak{o}_{-}}\omega[\mathfrak{p}(\beta_{k}^{2}),\infty_{+}]=\gamma_{k},~{}~{}~{}\gamma_{1}=4.80874,~{}~{}\gamma_{2}=4.96645,~{}~{}\gamma_{3}=5.03085,\end{split}$
(5.11)

where we have taken $\mathfrak{p}_{0}=(-3.0,45.9565)$ in (5.8). The quasi-
periodic evolution of $W_{2}(m_{1},m_{2},m_{3})$ is shown in Figure 1.

(a)

(b)

(c)

(d)

Figure 1: Shape and motion of $W_{2}(m_{1},m_{2},m_{3})$ given in (5.9b) for
$\\{\zeta_{j}\\}$ in (5.10) and $\mathfrak{p}_{0}=(-3.0,45.9565)$. (a) 3D plot
of $W_{2}(m_{1},m_{2},0)$. (b) 2D plot of $W_{2}(m_{1},0,0)$. (c) 2D plot of
$W_{2}(0,m_{2},0)$. (d) 2D plot of $W_{2}(0,0,m_{3})$.

One can see a periodic wave coupled with an apparent linear background that is
different from $W_{1}(m_{1},m_{2},m_{3})$. This is because in our example all
$\\{\Omega_{k}\\}$ and $B_{11}$ are pure imaginary and Jacobi’s function
$\vartheta_{3}(z|B_{11})$ has a $z$-dependent periodic multiplier $e^{-\pi
iB_{11}}e^{-2\pi iz}$ with respect to $B_{11}$, i.e.

$\vartheta_{3}(z+B_{11}\mid B_{11})=e^{-\pi iB_{11}}e^{-2\pi
iz}\vartheta_{3}(z\mid B_{11}).$

It is the periodic multiplier to give rise to the linear background when
$W_{2}(m_{1},m_{2},m_{3})$ evolves with respect to $\\{m_{k}\\}$ via the
formula (5.9b).

##  6 Concluding remarks

In this paper we constructed algebro-geometric solutions (4.1) to the lpmKP
equation (1.2). The KN spectral problem (1.4) was employed as an associated
spectral problem, of which the Darboux transformation gives rise to the Lax
triad (2.6) for the lpmKP equation. Compared with the known one (e.g.
Eq.(3.113) in [31]), this Lax triad is not explicit, as the discrete potential
function $W$ is defined (via $Z^{(j)}$) by the KN functions $(u,v)$. Discrete
evolutions are introduced by regarding the Darboux transformation as a map
(3.10) to generate discrete flows. The map is shown to be symplectic and
integrable, sharing the same integrals with the continuous Hamiltonian system
(3.1) which is obtained through the so-called nonlinearisation of the KN
spectral problem. Then, employing algebro-geometric techniques, using the
Baker–Akhiezer functions and Abel map we introduced the Riemann theta function
and finally obtained the algebro-geometric solutions (4.1) for the lpmKP
equation. As an example, in Sec.5 we presented an explicit solution for the
$g=1$ case and illustrated it in Fig.1.

Based on a series of work [9, 10, 12, 13, 60, 61, 62], in Sec.1 we have
summarized a framework of the approach for constructing algebro-geometric
solutions to multidimensionally consistent systems. The approach has proved
effective and this paper added one more important successful example.
Reviewing the series of work [9, 10, 12, 13, 60, 61, 62] and the present
paper, there are several related problems that are interesting and remain
open. Let us raise them below.

One is to extend solutions to full space. In fact, in the solutions (4.1) all
$m_{i}\geq 0$, i.e. the solutions are defined on the first octant. The same
thing happened to the lpKP equation [10]. For those quadrilateral equations
studied in [9, 12, 13, 60, 61, 62], the obtained algebro-geometric solutions
are defined on the first quadrant. This is because in this approach discrete
flows are generated by iterating maps (e.g. (3.10)) towards one direction. One
may develop the approach to obtain solutions in full space.

The second is to construct algebro-geometric solutions containing two soliton
parameters for 3D lattice equations. Let us explain the problem below. In
general, an integrable 3D equation admits a plane-wave factor (PWF) with two
independent soliton parameters. For example, for the discrete AKP equation,
its PWF reads [46]

$\rho_{i}=\left(\frac{\beta_{1}-p_{i}}{\beta_{1}-q_{i}}\right)^{m_{1}}\left(\frac{\beta_{2}-p_{i}}{\beta_{2}-q_{i}}\right)^{m_{2}}\left(\frac{\beta_{3}-p_{i}}{\beta_{3}-q_{i}}\right)^{m_{3}},$

where $\beta_{1},\beta_{2},\beta_{3}$ are spacing parameters and $p_{i}$ and
$q_{i}$ are referred to as soliton parameters. By imposing constraints on
$(p_{i},q_{i})$ one can have a PWF for reduced (2D) equations. Both the lpKP
and lpmKP equations allow a PWF with two independent soliton parameters (§9.7
of [31]), and for 2D lattice equations, e.g. the ABS equations, their PWF
usually reads [32, 49]

$\rho_{i}=\left(\frac{\beta_{1}-p_{i}}{\beta_{1}+p_{i}}\right)^{m_{1}}\left(\frac{\beta_{2}-p_{i}}{\beta_{2}+p_{i}}\right)^{m_{2}},$

which contains only one soliton parameter $p_{i}$. In the present paper, the
lpmKP equation (1.2) is reconstructed in (2.15), as a consequence of 3D
consistency of the ldNLS equation (2.11). Note that this does not mean the
lpmKP equation (1.2) is a fake 3D equation, because it does allows PWF with
two independent soliton parameters (§9.7 of [31]). However, this indicates
that the solutions we constructed in the paper, by using 3D consistency of the
2D ldNLS equation, is a special solution in which PWF contains a single
soliton parameter. One may develop an approach for 3D lattice equations to
construct their algebro-geometric solutions in which the PWF contains two
independent soliton parameters and therefore allows reductions.

The third problem is to apply the scheme to other ABS equations and 3D lattice
equations that are 4D consistent (including octahedron-type equations [2] and
the discrete BKP and Schwarzian BKP equation [1]). The approach has proved
effective and so far H1, H3(0) and Q${}_{1}(\delta)$, lpKP and lpmKP have been
solved with this approach. However, as we mentioned in Section 1, for other
equations, the associated continuous spectral problems are still unknown. It
is also notable that, as we pointed out in Section 1, for H1, H3(0) and
Q${}_{1}(0)$, each equation can have more than one associated continuous
spectral problems. For the same equation, these continuous spectral problems
may lead to either same (for H1 and Q${}_{1}(0)$, cf.[9, 62], and cf. [61,
62]) or different hyperelliptic curves (for H${}_{3}(0)$, cf.[12, 62]), but
for each equation the obtained solutions have different formulations.

The final problem is finite-gap integration based on theory of trigonal curves
for discrete integrable systems. So far the hyperelliptic curves associated
with the two-sheeted Riemann surfaces were employed in our scheme, cf.[9, 12,
13, 60, 61, 62]. There were already some exciting developments in the finite-
gap integration theory based on trigonal curves in continuous case, e.g. [19,
24, 25, 26]. It would be very meaningful to develop the theory in discrete
case to study lattice equations related to third-order spectral problem, e.g.
the discrete Boussinesq equations [33].

### Acknowledgments

The authors are grateful to the referees for their invaluable comments. Our
sincere thanks are also extended to Dr. Xing Li for sharing her expertise of
computing path integrations on torus and providing figures of this paper. This
work is supported by the National Natural Science Foundation of China (grant
nos. 11631007, 11875040).

##  Appendix A The semi-discrete lpmKP and ldNLS equations

Since the spectral problem (2.6c) arises from the Darboux transformation
(2.1), which commutes with the KN spectral problem and may serve as a Darboux
transformation for the whole KN hierarchy, we can have more equations in this
frame, which compose a hierarchy of the lpmKP family, in terms of the number
of discrete independent variables.

The first semi-discrete lpmKP equation (with two discrete independent
variables) is

$\Xi^{(1,2)}\equiv(\widetilde{W}-\overline{W})_{x}+\beta_{1}^{2}(e^{-\overline{\widetilde{W}}+\overline{W}}-e^{-\widetilde{W}+W})-\beta_{2}^{2}(e^{-\widetilde{\overline{W}}+\widetilde{W}}-e^{-\overline{W}+W})=0,$
(A.1)

which is a consequence of the compatibility of (1.4) (2.6c) and (2.6f), where
(2.10a), (2.10b) and (2.9) should be used. Alternatively, equation (A.1) is
obtained from (2.12) by substituting (2.9) into the equation. Note that (A.1)
was also found in [50] (Eq.(4.27)) as a continuum limit of the lpmKP equation
(1.2).

The second semi-discrete lpmKP equation (with one discrete independent
variable),

$\Xi^{(2,1)}\equiv(\widetilde{W}+W)_{xx}-(\widetilde{W}_{x}^{2}-W_{x}^{2})-2\beta_{1}^{2}(e^{-\widetilde{W}+W})_{x}-(\widetilde{W}-W)_{y}=0,$
(A.2)

is obtained from the compatibility of (1.4), (2.6c) and the following linear
problem

$\partial_{y}\chi=U_{2}\chi,\quad
U_{2}=\lambda^{2}U_{1}+\left(\begin{array}[]{cc}\lambda^{2}(-uv)&\lambda(u_{x}-2u^{2}v)\\\
\lambda(-v_{x}-2uv^{2})&-\lambda^{2}(-uv)\end{array}\right).$ (A.3)

The calculation is complicated. Let us sketch it below. First, the
compatibility of (1.4) and (A.3) yields the dNLS equations

$\Xi^{(2,0)}_{1}\equiv
u_{y}-u_{xx}+2(u^{2}v)_{x}=0,~{}\quad\Xi^{(2,0)}_{2}\equiv
v_{y}+v_{xx}+2(uv^{2})_{x}=0,$ (A.4)

and the compatibility of (1.4) and (A.3) yields relations (2.3) with
formulation (2.4), where we need to replace $Z$ with $Z^{(1)}$. In particular,
(2.3b) and (2.3c) indicate that

$B_{1}\doteq(Z^{(1)}u)_{x}-\widetilde{u}-\beta_{1}^{2}u=0,~{}\quad
B_{2}\doteq(Z^{(1)}\widetilde{v})_{x}+v+\beta_{1}^{2}\widetilde{v}=0.$ (A.5)

Then, making use of (2.9), (A.4) and (A.5), the compatibility of (2.6c) and
(A.3) yields

$\mathbf{0}=\partial_{y}D^{(\beta_{1})}-\widetilde{U}_{2}D^{(\beta_{1})}+D^{(\beta_{1})}U_{2}=\left(\begin{array}[]{cc}-\lambda^{2}Z^{(1)}\Xi^{(2,1)}&\lambda\partial^{-1}_{x}\partial_{y}B_{1}\\\
\lambda^{3}\partial^{-1}_{x}\partial_{y}B_{2}&0\end{array}\right),$

which gives rise to equation (A.2). Note that (A.2) can also be derived in the
following alternative way. From the first equation in (A.4) and the shifted
second equation
$\widetilde{v}_{y}=-\widetilde{v}_{xx}-2(\widetilde{u}\widetilde{v}^{2})_{x}$,
one can have

$\displaystyle(u\widetilde{v})_{y}=(u_{x}\widetilde{v}-u\widetilde{v}_{x})_{x}-2u\widetilde{v}\big{[}(u_{x}v+\widetilde{u}\widetilde{v}_{x})+(\widetilde{u}\widetilde{v}+uv)_{x}\big{]},$
(A.6)

which can lead to equation (A.2) by using (A.5) and relation
$(u\widetilde{v})_{y}=(\widetilde{W}-W)_{y}(2-Z^{(1)})/(Z^{(1)})^{2}$ due to
$Z^{(1)}=(Z^{(1)})^{2}u\widetilde{v}+1$. We also note that equation (A.2) can
be converted to its non-potential form

$-s\Xi^{(2,1)}\equiv s_{xx}+2\beta_{1}^{2}ss_{x}+2[s(\Delta^{-1}\ln
s)_{x}]_{x}-s_{y}=0,$ (A.7)

where $s=\ln(W-\widetilde{W})$ and $\Delta f=\widetilde{f}-f$. This equation
was obtained in [55] as a gauge equivalence of a semi-discrete KP equation.

We have derived the lpmKP equation $\Xi^{(0,3)}=0$ and two semi-discrete lpmKP
equations $\Xi^{(1,2)}=0$ and $\Xi^{(2,1)}=0$. All these equations have the
pmKP equation (1.3), $\Xi^{(3,0)}=0$, as continuum limit. Let us replace
$\beta_{k}$ by $\beta_{k}^{-2}=\varepsilon_{k}=c_{k}\varepsilon,\,k=1,2,3$,
with non-zero and distinct constants $c_{1},c_{2},c_{3}$, and consider
continuum limits in terms of Miwa’s variables
$\vec{t}=(t_{1},t_{2},t_{3},\cdots)$, where [46]

$t_{k}=-\frac{1}{k}\sum_{i}\varepsilon_{i}^{k}m_{i}.$

This indicates that

$T_{1}^{j_{1}}T_{2}^{j_{2}}T_{3}^{j_{3}}W(m_{1},m_{2},m_{3})=W\left(x-\sum^{3}_{i=1}j_{i}\varepsilon_{i},\,y-\frac{1}{2}\sum^{3}_{i=1}j_{i}\varepsilon_{i}^{2},\,t-\frac{1}{3}\sum^{3}_{i=1}j_{i}\varepsilon_{i}^{3}\right),$

where $x=t_{1},y=t_{2},t=t_{3}$ and $j_{i}\geq 0$ for $i=1,2,3$; and for
differential-difference case,

$T_{1}^{j_{1}}T_{2}^{j_{2}}W(x^{\prime},m_{1},m_{2})=W\left(x-\sum^{2}_{i=1}j_{i}\varepsilon_{i},\,y-\frac{1}{2}\sum^{2}_{i=1}j_{i}\varepsilon_{i}^{2},\,t-\frac{1}{3}\sum^{2}_{i=1}j_{i}\varepsilon_{i}^{3}\right),$

where $x=x^{\prime}+t_{1},y=t_{2},t=t_{3}$ and $j_{i}\geq 0$ for $i=1,2$;

$T_{1}^{j_{1}}W(x^{\prime},y^{\prime},m_{1})=W\left(x-j_{1}\varepsilon_{1},\,y-j_{1}\frac{\varepsilon_{1}^{2}}{2},\,t-j_{1}\frac{\varepsilon_{1}^{3}}{3}\right),$

where $x=x^{\prime}+t_{1},y=y^{\prime}+t_{2},t=t_{3}$ and $j_{1}\geq 0$. It
turns out that

$\displaystyle\Xi^{(2,1)}(x^{\prime},y^{\prime},m_{1})=\Xi^{(3,0)}\frac{2}{3}c_{1}^{2}\varepsilon^{2}+O(\varepsilon^{3}),$
$\displaystyle\Xi^{(1,2)}(x^{\prime},m_{1},m_{2})=\Xi^{(3,0)}\frac{1}{3}c_{1}c_{2}(c_{1}-c_{2})\varepsilon^{3}+O(\varepsilon^{4}),$
$\displaystyle\Xi^{(0,3)}(m_{1},m_{2},m_{3})=\Xi^{(3,0)}\frac{1}{3}\big{(}c_{1}c_{2}(c_{1}-c_{2})+c_{2}c_{3}(c_{2}-c_{3})+c_{3}c_{1}(c_{3}-c_{1})\big{)}\varepsilon^{3}+O(\varepsilon^{4}).$

Similarly, equations (2.5) and (2.11), up to some transformations, yield the
dNLS equations (A.4) in continuum limit. In fact, for equation (2.5), we first
replace $u$ and $v$ by $(-\beta_{1}^{2})^{m_{1}}u$, and
$(-\beta_{1}^{2})^{-m_{1}}v$, respectively, and rewrite (2.5) as

$\displaystyle\Xi_{1}^{{}^{\prime}(1,1)}(x,m_{1})\equiv
u_{x}+(\widetilde{u}\widetilde{v}-uv)u+\frac{1}{2}\Big{(}1+\sqrt{1+4\beta_{1}^{-2}u\widetilde{v}}\,\Big{)}\beta_{1}^{2}(\widetilde{u}-u)=0,$
$\displaystyle\Xi_{2}^{{}^{\prime}(1,1)}(x,m_{1})\equiv\widetilde{v}_{x}+(\widetilde{u}\widetilde{v}-uv)\widetilde{v}+\frac{1}{2}\Big{(}1+\sqrt{1+4\beta_{1}^{-2}u\widetilde{v}}\,\Big{)}\beta_{1}^{2}(\widetilde{v}-v)=0,$

which is available for taking continuum limit. Then, let
$\beta_{1}^{-2}=\varepsilon_{1}$ and define

$T_{1}^{j}{f}(x^{\prime},m_{1})=f(x-j\varepsilon_{1},\,y-j\varepsilon_{1}^{2}/2),$

where $x=x^{\prime}+t_{1},y=t_{2}$ and $j\geq 0$. Then we have
$(\varepsilon_{1}\sim 0)$

$\displaystyle\Xi_{1}^{{}^{\prime}(1,1)}(x^{\prime},m_{1})=-\frac{\varepsilon_{1}}{2}\Xi_{1}^{(2,0)}+O(\varepsilon_{1}^{2}),$
$\displaystyle\Xi_{2}^{{}^{\prime}(1,1)}(x^{\prime},m_{1})=-\frac{\varepsilon_{1}}{2}\Xi_{2}^{(2,0)}+O(\varepsilon_{1}^{2}),$

where $\Xi_{i}^{(2,0)}$ are given in (A.4). Finally, in (2.11), replace $u$ by
$(-\beta_{1}^{2})^{m_{1}}(-\beta_{2}^{2})^{m_{2}}u$ and $v$ by
$(-\beta_{1}^{2})^{-m_{1}}(-\beta_{2}^{2})^{-m_{2}}v$, and rewrite the
equation as

$\displaystyle\Xi_{1}^{{}^{\prime}(0,2)}\equiv\beta_{1}^{2}(\widetilde{Z}^{(2)}\widetilde{u}-Z^{(2)}u)-\beta_{2}^{2}(\overline{Z}^{(1)}\overline{u}-Z^{(1)}u)=0,$
$\displaystyle\Xi_{2}^{{}^{\prime}(0,2)}\equiv\beta_{1}^{2}(\overline{Z}^{(2)}\widetilde{\overline{v}}-Z^{(2)}\overline{v})-\beta_{2}^{2}(\overline{Z}^{(1)}\overline{\widetilde{v}}-Z^{(1)}\widetilde{v})=0.$

Let $\beta_{i}^{-2}=\varepsilon_{i}=c_{i}\varepsilon,$ and introduce

$T_{1}^{j_{1}}T_{2}^{j_{2}}f(m_{1},m_{2})=f\left(x-\sum^{2}_{i=1}j_{i}\varepsilon_{i},\,y-\frac{1}{2}\sum^{2}_{i=1}j_{i}\varepsilon_{i}^{2}\right),$

where $x=t_{1},y=t_{2}$ and $j_{i}\geq 0$ for $i=1,2$. The continuum limit
yields

$\displaystyle\Xi_{1}^{{}^{\prime}(0,2)}=-\frac{(c_{1}-c_{2})\varepsilon}{2}\Xi_{1}^{(2,0)}+O(\varepsilon^{2}),$
$\displaystyle\Xi_{2}^{{}^{\prime}(0,2)}=-\frac{(c_{1}-c_{2})\varepsilon}{2}\Xi_{2}^{(2,0)}+O(\varepsilon^{2}).$

In the rest part of this section, we review some links between
$(1+1)$-dimensional and $(2+1)$-dimensional integrable systems, which maybe
helpful for understanding the connections in the discrete case. Let us start
with the continuous mKP hierarchy, which are generated from the compatibility
of (see [39])

$\displaystyle
L\varphi=\lambda\varphi,~{}~{}L=\partial_{x}+w_{0}+w_{1}\partial_{x}^{-1}+w_{2}\partial_{x}^{-2}+\cdots,$
(A.8a) $\displaystyle\varphi_{t_{j}}=A_{j}\varphi,$ (A.8b)

where $w_{j}=w_{j}(x=t_{1},y=t_{2},t=t_{3},t_{4},\cdots)$,
$\partial_{x}=\partial/\partial x$, $A_{j}=(L^{j})_{\geq 1}$ stands for the
pure differential part of $L^{j}$. One can express $w_{j}$ for $j\geq 1$ in
terms of $w_{0}\equiv w$ using the compatibility between (A.8a) and
$\varphi_{y}=A_{2}\varphi$, and then the mKP hierarchy $w_{t_{j}}=P_{j}(w)$
arise from the compatibility of $\varphi_{y}=A_{2}\varphi$ and (A.8b). By
$A_{j}^{*}$ we denote the operator adjoint of $A_{j}$. Introduce

$\psi_{t_{j}}=-A_{j}^{*}\psi.$ (A.9)

It can be verified that for the above eigenfunctions $\varphi$ and $\psi$, the
mKP hierarchy admits a symmetry $\sigma=(\varphi\psi)_{x}$. Consider the
symmetry constraint $w=\varphi\psi$. In the following we denote $\varphi=u$
and $\psi=v$, thus we have (cf. $w=W_{x}=uv$ in (2.9))

$w=uv.$ (A.10)

With this constraint, the spectral problem (A.8a) is converted to the KN
spectral problem (1.4) (up to gauge transformation), and the coupled system
(A.8b) and (A.9) (with $\varphi=u,~{}\psi=v$) give rise to the KN hierarchy
(up to gauge transformation)

$\mathbf{u}_{t_{j}}=K_{j}(\mathbf{u}),~{}~{}\mathbf{u}=(u,v)^{T}.$ (A.11)

For more details, one may refer to [15]. Note that the KN hierarchy (A.11)
also result from the compatibility of the KN spectral problem (1.4) and the
time evolution $\chi_{t_{j}}=U_{j}\chi$. In this context, it is not surprised
that the potential mKP equation (1.3) can be obtained from the compatibility
of the triad $\chi_{t_{j}}=U_{j}\chi$ for $j=1,2,3$ and with
$t_{1}=x,t_{2}=y,t_{3}=t$ and $W_{x}=uv$. Such a link between
$(1+1)$-dimensional and $(2+1)$-dimensional integrable systems was first
revealed in [8, 18, 40] in the early 1990s for the ZS-AKNS and KP systems. For
the differential-difference case with one discrete independent variable, the
analogue results build connections between the semi-discrete ZS-AKNS and
differential-difference KP systems (see [16]). The obtained semi-discrete ZS-
AKNS spectral problem is nothing but the Darboux transformation of the ZS-AKNS
spectral problem (1.1), and it has been used to derive algebro-geometry
solutions for the lpKP equation [10]. However, for the differential-difference
mKP hierarchy, its eigenfunction symmetry constraint gives rise the
relativistic Toda spectral problem (see [17]), which is not the Darboux
transformation (2.1) that we used in this paper. Anyway, as we can see, in our
approach it is an important step to establish the link between the lpmKP and
KN equations. The eigenfunction symmetry constraint can provide some insights
but not always as direct as expected.

##  Appendix B Riemann theta function associated with hyperelliptic curve

In this section we introduce how a Riemann theta function arises from a
hyperelliptic curve. For more details one can refer to [23, 30, 47]. This
section also provides some complex algebraic geometry preliminaries that will
be used in our approach.

For a generic hyperelliptic curve $\mathcal{R}$: $\xi^{2}=R(\zeta)$ with genus
$g$ (for example, (3.8)), the following Abelian differentials of the first
kind constitute the basis of holomorphic differentials of $\mathcal{R}$:

$\tilde{\omega}_{j}=\displaystyle\frac{\zeta^{g-j}\mathrm{d}\zeta}{2\sqrt{R(\zeta)}},\
\ j=1,\ldots,g.$ (B.1)

Let the closed loops $a_{1},\cdots,a_{g},b_{1},\cdots,b_{g}$ be the canonical
basis of the homology group $\mathrm{H}_{1}(\mathcal{R})$, with intersection
numbers

$a_{j}\circ b_{k}=\delta_{jk},~{}~{}a_{j}\circ a_{k}=0,~{}~{}b_{j}\circ
b_{k}=0,~{}~{}j,k=1,2,\cdots,g.$

Note that here we adopt the conventional notations $\\{a_{j},b_{j}\\}$ for
these canonical basis, without making confusion with the discrete potentials
$(a_{m},b_{m})$ that we used in Sec.4. The basis of holomorphic differentials
can be normalized as follows:

$\displaystyle\begin{split}\vec{\omega}\doteq(\omega_{1},\cdots,\omega_{g})^{T}&=C(\tilde{\omega}_{1},\cdots,\tilde{\omega}_{g})^{T}\\\
&=(C_{1}\zeta^{g-1}+C_{2}\zeta^{g-2}+\ldots
C_{g})\displaystyle\frac{\mathrm{d}\zeta}{2\sqrt{R(\zeta)}},\end{split}$ (B.2)

where

$C=(A_{jk})^{-1}_{g\times g},\ \ A_{jk}=\int_{a_{k}}\tilde{\omega}_{j},$ (B.3)

and $C_{l}$ stands for the _l_ -th column vector of $C$. For this normalized
basis $\vec{\omega}$, we have

$\int_{a_{k}}\omega_{j}=\delta_{jk},\ \ \int_{b_{k}}\omega_{j}=B_{jk},$ (B.4)

where the matrix $B=(B_{jk})_{g\times g}$ is symmetric with positive definite
imaginary part. $B$ can be used as a periodic matrix to defined the Riemann
theta function (which is holomorphic):

$\theta(z;B)=\sum_{z^{\prime}\in\mathbb{Z}^{g}}\exp\pi\sqrt{-1}(<Bz^{\prime},z^{\prime}>+2<z,z^{\prime}>),\
\ z\in\mathbb{C}^{g},$ (B.5)

where by $<\cdot,\cdot>$ we denote the scalar product in $\mathbb{C}^{g}$.

Next, let us reveal features of zeros of the above Riemann theta function. Let
$\mathcal{T}$ be a period lattice $\mathcal{T}=\\{z\in\mathbb{C}^{g}\mid
z=\vec{m}+B\vec{n},~{}\vec{m},\vec{n}\in\mathbb{Z}^{g}$}. The quotient space
$J(\mathcal{R})=\mathbb{C}^{g}/\mathcal{T}$ is called the Jacobian variety of
$\mathcal{R}$. Introduce the Abel map (also called Abel-Jacobi map)
$\mathcal{A}:\mathcal{R}\rightarrow J(\mathcal{R})$, by

$\mathcal{A}(\mathfrak{p})=\int_{\mathfrak{p}_{0}}^{\mathfrak{p}}\vec{\omega}=\left(\int_{\mathfrak{p}_{0}}^{\mathfrak{p}}\omega_{1},\int_{\mathfrak{p}_{0}}^{\mathfrak{p}}\omega_{2},~{}\cdots,\int_{\mathfrak{p}_{0}}^{\mathfrak{p}}\omega_{g}\right)^{T},$
(B.6)

where $\mathfrak{p},\mathfrak{p}_{0}\in\mathcal{R}$ with $\mathfrak{p}_{0}$
being some choosing fixed point. The map can be extended to the divisors by
defining

$\mathcal{A}\left(\sum_{k}n_{k}\mathfrak{p}_{k}\right)=\sum_{k}n_{k}\mathcal{A}(\mathfrak{p}_{k}).$
(B.7)

Consider the function

$\theta(z(\mathfrak{p},D);B)=\theta(-\mathcal{A}(\mathfrak{p})+\mathcal{A}(D)+\vec{K});B),~{}\
\ \mathfrak{p}\in\mathcal{R},$ (B.8)

where the divisor $D=\sum_{k=1}^{g}\mathfrak{p}_{k}$ with
$\\{\mathfrak{p}_{k}\\}$ being $g$ distinct points on $\mathcal{R}$, and the
vector $\vec{K}$ of Riemann constants is defined as

$\vec{K}=-\sum_{k=1}^{g}\bigg{[}\int_{a_{k}}\mathcal{A}\,\omega_{k}-\bigg{(}\displaystyle\frac{B_{kk}}{2}+\mathcal{A}_{k}(\mathfrak{p}^{\prime})\bigg{)}\int_{a_{k}}\vec{\omega}\bigg{]},$
(B.9)

where $\mathcal{A}_{k}$ is the $k$-th component of the Abel map $\mathcal{A}$,
and $\mathfrak{p}^{\prime}$ is the base point of the fundamental group
$\pi_{1}(\mathcal{R})$ (see [23]). By choosing
$\mathfrak{p}^{\prime}=\mathfrak{p}_{0}$, the above formula reduces to

$\vec{K}=-\sum_{k=1}^{g}\bigg{(}\int_{a_{k}}\mathcal{A}\omega_{k}-\displaystyle\frac{B_{kk}}{2}\int_{a_{k}}\vec{\omega}\bigg{)}.$
(B.10)

According to the Riemann vanishing theorem, the zeros of
$\theta(z(\mathfrak{p},D),B)$ are $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{g}$.
Due to this fact, the Riemann theta function (B.8) is often used to
characterize meromorphic functions with certain zeros and poles.

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