#  New Construction of Algebro-Geometric Solutions to the Modified Kadomtsev-
Petviashvili Hierarchy

Peng Zhao 111College of Art and Sciences, Shanghai Maritime University,
Shanghai 201306, P. R. China Engui Fan 222School of Mathematical Sciences,
Institute of Mathematics and Key Laboratory of Mathematics for Nonlinear
Science, Fudan University, Shanghai 200433, P. R. China 333Corresponding
author and e-mail address<EMAIL_ADDRESS>
###### Abstract

We extend Gesztesy-Holden’s method to 2+1 dimensional case to obtain a unified
construction to the algebro-geometric solutions of the whole modified
Kadomtsev-Petviashvili (mKP) hierarchy. Our tools include the relations
between solutions of the Gerdjikov-Ivanov (GI) and mKP hierarchy, the Baker-
Akhiezer functions in 2+1 dimensions, a special function
$\psi_{1}(P)\psi_{2}(P^{*})$ on $X\times\mathbb{R}^{3}$ and Dubrovin-type
equations for auxiliary divisors.

Key words: modified Kadomtsev-Petviashvili hierarchy, Baker-Akhiezer function,
algebro-geometric method, Riemann theta function, algebro-geometric solutions.

##  1 Introduction

The KP hierarchy and its counterpart, for example, the mKP hierarchy, plays an
important role in a variety of different fields including modern string theory
and in connection with the solution of the Schottky problem of compact Riemann
surfaces [40], [47]. It is generally believed that all integrable hierarchies
of (1+1)-dimensional equations by means of the inverse scattering method can
be represented as certain reductions of a universal Kadomtsev-Petviashvili
(KP) hierarchy of (2+1)-dimensional equations and/or of its extentions to
modified and multicomponent cases [20], [45]. This hypothesis originates from
a unifying Sato theory that describes the KP hierarchy in terms of the
pseudodifferential operator

$\mathscr{L}=\frac{\partial}{\partial
x}+\sum_{j=1}^{\infty}u_{j+1}(\frac{\partial}{\partial x})^{-j}$

with functions $u_{j}=u_{j}(t_{1},t_{2},\ldots)$ that depends on an infinite
number of independent variables $t_{j},j\in\mathbb{N}$ [6], [7], [23], [46],
which indicates that by introducing an infinite set of time variables one can
also treat the integrable equations as flows on infinite-dimensional
Grassmannian manifolds [23]. Based on the treatment of partial differential KP
equations as dynamical systems on the infinite-dimensional algebra of pseudo-
differential operators [1], the KP equation [7], [23]

$u_{t}=\frac{1}{4}u_{xxx}+3uu_{x}+\frac{3}{4}\partial^{-1}_{x}u_{yy},$ (1.1)

is considered as the simplest member of a hierarchy of equations that can be
brought to bilinear form and solved by the $\tau$-funtion approach. This
theory reveals deep interrelations between the Hamiltonian structures of the
KP hierarchy and two-dimensional conformal field theory as well as W1+∞
algebras [49].

The problem of constructing the quasiperiodic solutions (QPS) is one of the
most challenging problems of the theory of integrable systems, and many
mathematicians and physicists spent much efforts to obtain the QPS for almost
all equations that are known to be integrable. Among the most powerful
solution generating methods are techniques from algebraic geometry which lead
to solutions in terms of Riemann theta functions on certain Riemann surfaces,
that is, the so-called algebro-geometric solutions. This kinds of solutions
were originally studied on the KdV equation based on the inverse spectral
theory and algebro-geometric method developed by pioneers such as Novikov
[10], Dubrovin _et al._ [11], [12], Its & Matveev [21], [22], Lax [36], and
McKean & van Moerbeke [38] for 1 + 1 systems, and extended by Krichever in
1976 for 2 + 1 systems like KP [32], [33] in the late 1970s. Later this theory
has been developed to the whole hierarchies of nonlinear integrable equations
by Gesztesy, Holden _et al._ using polynomial recursion method [8], [16],
[17], [18]. Another breakthrough in this area was made by Mumford [41] in
early 1980s, who observed that integrable equations like KdV, KP or sine-
Gordon, are hidden in Fay’s trisecant formula. Mumford’s approach are based on
degenerated versions of Fay’s identity, which reveals the relations between
algebro-geometric solutions of integrable equations and a purely algebro-
geometric identity [24], [26], [27]. A detailed introduction and recent
development about this subject can be found in the survey article [37].

The study of higher dimensional integrable equations is usually considered a
more difficult problem than (1+1)-dimensional equations and by now, much work
has been done on KP and mKP equation. An important development of the algebro-
geometric method was the passage from 1+1 systems to the integration of 2+1
KP-like systems, realized by Krichever in 1976 [32], who constructed algebro-
geometric solutions of the KP equation on a basis of a purely algebraic
formulation of algebro-geometric approach. A generalized Miura transformation
between the KP equation and mKP equation was discussed in [14]. Later it has
been shown that the mKP hierarchy can be constructed on the basis of the Sato
approach by means of the gauge transformation of the KP hierarchy which
results in modification of the pseudo-differential operator $\mathscr{L}$
[43]. In ref. [19], the authors obtain new Bäcklund transformations for the KP
hierarchy and the possibility of transferring classes of KP solutions into
those of mKP solutions. A explicit theta function solution of the mKP equation
is derived by the technique of nonlinearization of Lax pairs and Abel
inversion [4]. However, within the knowledge of the authors, the algebro-
geometric solutions of the entire mKP hierarchy has not been considered so
far.

In this paper, using inherent relations between the mKP hierarchy and GI
hierarchy, we shall improve the method in [18] to obtain a unified
construction to the algebro-geometric solutions of the whole hierarchy. We
extend the concepts the Baker-Ahkiezer function to the mKP hierarchy and find
a class of quasiperiodic solutions by algebro-geometric method. The
construction exploits explicit form of the Baker-Ahkiezer function and involve
also a fundamental meromorphic function, which allows for the analytic
properties of the Baker-Ahkiezer function. The theta function representation
for algebro-geometric solutions of the mKP hierarchy are then verified
directly using formulae for the Baker-Ahkiezer function and its
$x$-derivative. In our construction, theoretical analysis of algebro-geometric
solution for the whole mKP hierarchy are carried out from a different angle of
view and the whole approach discussed in the present paper is a general one
and gives similar results for other 2+1 dimensional and higher dimensional
soliton equations.

This paper is organized as follows. In section 2, we first introduce the GI
hierarchy and mKP hierarchy and then discuss the Burchnall-Chaundy polynomial
in connection with a basic initial value problem and underlying hyperelliptic
curve. In section 3, the dynamics of auxiliary spectral points
$\\{\mu_{j}\\}_{j=1}^{n},\\{\nu_{j}\\}_{j=1}^{n}$ and corresponding trace
formula is considered. In section 4, we present explicit representation for
the Baker-Akhiezer functions $\psi_{1}(P),\psi_{2}(P)$, whose analytic
properties can be derived by studying an important function
$\psi_{1}(P)\psi_{2}(P^{*})$. In section 5, we shall obtain Riemann theta
function representation for
$\psi_{1}(P),\psi_{2}(P),\psi_{1}(P)\psi_{2}(P^{*})$, and especially for the
algebro-geometric solutions $q,r$ of the whole mKP hierarchy.

##  2 mKP Hierarchy, GI Hierarchy, Burchnall-Chaundy Polynomial and Basic
Initial Value Problem

To make this paper self-contained, we shall first introduce the mKP hierarchy
in standard literature and then provide the construction of the GI hierarchy
and derive the associated sequence of Lax pairs using a polynomial recursion
formalism. Moreover, we obtain the Burchnall-Chaundy polynomial in connection
with the modified KP hierarchy and underlying hyperelliptic curve.

Throughout this section we make the following hypothesis.

######  Hypothesis 2.1.

Suppose that $q,r:\mathbb{R}^{3}\rightarrow\mathbb{C}$ satisfy

$\begin{split}&q(\cdot,y,t_{p})\in
C^{\infty}(\mathbb{R}),~{}~{}t_{p}\in\mathbb{R},\quad\\\
&q(x,y,\cdot),r(x,y,\cdot),q(x,\cdot,t_{p}),r(x,\cdot,t_{p})\in
C^{1}(\mathbb{R}),~{}~{}x,y\in\mathbb{R},\\\ &q(x,y,t_{p})\neq
0,~{}~{}r(x,y,t_{p})\neq 0,~{}~{}(x,y,t_{p})\in\mathbb{R}^{3}.\end{split}$
(2.1)

The KP and mKP equation, e.g. [23], [28], [29], [30]) and the algebraic
framework of the KP and mKP hierarchy can be found in [1], [2], [9], [15],
[39], [44], [35], [48]. In Sato theory, the mKP hierarchy is described by the
isospectral deformations of the eigenvalue problem

$\begin{split}&L\psi=\lambda\psi,~{}~{}\lambda\in\mathbb{C},\end{split}$ (2.2)

where the pseudodifferential operator $L$ is given by

$L=\partial+\sum_{j=0}^{\infty}u_{j+1}\partial^{-j},~{}~{}\partial=\frac{\partial}{\partial
x},$ (2.3)

and $u_{j}$ are functions in infinitely many variables $(x,t_{1},t_{2}\ldots)$
with $t_{1}=y.$ We denote by $B_{m}$ the differential part of $L^{m}:$

$\displaystyle B_{m}=(L^{m})_{+}=\sum_{j=1}^{m}b_{m,j}\partial^{j}.$ (2.4)

The coefficients $b_{m,j}$ in (2.4) can be uniquely determined by the
coordinates $u_{j}$, and their $x$ derivatives. Explicitly,

$\displaystyle B_{1}=$ $\displaystyle~{}\partial,$ $\displaystyle B_{2}=$
$\displaystyle~{}\partial^{2}+2u_{1}\partial,$ $\displaystyle B_{3}=$
$\displaystyle~{}\partial^{3}+3u_{1}\partial^{2}+3(u_{2}+u_{1,x}+u_{1}^{2})\partial,$
$\displaystyle B_{4}=$
$\displaystyle~{}\partial^{4}+4u_{1}\partial^{3}+(4u_{2}+6u_{1,x}+6u_{1}^{2})\partial^{2}+(4u_{3}+6u_{2,x}$
$\displaystyle+4u_{1,xx}+12u_{1}u_{2}+12u_{1}u_{1,x}+4u_{1}^{3})\partial,~{}~{}\textrm{etc.}$

From the compatibility conditions of (2.2) and

$\phi_{t_{m}}=B_{m}\phi,$ (2.5)

we have

$\displaystyle L_{t_{m}}=[B_{m},L],$ (2.6)

or equivalently,

$(B_{m})_{t_{n}}-(B_{n})_{t_{m}}=[B_{n},B_{m}].$ (2.7)

The mKP hierarchy is obtained from (2.6) or (2.7) for infinite coordinates
$\\{u_{j}\\}_{j=2}^{\infty}.$ For example, from (2.6), one obtains

$\displaystyle u_{1,t_{2}}=$
$\displaystyle~{}2u_{2,x}+u_{1,xx}+2u_{1}u_{1,x},$ (2.8) $\displaystyle
u_{2,t_{2}}=$ $\displaystyle~{}2u_{3,x}+u_{2,xx}+2(u_{1}u_{2})_{x},$ (2.9)
$\displaystyle u_{3,t_{2}}=$
$\displaystyle~{}2u_{4,x}+u_{3,xx}+2u_{1}u_{3,x}+4u_{1,x}u_{3}-2u_{1,xx}u_{2},$
(2.10) $\displaystyle\ldots\ldots$ $\displaystyle u_{1,t_{3}}=$
$\displaystyle~{}3u_{3,x}+3u_{2,xx}+u_{1,xxx}+6(u_{1}u_{2})_{x}+3(u_{1}u_{1,x})_{x}+3u_{1}^{2}u_{1,x},$
(2.11) $\displaystyle u_{2,t_{3}}=$
$\displaystyle~{}3u_{4,x}+3u_{3,xx}+u_{2,xxx}+6(u_{1}u_{3})_{x}+3(u_{1}u_{2,x})_{x}+6u_{2}u_{2,x}$
$\displaystyle~{}+3(u_{1}^{2}u_{2})_{x},$ (2.12) $\displaystyle\ldots\ldots$
$\displaystyle u_{1,t_{4}}=$
$\displaystyle~{}4u_{4,x}+6u_{3,xx}+4u_{2,xxx}+u_{1,xxxx}+12(u_{1}u_{3})_{x}+12u_{2}u_{2,x}$
$\displaystyle~{}+12(u_{1}u_{2,x})_{x}+6(u_{1,x}u_{2})_{x}+4(u_{1}u_{1,xx})_{x}+6u_{1,x}u_{1,xx}$
$\displaystyle~{}+12(u_{1}^{2}u_{2})_{x}+6(u_{1}^{2}u_{1,x})_{x}+4u_{1}^{3}u_{1,x}$
(2.13) $\displaystyle\ldots\ldots$

Eliminating $u_{2}$, $u_{3}$, $u_{4}$ from (2.8), (2.9), (2.10) and taking
into account (2.11), (2.13), one obtains

$\displaystyle u_{t_{3}}=$
$\displaystyle~{}\frac{1}{4}u_{xxx}-\frac{3}{2}u^{2}u_{x}+\frac{3}{2}u_{x}\partial^{-1}u_{y}+\frac{3}{4}\partial^{-1}u_{yy}$
(2.14) $\displaystyle u_{t_{4}}=$
$\displaystyle~{}\frac{1}{2}u_{xxy}-2u^{2}u_{y}+u\partial^{-1}u_{yy}+2u_{y}\partial^{-1}u_{y}-\partial^{-1}(uu_{y})_{y}-2u_{x}\partial^{-1}uu_{y}$
$\displaystyle~{}+u_{x}\partial^{-2}u_{yy}+\frac{1}{2}\partial^{-2}u_{yyy}.$
(2.15)

where we denote by $u=u_{1}.$ Equation (2.14) is the mKP equation and equation
(2.15) is the first higher order flows of the mKP hierarchy. Similarly, the
mKP and higher-order mKP equation can also be derived from (2.7) with
$n=2,m=3$ and $n=2,m>3$, respectively.

Next we construct the GI hierarchy. To this end one has to consider the
following one-dimensional $2\times 2$ matrix-valued differential expression

$M=\left(\begin{array}[]{cc}-\partial+\frac{1}{4}qr&-qz\\\
rz&\partial-\frac{1}{4}qr\\\ \end{array}\right),$ (2.16)

and another $2\times 2$ matrix-valued differential expression of order $2n+2$
with respect to $z,$ denoted by $Q_{2n+2},n\in\mathbb{N}_{0},$ which is
defined recursively in the following. Defines the sequences of differential
polynomials
$\\{f_{\ell}\\}_{\ell\in\mathbb{N}_{0}},\\{g_{\ell}\\}_{\ell\in\mathbb{N}_{0}},$
and $\\{h_{\ell}\\}_{\ell\in\mathbb{N}_{0}},$ recursively by

$\displaystyle
g_{2\ell+1}=~{}f_{2\ell}=h_{2\ell}=0,~{}~{}\ell\in\mathbb{N}_{0},$ (2.17)
$\displaystyle f_{1}=~{}2q,~{}g_{0}=2,~{}h_{1}=-2r,$ (2.18) $\displaystyle
2f_{2\ell+1}=-f_{2\ell-1,x}+2qg_{2\ell}+(1/2)qrf_{2\ell-1},~{}\ell\in\mathbb{N}_{0},$
(2.19) $\displaystyle
2h_{2\ell+1}=~{}h_{2\ell-1,x}-2rg_{2\ell}+(1/2)qrh_{2\ell-1},~{}\ell\in\mathbb{N}_{0},$
(2.20) $\displaystyle
g_{2\ell,x}=~{}rf_{2\ell+1}+qh_{2\ell+1},~{}\ell\in\mathbb{N}_{0}.$ (2.21)

Here $q,r$ should be considered as functions of infinitely many variables
$(t_{1},t_{2},t_{3},\ldots)$. Explicitly, one computes

$\displaystyle f_{1}=$ $\displaystyle~{}2q,$ $\displaystyle f_{3}=$
$\displaystyle~{}-(1/2)q^{2}r-q_{x}+c_{1}(2q),$ $\displaystyle f_{5}=$
$\displaystyle~{}(1/8)q^{3}r^{2}+(3/4)qq_{x}r-(1/4)q^{2}r_{x}+(1/2)q_{xx}$
$\displaystyle~{}+c_{1}(-q^{2}r/2-q_{x})+c_{2}(2q),$
$\displaystyle\ldots\ldots,$ $\displaystyle g_{0}=$ $\displaystyle~{}2,$
$\displaystyle g_{2}=$ $\displaystyle~{}-qr+2c_{1},$ $\displaystyle g_{4}=$
$\displaystyle~{}(1/4)q^{2}r^{2}+(1/2)(q_{x}r-qr_{x})+c_{1}(-qr)+2c_{2},$
$\displaystyle g_{6}=$
$\displaystyle~{}-q^{3}r^{3}+3q^{2}rr_{x}-3qq_{x}r^{2}-q_{xx}r-qr_{xx}+q_{x}r_{x}$
$\displaystyle+((1/4)q^{2}r^{2}+(1/2)(q_{x}r-qr_{x}))c_{1}+c_{2}(-qr)+2c_{3},$
$\displaystyle\ldots\ldots,$ $\displaystyle h_{1}=$ $\displaystyle~{}-2r,$
$\displaystyle h_{3}=$ $\displaystyle~{}(1/2)qr^{2}-r_{x}+c_{1}(-2r),$
$\displaystyle h_{5}=$
$\displaystyle~{}-(1/8)q^{2}r^{3}+(3/4)qrr_{x}-(1/4)q_{x}r^{2}-(1/2)r_{xx}$
$\displaystyle+c_{1}(qr^{2}/2-r_{x})+c_{2}(-2r),~{}~{}\textrm{etc.,}$

where $\\{c_{\ell}\\}_{\ell=0}^{\infty}\subset\mathbb{C}$ are integration
constants. Subsequently, we also introduce the corresponding homogeneous
coefficients $\hat{f}_{\ell},\hat{g}_{\ell},$ and $\hat{h}_{\ell}$, defined by
vanishing of the integration constants $c_{k},$ for $k=1,\ldots,\ell,$

$\displaystyle\hat{f}_{1}=2q,~{}~{}\hat{f}_{2\ell+1}=f_{2\ell+1}|_{c_{k}=0,k=1,\ldots,\ell},~{}~{}\ell\in\mathbb{N},$
(2.22)
$\displaystyle\hat{g}_{0}=2,~{}~{}\hat{g}_{2\ell}=g_{2\ell}|_{c_{k}=0,k=1,\ldots,\ell},~{}~{}\ell\in\mathbb{N},$
(2.23)
$\displaystyle\hat{h}_{1}=-2r,~{}~{}\hat{h}_{2\ell+1}=h_{2\ell+1}|_{c_{k}=0,k=1,\ldots,\ell},~{}~{}\ell\in\mathbb{N}.$
(2.24)

Hence,

$\begin{split}&f_{2\ell+1}=\sum_{k=0}^{\ell}c_{\ell-k}\hat{f}_{2k+1},~{}~{}g_{2\ell}=\sum_{k=0}^{\ell}c_{\ell-k}\hat{g}_{2k},\\\
&h_{2\ell+1}=\sum_{k=0}^{\ell}c_{\ell-k}\hat{h}_{2k+1},\end{split}$ (2.25)

introducing $c_{0}=1.$ Next we construct the stationary GI hierarchy. Now
$q,r$ are considered as functions of the variable $x$ and the $2\times 2$
matrix-valued differential expression $Q_{2n+2}$ is introduced by

$Q_{2n+2}=\sum_{j=0}^{n}\left(\begin{array}[]{cc}-g_{2\ell}z^{2}&f_{2\ell+1}z\\\
-h_{2\ell+1}z&g_{2\ell+2}z^{2}\\\
\end{array}\right)M^{n-\ell}+\left(\begin{array}[]{cc}\frac{1}{2}g_{2n+2}&0\\\
0&-\frac{1}{2}g_{2n+2}\\\ \end{array}\right),~{}n\in\mathbb{N}_{0},$ (2.26)

Using the recursion relations (2.17)-(2.21), we rewrite the commutator of
$Q_{2n+2}$ and $M$ as

$\begin{split}[Q_{2n+2},M]=&~{}[Q_{2n+2},M]|_{\textrm{ker}(M-z^{2})}\\\
=&\left(\begin{array}[]{cc}-\frac{1}{2}g_{2n+2,x}&-2f_{2n+3}+qg_{2n+2}\\\
-2h_{2n+3}-rg_{2n+2}&\frac{1}{2}g_{2n+2,x}\\\
\end{array}\right),~{}n\in\mathbb{N}_{0},\end{split}$ (2.27)

where

$\textrm{ker}(M-z^{2})=\\{\Psi=\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{2}\\\
\end{array}\right):~{}\mathbb{R}\rightarrow\mathbb{C}_{\infty}^{2}|(M-z^{2})\Psi=0\\},~{}~{}z\in\mathbb{C},$
(2.28)

denotes the two-dimensional kernel of $M-z^{2}$. The stationary GI hierarchy
can be constructed in terms of the vanishing of the commutator of $Q_{2n+2}$
and $M$ in (2.27), that is,

$[Q_{2n+2},M]=0,~{}~{}n\in\mathbb{N}_{0},$ (2.29)

or equivalently,

$\left(\begin{array}[]{cc}-\frac{1}{2}g_{2n+2,x}&-2f_{2n+3}+qg_{2n+2}\\\
-2h_{2n+3}-rg_{2n+2}&\frac{1}{2}g_{2n+2,x}\\\
\end{array}\right)=0,~{}~{}n\in\mathbb{N}_{0}.$ (2.30)

From $2f_{2n+3}-qg_{2n+2}=0$ and $2h_{2n+3}+rg_{2n+2}=0,$ we get

$2rf_{2n+3}-qrg_{2n+2}=0,~{}~{}2qh_{2n+3}+qrg_{2n+2}=0,$ (2.31)

that is,

$2(rf_{2n+3}+qh_{2n+3})=0,$ (2.32)

which implies $\frac{1}{2}g_{2n+2,x}=0.$ Thus, varying $n\in\mathbb{N}_{0},$
stationary GI hierarchy (2.30) can be written as

$\textrm{s-GI}_{n}(q,r)=\left(\begin{array}[]{c}2f_{2n+3}-qg_{2n+2}\\\
-2h_{2n+3}-rg_{2n+2}\\\ \end{array}\right)=0,~{}~{}n\in\mathbb{N}_{0}.$ (2.33)

Explicitly,

$\displaystyle\textrm{s-GI}_{0}(q,r)=$
$\displaystyle~{}\left(\begin{array}[]{c}-2q_{x}+2c_{1}q\\\ 2r_{x}+2c_{1}r\\\
\end{array}\right)=0,$ (2.36) $\displaystyle\textrm{s-GI}_{1}(q,r)=$
$\displaystyle~{}\left(\begin{array}[]{c}q_{xx}+qq_{x}r+c_{1}(-2q_{x})+2c_{2}q\\\
r_{xx}-qrr_{x}+2c_{1}r_{x}+2c_{2}r\\\ \end{array}\right)=0,$ (2.39)
$\displaystyle\textrm{s-GI}_{2}(q,r)=$
$\displaystyle~{}\left(\begin{array}[]{c}-\frac{1}{2}q_{xxx}-\frac{3}{4}qq_{xx}r-\frac{3}{8}q^{2}r^{2}q_{x}-\frac{3}{4}q_{x}^{2}r\\\
\frac{1}{2}r_{xxx}-\frac{3}{4}qrr_{xx}+\frac{3}{8}q^{2}r^{2}r_{x}-\frac{3}{4}qr_{x}^{2}\\\
\end{array}\right)+c_{1}\left(\begin{array}[]{c}q_{xx}+qq_{x}r\\\
r_{xx}-2qrr_{x}\\\ \end{array}\right)$ (2.44)
$\displaystyle~{}~{}+c_{2}\left(\begin{array}[]{c}-2q_{x}\\\ 2r_{x}\\\
\end{array}\right)+c_{3}\left(\begin{array}[]{c}2q\\\ 2r\\\
\end{array}\right)=0,~{}~{}\textrm{etc.},$ (2.49)

represent the first few equations of the stationary GI hierarchy.

In the following we shall frequently make the assumption that $q,r$ satisfy
the $n$th stationary GI equations, that is, they satisfy one of the $n$th
stationary GI equations after a particular choice of integration constants
$c_{\ell}\in\mathbb{C},\ell=1,\ldots,n+1,n\in\mathbb{N}_{0}.$

Next we introduce polynomials $F_{2n+1},$ $G_{2n+2}$ and $H_{2n+1}$ with
respect to the spectral parameter $z\in\mathbb{C}$,

$\displaystyle F_{2n+1}(z)=$
$\displaystyle~{}\sum_{\ell=0}^{n}f_{2\ell+1}z^{2(n-\ell)+1},$ (2.50)
$\displaystyle G_{2n+2}(z)=$
$\displaystyle\sum_{\ell=0}^{n}g_{2\ell}z^{2(n-\ell)+2}+\frac{1}{2}g_{2n+2},$
(2.51) $\displaystyle H_{2n+1}(z)=$
$\displaystyle~{}\sum_{\ell=0}^{n}h_{2\ell+1}z^{2(n-\ell)+1},$ (2.52)

and corresponding homogeneous polynomials

$\displaystyle\widehat{F}_{0}(z)=F_{0}(z)=2q,~{}~{}$
$\displaystyle\widehat{F}_{2\ell+1}(z)=F_{2\ell+1}(z)|_{c_{k}=0,k=1,\ldots,\ell}=\sum_{k=0}^{\ell}\hat{f}_{2k+1}z^{2(\ell-k)+1},~{}~{}\ell\in\mathbb{N},$
$\displaystyle\widehat{G}_{0}(z)=G_{0}(z)=2,~{}~{}$
$\displaystyle\widehat{G}_{2\ell+2}(z)=G_{2\ell+2}(z)|_{c_{k}=0,k=1,\ldots,\ell+1}=\sum_{k=0}^{\ell}\hat{g}_{2k}z^{2(n-\ell)+2}+\frac{1}{2}\hat{g}_{2n+2},~{}~{}\ell\in\mathbb{N},$
$\displaystyle\widehat{H}_{0}(z)=H_{0}(z)=-2r,~{}~{}$
$\displaystyle\widehat{H}_{2\ell+1}(z)=H_{2\ell+1}(z)|_{c_{k}=0,k=1,\ldots,\ell}=\sum_{k=0}^{\ell}\hat{h}_{2k+1}z^{2(\ell-k)+1},~{}~{}\ell\in\mathbb{N}.$

Noting (2.17)-(2.21), one finds (2.29), or equivalently, (2.33) becomes

$\displaystyle\widetilde{G}_{2n+2,x}=$
$\displaystyle~{}\tilde{r}z\widetilde{F}_{2n+1}+\tilde{q}z\widetilde{H}_{2n+1},$
(2.53) $\displaystyle\widetilde{F}_{2n+1,x}=$
$\displaystyle~{}2\tilde{q}z\widetilde{G}_{2n+2}-(2z^{2}-\frac{1}{2}\tilde{q}\tilde{r})\widetilde{F}_{2n+1},$
(2.54) $\displaystyle\widetilde{H}_{2n+1,x}=$
$\displaystyle~{}2\tilde{r}z\widetilde{G}_{2n+2}+(2z^{2}-\frac{1}{2}\tilde{q}\tilde{r})\widetilde{H}_{2n+1},$
(2.55)

if we identifying $\tilde{q},\tilde{r}$ with $-q,-r$, respectively. Here

$\displaystyle\widetilde{F}_{2n+1}(z)=$
$\displaystyle~{}\sum_{\ell=0}^{n}\tilde{f}_{2\ell+1}z^{2(n-\ell)+1},$ (2.56)
$\displaystyle\widetilde{G}_{2n+2}(z)=$
$\displaystyle\sum_{\ell=0}^{n}\tilde{g}_{2\ell}z^{2(n-\ell)+2}+\frac{1}{2}\tilde{g}_{2n+2},$
(2.57) $\displaystyle\widetilde{H}_{2n+1}(z)=$
$\displaystyle~{}\sum_{\ell=0}^{n}\tilde{h}_{2\ell+1}z^{2(n-\ell)+1},$ (2.58)

and
$\\{\tilde{f}_{2\ell+1}\\}_{\ell=0,\ldots,n},\\{\tilde{g}_{2\ell}\\}_{\ell=0,\ldots,n+1},\\{\tilde{h}_{2\ell+1}\\}_{\ell=0,\ldots,n}$
satisfy the same recursion relations as
$\\{f_{2\ell+1}\\}_{\ell=0,\ldots,n},\\{g_{2\ell}\\}_{\ell=0,\ldots,n+1},\\{h_{2\ell+1}\\}_{\ell=0,\ldots,n}$
in (2.17)-(2.21), that is,

$\displaystyle\tilde{g}_{2\ell+1}=~{}\tilde{f}_{2\ell}=\tilde{h}_{2\ell}=0,~{}~{}\ell\in\mathbb{N}_{0},$
(2.59)
$\displaystyle\tilde{f}_{1}=~{}2\tilde{q},~{}\tilde{g}_{0}=2,~{}\tilde{h}_{1}=-2\tilde{r},$
(2.60) $\displaystyle
2\tilde{f}_{2\ell+1}=-\tilde{f}_{2\ell-1,x}+2\tilde{q}\tilde{g}_{2\ell}+(1/2)\tilde{q}\tilde{r}\tilde{f}_{2\ell-1},~{}\ell\in\mathbb{N}_{0},$
(2.61) $\displaystyle
2\tilde{h}_{2\ell+1}=~{}\tilde{h}_{2\ell-1,x}-2\tilde{r}\tilde{g}_{2\ell}+(1/2)\tilde{q}\tilde{r}\tilde{h}_{2\ell-1},~{}\ell\in\mathbb{N}_{0},$
(2.62)
$\displaystyle\tilde{g}_{2\ell,x}=~{}\tilde{r}\tilde{f}_{2\ell+1}+\tilde{q}\tilde{h}_{2\ell+1},~{}\ell\in\mathbb{N}_{0}.$
(2.63)

Moreover, (2.53)-(2.55) yield

$(\widetilde{G}_{2n+2}^{2}-\widetilde{F}_{2n+1}\widetilde{H}_{2n+1})_{x}=0,$
(2.64)

which implies
$\widetilde{G}_{2n+2}^{2}-\widetilde{F}_{2n+1}\widetilde{H}_{2n+1}$ is
$x$-independent and hence

$\widetilde{G}_{2n+2}^{2}-\widetilde{F}_{2n+1}\widetilde{H}_{2n+1}=R_{4n+4},$
(2.65)

where the integration constant $R_{4n+4}$ is a polynomial of degree $4n+4$
with respect to $z.$ Let $\\{E_{m}^{2}\\}_{m=0,\ldots,2n+1}$ be its zeros,
then

$R_{4n+4}(z)=4\prod_{m=0}^{2n+1}(z^{2}-E_{m}^{2}),~{}~{}\\{E_{m}^{2}\\}_{m=0,\ldots,2n+1}\in\mathbb{C}.$
(2.66)

Next we study the restriction of the differential expression (2.28) to the
two-dimensional kernel of $M-z^{2}.$ More precisely, (2.26) implies

$Q_{2n+2}|_{\textrm{ker}(M-z^{2})}=\left(\begin{array}[]{cc}-G_{2n+2}&F_{2n+1}\\\
-H_{2n+2}&G_{2n+2}\\\ \end{array}\right)\Big{|}_{\textrm{ker}(M-z^{2})}.$
(2.67)

We emphasize that the result (2.67) is valid independently of whether or not
$Q_{2n+2}$ and $M$ commute. However, if one makes the additional assumption
$Q_{2n+2}$ and $M$ commute, we can prove that there exists a algebraic
relationship between $Q_{2n+2}$ and $M$ and this relation will be shown in the
following results.

######  Theorem 2.2.

Assume that $Q_{2n+2}$ and $M$ commute, $[Q_{2n+2},M]=0,$ or equivalently,
suppose s-GI${}_{n}(q,r)=0$ for some fixed $n\in\mathbb{N}_{0}.$ Then $M$ and
$Q_{2n+2}$ satisfy an algebraic relationship of type

$\begin{split}\mathcal{F}_{n}(M,Q_{2n+2})=Q^{2}_{2n+2}-R_{4n+4}(M)=0,\\\
R_{4n+4}=4\prod_{m=0}^{2n+1}(z^{2}-E_{m}^{2}),~{}~{}z\in\mathbb{C}.\end{split}$
(2.68)

###### Proof.

From the commutativity of $Q_{2n+2}$ and $M$, the definition of $R_{4n+4}$
(2.65) and the relations

$\widetilde{G}_{2n+2}=G_{2n+2},\quad\widetilde{F}_{2n+1}=-F_{2n+1},\quad\widetilde{H}_{2n+1}=-H_{2n+1},$

as well as the expression for $Q_{2n+2}$ on the kernel of $M-z^{2}$ (2.67),
one infers

$\displaystyle Q^{2}_{2n+2}|_{\textrm{ker}(M-z^{2})}=$
$\displaystyle~{}\left(\begin{array}[]{cc}G_{2n+2}^{2}-F_{2n+1}H_{2n+1}&0\\\
0&G_{2n+2}^{2}-F_{2n+1}H_{2n+1}\\\
\end{array}\right)\Big{|}_{\textrm{ker}(M-z^{2})}$ $\displaystyle=$
$\displaystyle~{}\left(\begin{array}[]{cc}R_{4n+4}&0\\\ 0&R_{4n+4}\\\
\end{array}\right)\Big{|}_{\textrm{ker}(M-z^{2})}$ $\displaystyle=$
$\displaystyle~{}R_{4n+4}(M)|_{\textrm{ker}(M-z^{2})}.$

Thus, $Q_{2n+2}^{2}$ and $R_{4n+4}$ coincide on $\textrm{ker}(M-z^{2}).$ Since
$z^{2}\in\mathbb{C}^{2}$ is arbitrary, we finally obtain (2.68). ∎

One calls $\mathcal{F}_{n}(M,Q_{2n+2})$ the Burchnall-Chaundy polynomial of
the pair $[M,Q_{2n+2}].$ The relation (2.68) naturally leads to the
hyperelliptic curve $\mathcal{K}_{n},$ where

$\begin{split}&\mathcal{K}_{n}:~{}\mathcal{F}_{n}(z,y)=y^{2}-R_{4n+4}(z)=0,\\\
&R_{4n+4}(z)=\prod_{m=0}^{2n+1}(z^{2}-E_{m}^{2}),~{}~{}\\{E_{m}^{2}\\}_{m=0,\ldots,2n+1}\in\mathbb{C}.\end{split}$
(2.69)

Next section we introduce the notations
$\eta=z^{2},\widetilde{E}_{m}=E_{m}^{2}$, (2.69) changes to the hyperelliptic
curve of arithmetic genus $n\in\mathbb{N}_{0}$ (possibly with a singular
affine part), where

$\begin{split}&\mathcal{K}_{n}:~{}\mathcal{F}_{n}(\eta,y)=y^{2}-R_{4n+4}(\eta)=0,\\\
&R_{4n+4}(\eta)=\prod_{m=0}^{2n+1}(\eta-\widetilde{E}_{m}),~{}~{}\\{\widetilde{E}_{m}\\}_{m=0,\ldots,2n+1}\in\mathbb{C}.\end{split}$
(2.70)

Next we introduce the time-dependent GI hierarchy. This means that $q,r$ are
now considered an functions of both space and time. For each equation in the
hierarchy, that is, for each $n$, we introduce a deformation parameter
$t_{n}\in\mathbb{R}$ in $q$ and $r,$ replacing $q(x),r(x)$ by $q(x,t_{n})$,
$r(x,t_{n})$. The matrix differential expression $M$ now becomes

$M=\left(\begin{array}[]{cc}\partial+\frac{1}{4}q(\cdot,t_{n})r(\cdot,t_{n})&-zq(\cdot,t_{n})\\\
zr(\cdot,t_{n})&-\partial+\frac{1}{4}q(\cdot,t_{n})r(\cdot,t_{n})\\\
\end{array}\right).$ (2.71)

The quantities
$\\{f_{2\ell+1}\\}_{\ell\in\mathbb{N}_{0}},\\{g_{2\ell}\\}_{\ell\in\mathbb{N}_{0}}$
and $\\{h_{2\ell+1}\\}_{\ell\in\mathbb{N}_{0}}$ and $Q_{2n+2},n\in\mathbb{N},$
are still defined by (2.17)-(2.21) and (2.26), respectively. The time-
dependent GI hierarchy is obtained by imposing the Lax commutator equations

$\frac{d}{dt_{n}}(-M)-[-Q_{2n+2},-M]=0,~{}~{}t_{n}\in\mathbb{R},$ (2.72)

or equivalently,

$\displaystyle\left(\begin{array}[]{cc}-\frac{1}{4}(qr)_{t_{n}}+\frac{1}{2}g_{2n+2,x}&q_{t_{n}}-2f_{2n+3}+qg_{2n+2}\\\
-r_{t_{n}}+2h_{2n+3}+rg_{2n+2}&\frac{1}{4}(qr)_{t_{n}}+\frac{1}{2}g_{2n+2,x}\\\
\end{array}\right)=0.$ (2.75)

From (2.75), we obtain

$\displaystyle\frac{1}{4}(qr)_{t_{n}}=$
$\displaystyle~{}\frac{1}{4}(rq_{t_{n}}+qr_{t_{n}})$ $\displaystyle=$
$\displaystyle~{}\frac{1}{4}[r(2f_{2n+3}-qg_{2n+2})+q(2h_{2n+3}+rg_{2n+2})]$
$\displaystyle=$ $\displaystyle~{}\frac{1}{2}g_{2n+2,x},$

that is,

$\frac{1}{2}g_{2n+2,x}-\frac{1}{4}(qr)_{t_{n}}=0.$ (2.76)

Thus, varying $n$, the time-dependent GI hierarchy can be written as

$\textrm{GI}_{n}(q,r)=\left(\begin{array}[]{c}q_{t_{n}}-2f_{2n+3}+qg_{2n+2}\\\
r_{t_{n}}-2h_{2n+3}-rg_{2n+2}\\\ \end{array}\right)=0,\quad
t_{n}\in\mathbb{R},\quad n\in\mathbb{N}_{0}.$ (2.77)

Explicitly,

$\displaystyle\textrm{GI}_{0}(q,r)=$
$\displaystyle~{}\left(\begin{array}[]{c}q_{t_{0}}+2q_{x}-2c_{1}q\\\
r_{t_{0}}+2r_{x}+2c_{1}r\\\ \end{array}\right)=0,$ (2.80)
$\displaystyle\textrm{GI}_{1}(q,r)=$
$\displaystyle~{}\left(\begin{array}[]{c}q_{t_{1}}-q_{xx}-qq_{x}r+2c_{1}q_{x}-2c_{2}q\\\
r_{t_{1}}+r_{xx}-qrr_{x}+2c_{1}r_{x}+2c_{2}r\\\ \end{array}\right)=0,$ (2.83)
$\displaystyle\textrm{GI}_{2}(q,r)=$
$\displaystyle~{}\left(\begin{array}[]{c}q_{t_{2}}+\frac{1}{2}q_{xxx}+\frac{3}{4}qq_{xx}r+\frac{3}{8}q^{2}r^{2}q_{x}+\frac{3}{4}q_{x}^{2}r\\\
r_{t_{2}}+\frac{1}{2}r_{xxx}-\frac{3}{4}qrr_{xx}+\frac{3}{8}q^{2}r^{2}r_{x}-\frac{3}{4}qr_{x}^{2}\\\
\end{array}\right)+c_{1}\left(\begin{array}[]{c}-q_{xx}-qq_{x}r\\\
r_{xx}-2qrr_{x}\\\ \end{array}\right)$ (2.88)
$\displaystyle~{}~{}+c_{2}\left(\begin{array}[]{c}2q_{x}\\\ 2r_{x}\\\
\end{array}\right)+c_{3}\left(\begin{array}[]{c}-2q\\\ 2r\\\
\end{array}\right)=0,~{}~{}\textrm{etc.},$ (2.93)

are the first few equations of the time-dependent GI hierarchy. The system of
equation $\textrm{GI}_{1}(q,r)=0$ with $c_{1}=c_{2}=0$ represents the GI
system.

######  Remark 2.3.

An alternative construction of the GI hierarchy can be introduced by
developing its zero-curvature formulism. One defines

$\displaystyle U(z)=$
$\displaystyle~{}\left(\begin{array}[]{cc}-z^{2}+\frac{1}{4}qr&qz\\\
rz&z^{2}-\frac{1}{4}qr\\\ \end{array}\right),~{}~{}z\in\mathbb{C},~{}~{}$
(2.96) $\displaystyle V_{2n+2}(z)=$
$\displaystyle\left(\begin{array}[]{cc}-G_{2n+2}(z)&F_{2n+1}(z)\\\
-H_{2n+2}(z)&G_{2n+2}(z)\\\ \end{array}\right),\quad n\in\mathbb{N}_{0},$
(2.99)

where $G_{n+1},F_{n},H_{n}$ are polynomials defined in (2.50)-(2.52). Hence,
the stationary part of this section, being a consequence of $[Q_{2n+2},M]=0$,
can equivalently be based on the stationary zero-curvature equation

$\displaystyle 0=$ $\displaystyle~{}-V_{2n+2,x}(z)+[U(z),V_{2n+2}(z)]$
$\displaystyle=$
$\displaystyle~{}\left(\begin{smallmatrix}G_{2n+2,x}-zrF_{2n+1}-qzH_{2n+1}&-F_{2n+1,x}+2zqG_{2n+2}+(-2z^{2}+\frac{1}{2}qr)F_{2n+1}\\\
H_{2n+1,x}-2rzG_{2n+2}+(-2z^{2}+\frac{1}{2}qr)H_{2n+1}&-G_{2n+2,x}+zrF_{2n+1}+qzH_{2n+1}\\\
\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle~{}\left(\begin{array}[]{cc}\frac{1}{2}g_{2n+2,x}&(-qg_{2n+2}+2f_{2n+3})z\\\
(rg_{2n+2}+2h_{2n+3})z&-\frac{1}{2}g_{2n+2,x}\\\ \end{array}\right).$ (2.102)

In particular, the hyperelliptic curve $\mathcal{K}_{n}$ can also be obtained
from the characteristic equation of $V_{2n+2}$ by

$\displaystyle\textrm{det}(yI_{2}-V_{2n+2}(z))=$
$\displaystyle~{}y^{2}-\textrm{det}(V_{2n+2})$ $\displaystyle=$
$\displaystyle~{}y^{2}-G_{2n+2}^{2}(z)+F_{2n+1}(z)H_{2n+1}(z)$
$\displaystyle=$ $\displaystyle~{}y^{2}-\mathscr{R}_{4n+4}(z)=0.$ (2.103)

Similarly, the time-dependent part (2.72), (2.75), (2.77), being based on the
Lax equation (2.72), can equivalently developed from the zero-curvature
equation

$\displaystyle 0=$ $\displaystyle~{}U_{t_{n}}-V_{2n+2,x}+[U,V_{2n+2}]$
$\displaystyle=$
$\displaystyle~{}\left(\begin{smallmatrix}(qr)_{t_{n}}+G_{2n+2,x}-zrF_{2n+1}-qzH_{2n+1}&q_{t_{n}}z-F_{2n+1,x}+2zqG_{2n+2}+(-2z^{2}+\frac{1}{2}qr)F_{2n+1}\\\
r_{t_{n}}z+H_{2n+1,x}-2rzG_{2n+2}+(-2z^{2}+\frac{1}{2}qr)H_{2n+1}&-(qr)_{t_{n}}-G_{2n+2,x}+zrF_{2n+1}+qzH_{2n+1}\\\
\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle~{}\left(\begin{array}[]{cc}(qr)_{t_{n}}+\frac{1}{2}g_{2n+2,x}&(q_{t_{n}}-qg_{2n+2}+2f_{2n+3})z\\\
(r_{t_{n}}+rg_{2n+2}+2h_{2n+3})z&-(qr)_{t_{n}}-\frac{1}{2}g_{2n+2,x}\\\
\end{array}\right).$ (2.106)

Relations between the mKP hierarchy and the GI hierarchy can be summarized as
follows.

######  Theorem 2.1 (see [5] or [31]).

Assume $q,r$ is a compatible solution of the system

$\displaystyle\widehat{\textrm{GI}}_{1}(q,r)=0,$ (2.107)
$\displaystyle\widehat{\textrm{GI}}_{p}(q,r)=0,~{}~{}p\geq 2,$ (2.108)

where we denote by

$\widehat{\textrm{GI}}_{n}(q,r)=\textrm{GI}_{n}(q,r)|_{c_{k}=0,k=1,\ldots,n+1},~{}~{}n\in\mathbb{N}_{0},$
(2.109)

the corresponding homogeneous GI equations. Then

$u(x,y,t_{p})=-2^{p-1}q(-2x,y,(-2)^{p-1}t_{p})r(-2x,y,(-2)^{p-1}t_{p})$

yields a solution of the $p$th KP equation.

Given these preparations, next we shall study the algebro-geometric solutions
of the KP hierarachy. Now $q,r$ are considered as functions of variables
$x,t_{1},t_{p}$ with $t_{1}=y$. Then we introduce the following auxiliary
linear problem 444One can also start from the following linear problems
$\begin{split}&\psi_{x}(z)=U(z)\psi(z),\\\
&\psi_{t_{m}}(z)=\widehat{V}_{2m+2}(z)\psi(z),~{}~{}z\in\mathbb{C},~{}~{}m=1,2,3,\ldots\end{split}$
(2.110) where $q,r$ are considered as functions of $x,y,t_{2},\ldots$ and
obtain similar results.

$\begin{split}&\psi_{x}(z)=U(z)\psi(z),\\\
&\psi_{y}(z)=\widehat{V}_{4}(z)\psi(z),\\\
&\psi_{t_{2}}(z)=\widehat{V}_{2p+2}(z)\psi(z),~{}~{}z\in\mathbb{C},~{}~{}p\geq
2,\end{split}$ (2.111)

where

$\widehat{V}_{2k+2}(z)=V_{2k+2}(z)|_{c_{\ell}=0,\ell=1,\ldots,k+1},$

and $\psi(z)=(\psi_{1}(z,x,y,t_{p}),\psi_{2}(z,x,y,t_{p}))^{T}.$ Let

$\psi^{\pm}(z)=(\psi_{1}^{\pm}(z,x,y,t_{p}),\psi_{2}^{\pm}(z,x,y,t_{p}))$

be two fundamental solutions of linear system (2.111) and three squared basis
functions $\mathscr{G},\mathscr{F},\mathscr{H}$ can be defined in term of
$\psi^{\pm}_{1}(z),\psi^{\pm}_{2}(z)$ by

$\begin{split}\mathscr{G}(z,x,y,t_{p})=&~{}\frac{1}{2}(\psi_{1}^{+}(z,x,y,t_{p})\psi_{2}^{-}(z,x,y,t_{p})\\\
&~{}+\psi_{1}^{-}(z,x,y,t_{p})\psi_{2}^{+}(z,x,y,t_{p})),\\\
\mathscr{F}(z,x,y,t_{p})=&~{}\psi_{1}^{+}(z,x,y,t_{p})\psi_{1}^{-}(z,x,y,t_{p}),\\\
\mathscr{H}(z,x,y,t_{p})=&~{}\psi_{2}^{+}(z,x,y,t_{p})\psi_{2}^{-}(z,x,y,t_{p}).\end{split}$

Using (2.96), (2.99) and (2.111), we derive the following linear system

$\displaystyle\mathscr{G}_{x}=rz\mathscr{F}+qz\mathscr{H},$ (2.112)
$\displaystyle\mathscr{G}_{y}=\widehat{F}_{3}\mathscr{H}-\widehat{H}_{3}\mathscr{F},$
(2.113)
$\displaystyle\mathscr{G}_{t_{p}}=\widehat{F}_{2p+1}\mathscr{H}-\widehat{H}_{2p+1}\mathscr{F},$
(2.114)
$\displaystyle\mathscr{F}_{x}=2qz\mathscr{G}+\left(-2z^{2}+\frac{1}{2}qr\right)\mathscr{F},$
(2.115)
$\displaystyle\mathscr{F}_{y}=2\widehat{F}_{3}\mathscr{G}-2\widehat{G}_{4}\mathscr{F},$
(2.116)
$\displaystyle\mathscr{F}_{t_{p}}=2\widehat{F}_{2p+1}\mathscr{G}-2\widehat{G}_{2p+2}\mathscr{F},$
(2.117)
$\displaystyle\mathscr{H}_{x}=2rz\mathscr{G}+\left(2z^{2}-\frac{1}{2}qr\right)\mathscr{H},$
(2.118)
$\displaystyle\mathscr{H}_{y}=2\widehat{G}_{4}\mathscr{H}-2\widehat{H}_{3}\mathscr{G},$
(2.119)
$\displaystyle\mathscr{H}_{t_{p}}=2\widehat{G}_{2p+2}\mathscr{H}-2\widehat{H}_{2p+1}\mathscr{G}.$
(2.120)

For each fixed $p$, solutions of linear system (2.112)-(2.120) are connected
with the following basic initial value problem of GI system (2.122).

######  Theorem 2.1.

Assume $q,r\in C^{\infty}(\mathbb{R}^{2+1})$. Moreover, if $(q,r)$ solves
(2.107), (2.108), then the set

$\mathcal{E}=\textrm{span}\\{(\mathscr{G},\mathscr{F},\mathscr{H})|~{}(G_{2j+2},F_{2j+1},H_{2j+1}),~{}j\in\mathbb{N}_{0}\\}$
(2.121)

forms a vector space on $\mathbb{C}$ for solutions of the system
(2.112)-(2.120). In particular, the functions $q,r$ which constituting
$(G_{2n+2},$ $F_{2n+1},H_{2n+1})$, $n\in\mathbb{N}_{0},$ are connected with
solutions of the following initial problem

$\displaystyle\begin{cases}q_{y}=-2f_{5}+qg_{4},\\\ r_{y}=-2h_{5}-rg_{4},\\\
q_{t_{p}}=-2f_{2p+3}+qg_{2p+2},\\\ r_{t_{p}}=-rg_{2p+2}-2h_{2p+3},\\\
2f_{2n+3}-qg_{2n+2}=0,\\\ rg_{2n+2}+2h_{2n+3}=0,~{}~{}p\geq 2.\end{cases}$
(2.122)

###### Proof.

It is not difficult to verify that (2.122) is equivalent to the following zero
curvature representation

$\begin{split}&\frac{\partial}{\partial x}V_{2j+2}(z)-\frac{\partial}{\partial
t_{j}}U(z)=[U(z),V_{2j+2}(z)],~{}j=2,p,n,\end{split}$ (2.123)

from remark 2.3. Next we shall show

$\begin{split}&[V_{2j+2}(z)+\frac{\partial}{\partial
t_{j}},V_{2k+2}(z)+\frac{\partial}{\partial t_{k}}]=0,~{}j,k=2,p,\\\
&[V_{2j+2}(z)+\frac{\partial}{\partial
t_{j}},V_{2n+2}(z)]=0,~{}j=2,\end{split}$ (2.124)

which can be proved as follows. First, taking into account the $2\times 2$
matrix-valued differential expression

$\displaystyle M=$
$\displaystyle~{}\left(\begin{array}[]{cc}\partial+\frac{1}{4}qr&-zq\\\
zr&-\partial+\frac{1}{4}qr\\\ \end{array}\right),$ (2.127) $\displaystyle
Q_{2n+2}=$
$\displaystyle~{}\sum_{j=0}^{n}\left(\begin{array}[]{cc}-g_{2\ell}z^{2}&f_{2\ell+1}z\\\
-h_{2\ell+1}z&g_{2\ell+2}z^{2}\\\
\end{array}\right)M^{n-\ell}+\left(\begin{array}[]{cc}\frac{1}{2}g_{2n+2}&0\\\
0&-\frac{1}{2}g_{2n+2}\\\ \end{array}\right),$ (2.132)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n\in\mathbb{N}_{0},~{}f_{-1}=h_{-1}=0,$
(2.133)

one infers (2.123) is also equivalent to

$\begin{split}&[Q_{2j+2}+\frac{\partial}{\partial
t_{j}},M]=0,~{}~{}j=2,~{}p,\\\ &[Q_{2n+2},M]=0.\end{split}$

Then by Corollary 2 of Theorem 4.2 in [34] it follows

$\displaystyle[Q_{2j+2}+\frac{\partial}{\partial
t_{j}},Q_{2k+2}+\frac{\partial}{\partial t_{k}}]=0,~{}~{}j,~{}k=2,p,$
$\displaystyle[Q_{2j+2}+\frac{\partial}{\partial
t_{j}},Q_{2n+2}]=0,~{}~{}j=2,p,$

and hence

$\displaystyle[V_{2j+2}(z)+\frac{\partial}{\partial
t_{j}},V_{2k+2}(z)+\frac{\partial}{\partial t_{k}}]$
$\displaystyle=[Q_{2j+2}+\frac{\partial}{\partial
t_{j}},Q_{k+1}+\frac{\partial}{\partial t_{k}}]|_{\textrm{ker}(M-z)}$
$\displaystyle=0,~{}~{}j,~{}k=2,p,$ (2.134)
$\displaystyle[V_{2j+2}(z)+\frac{\partial}{\partial t_{j}},V_{2n+2}(z)]$
$\displaystyle=[Q_{2j+2}+\frac{\partial}{\partial
t_{j}},Q_{2n+2}]|_{\textrm{ker}(M-z)}$ $\displaystyle=0,~{}~{}j=2,p,$ (2.135)

holds, where

$\textrm{ker}(M-z)=\\{\Psi=\left(\begin{array}[]{c}\psi_{1}\\\ \psi_{2}\\\
\end{array}\right):\mathbb{R}^{2+1}\rightarrow\mathbb{C}_{\infty}^{2}|(M-z)\Psi=0\\},~{}z\in\mathbb{C}.\textrm{\qed}$

######  Remark 2.2.

This proposition can also be proved by introducing a meromorphic function
$\phi$ and considering $q,r$ as functions of three variables $x,y,t_{p}$ (see
[50], Lemma 5.3).

##  3 Spectral Curve, Dubrovin-type Equations and Trace Formula

In the following, we shall introduce the spectral curve associated with mKP
hierarchy. To emphasize the difference between different solutions in (2.121),
we add a subscript in each of $\mathscr{G},\mathscr{F},\mathscr{H}$, that is,

$(\mathscr{G}_{2n+2},\mathscr{F}_{2n+1},\mathscr{H}_{2n+1})=(G_{2n+2},F_{2n+1},H_{2n+1}),~{}n\in\mathbb{N}.$

Then using (2.112)-(2.120) and theorem 2.1, we have

$(\mathscr{G}^{2}_{2n+2}-\mathscr{F}_{2n+1}\mathscr{H}_{2n+1})_{t_{j}}=0,~{}~{}n\in\mathbb{N},~{}j=1,2,p,$
(3.1)

and hence $\mathscr{G}^{2}_{n+1}-\mathscr{F}_{n}\mathscr{H}_{n}$ is
$x,y,t_{p}$-independent implying

$\mathscr{G}^{2}_{2n+2}-\mathscr{F}_{2n+1}\mathscr{H}_{2n+1}=\mathscr{R}_{4n+4},~{}~{}n\in\mathbb{N},$
(3.2)

where integration constant $\mathscr{R}_{4n+4}$ is a polynomial of degree
$4n+4$ in $z$. Let

$\begin{split}\mathscr{R}_{4n+4}&(z)=\sum_{j=0}^{2n+2-j}(-1)^{j}s_{j}z^{2n+2-j},\\\
&s_{0}=4,~{}s_{j}\in\mathbb{C},j=1,\ldots,2n,\end{split}$ (3.3)

and hyperelliptic curve associated with the $p$th mKP equation is then
introduced as follows:

$\begin{split}X:\mathscr{P}(\eta,y)=&~{}y^{2}-\frac{1}{4}\mathscr{R}_{4n+4}(\sqrt{\eta})\\\
=&~{}y^{2}-\frac{1}{4}\mathscr{G}_{2n+2}^{2}(\sqrt{\eta})+\frac{1}{4}\mathscr{F}_{2n+1}(\sqrt{\eta})\mathscr{H}_{2n+1}(\sqrt{\eta})\\\
=&~{}0.\end{split}$ (3.4)

The curve $X$ is compactified by joining two points $P_{\infty\pm}$ at
infinity but for notational simplicity the compactification is also denoted by
$X$. Points $P$ on $X\backslash\\{P_{\infty\pm}\\}$ are denote by pairs
$(\eta,y)$, where $y(\cdot)$ is the meromorphic function on $X$ satisfying
$\mathscr{P}(\eta,y)=0.$ The complex structure on $X$ is defined in the usual
way. Hence, $X$ becomes a two-sheeted hyperelliptic Riemann surface of
(arithmetic) genus $n$ in a standard manner. Moreover, we denote the upper and
lower sheets $\Pi_{\pm}$ by

$\Pi_{\pm}=\\{(\eta,\pm\frac{1}{2}\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})})\in
X|\eta\in\Pi\\},$

where $\Pi$ denotes the cut plane $\mathbb{C}\backslash\mathcal{C}$ and
$\mathcal{C}$ is the union of $n$ nonintersecting cuts joining two different
branches of $\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}$. The holomorphic sheet
exchange map on $X$ is defined by

$\displaystyle*:\quad X\rightarrow X,$ $\displaystyle
P=(\eta,\frac{1}{2}\sqrt{\mathscr{R}_{4n+4}(\eta)})\mapsto
P^{*}=(\eta,-\frac{1}{2}\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}),\quad$
$\displaystyle P_{\infty\pm}\mapsto P_{\infty\pm}^{*}=P_{\infty\mp}.$

Moreover, positive divisors on $X$ of degree $n$ are denoted by

$\mathcal{D}_{P_{1},\ldots,P_{n}}:\begin{cases}X\rightarrow\mathbb{N}_{0},\\\
P\rightarrow\mathcal{D}_{P_{1},\ldots,P_{n}}(P)=\begin{cases}\textrm{ $k$ if
$P$ occurs $k$ times in $\\{P_{1},\ldots,P_{n}\\},$}\\\ \textrm{ $0$ if
$P\notin$$\\{P_{1},\ldots,P_{n}\\}.$}\end{cases}\end{cases}$ (3.5)

In particular, the divisor $\left(\phi(\cdot)\right)$ of a meromorphic
function $\phi(\cdot)$ on X is defined by

$\displaystyle\left(\phi(\cdot)\right):X\rightarrow\mathbb{Z},\quad
P\mapsto\omega_{\phi}(P),$ (3.6)

where $\omega_{\phi}(P)=m_{0}\in\mathbb{Z}$ if
$(\phi\circ\zeta_{P}^{-1})(\zeta)=\sum_{n=m_{0}}^{\infty}c_{n}(P)\zeta^{n}$
for some $m_{0}\in\mathbb{Z}$ by using a chart $(U_{P},\zeta_{P})$ near $P\in
X.$ Finally, we introduce symmetric functions of $x_{1},\ldots,x_{n}$

$\begin{split}&\Psi_{0}(\underline{x})=1,~{}~{}\Psi_{1}(\underline{x})=\sum_{j=1}^{n}x_{j},~{}~{}\Psi_{2}(\underline{x})=\sum_{j,k=1,j<k}^{n}x_{j}x_{k},\\\
&\Psi_{3}(\underline{x})=\sum\nolimits_{j,k,\ell=1,j<k<\ell}^{n}x_{j}x_{k}x_{\ell},~{}~{}\textrm{etc.,}\end{split}$
(3.7)

where

$\underline{x}=(x_{1},\ldots,x_{n}).$

Now we turn to the parameter representations of
$\mathscr{G}_{2n+2},\mathscr{F}_{2n+1},\mathscr{H}_{2n+1}$, which are
described by the evolution of auxiliary spectrum points $\mu_{j}(x,y,t_{p}),$
$\nu_{j}(x,y,t_{p}),j=1,\ldots,n$. This procedure is standard, which is
similar with 1+1 dimensional case.

######  Theorem 3.1.

Solutions of (2.112)-(2.120) can also be expressed as

$\begin{split}&\mathscr{G}_{2n+2}=\sum_{j=0}^{n+1}(-1)^{j}\dot{g}_{2j}(x,y,t_{p})z^{2n+2-2j},~{}~{}\\\
&\mathscr{F}_{2n+1}=2q(x,y,t_{p})z\prod_{j=1}^{n}(z^{2}-\mu_{j}(x,y,t_{p})),\\\
&\mathscr{H}_{2n+1}=-2r(x,y,t_{p})z\prod_{j=1}^{n}(z^{2}-\nu_{j}(x,y,t_{p})),\end{split}$
(3.8)

where

$\displaystyle\dot{g}_{0}=$ $\displaystyle~{}2,~{}$
$\displaystyle\dot{g}_{2}=$ $\displaystyle~{}qr+\frac{s_{1}}{4},$ (3.9)
$\displaystyle\dot{g}_{2j}=$
$\displaystyle~{}\frac{1}{4}\Big{[}-\sum_{\nu=1}^{j-1}\dot{g}_{\nu}\dot{g}_{j-\nu}-4qr\sum_{\alpha+\beta=j}\Psi_{\alpha}(\underline{\mu})\Psi_{j-\beta}(\underline{\nu})+s_{j}\Big{]},~{}s=2,\ldots,n+1,$

and $\\{\mu_{j}(x,y,t_{p})\\}_{j=1}^{n},\\{\nu_{j}(x,y,t_{p})\\}_{j=1}^{n}$
are $n$ roots of $\mathscr{F}_{2n+1},\mathscr{H}_{2n+1}$, respectively.
Moreover, if $\\{\mu_{j}(x,y,t_{p})\\}_{j=1}^{n}$ are mutually distinct and
finite, then they satisfy the Dubrovin-type equations

$\displaystyle\mu_{j,x}=-\frac{\sqrt{\mu_{j}\mathscr{R}_{4n+4}(\sqrt{\mu_{j}})}}{\prod_{k\neq
j}(\mu_{j}-\mu_{k})},$ (3.10)
$\displaystyle\mu_{j,y}=-\frac{\widehat{F}_{5}(\sqrt{\mu_{j}})\sqrt{\mathscr{R}_{4n+4}(\sqrt{\mu_{j}})}}{q\prod_{k\neq
j}(\mu_{j}-\mu_{k})},$ (3.11)
$\displaystyle\mu_{j,t_{p}}=-\frac{\widehat{F}_{2p+1}(\sqrt{\mu_{j}})\sqrt{\mathscr{R}_{4n+4}(\sqrt{\mu_{j}})}}{q\prod_{k\neq
j}(\mu_{j}-\mu_{k})},$ (3.12)

and similar statement is also true for $\\{\nu_{j}(x,y,t_{p})\\}_{j=1}^{n}$,
where (3.10)-(3.12) change to

$\displaystyle\nu_{j,x}=-\frac{\sqrt{\mathscr{R}_{4n+4}(\sqrt{\nu_{j}})}}{\prod_{k\neq
j}(\nu_{j}-\nu_{k})},$ (3.13)
$\displaystyle\nu_{j,y}=-\frac{\widehat{H}_{5}(\sqrt{\mu_{j}})\sqrt{\mathscr{R}_{4n+4}(\sqrt{\mu_{j}})}}{r\prod_{k\neq
j}(\mu_{j}-\mu_{k})},$ (3.14)
$\displaystyle\nu_{j,t_{p}}=-\frac{\widehat{H}_{2p+1}(\sqrt{\mu_{j}})\sqrt{\mathscr{R}_{4n+4}(\sqrt{\mu_{j}})}}{r\prod_{k\neq
j}(\mu_{j}-\mu_{k})}.$ (3.15)

Finally, $q,r$ and $\mu_{j},\nu_{j}$ are connected by the following trace
formula

$\begin{split}&\sum_{j=1}^{n}\mu_{j}=\frac{1}{4}qr+\frac{1}{2}\frac{q_{x}}{q}+\frac{s_{1}}{8},\\\
&\sum_{j=1}^{n}\nu_{j}=\frac{1}{4}qr-\frac{1}{2}\frac{r_{x}}{r}+\frac{s_{1}}{8}.\end{split}$
(3.16)

###### Proof.

First, insertion of (3.8) into (3.2), (3.3) and a comparison powers of $z$
yields (3.1). Then by (2.115), (2.116), (2.117), (2.118), (2.119) and (2.120),
taking into account (3.4), (3.8), one derives (3.10)-(3.15). Moreover,
combining (2.51) with (3.1) yields

$\begin{split}c_{1}=-\frac{1}{8}s_{1}.\end{split}$ (3.17)

Finally, the formula (3.16) is the direct result of (3.8), (3.17) and theorem
2.1. ∎

##  4 Baker-Akhiezer Function

It is well known that the Baker-Akhiezer function plays a very important role
in finite gap integration of soliton equations and it permits us to obtain the
Riemann theta function representation for solutions of a given equation and
there are numerous articles have been devoted to this subject [3], [8], [17],
[25]. This section aims to provide the explicit form of ”Baker-Akhiezer
function” associated with the $p$th mKP equation, which consists of spectral
parameter $z$, potentials $q,r$, and then study its analytic properties.
Moreover, we will find the conservation relations of soliton equations play a
key role in the construction of Baker-Akhiezer function, which reveals
symmetry is the intrinsic character of classical integrable system.

Let us first study the Baker-Akhiezer function

$\psi(P)=(\psi_{1}(P,x,y,t_{p}),\psi_{2}(P,x,y,t_{p})),~{}P\in X,$

which satisfies

$\begin{split}&\psi_{x}(P)=~{}U(z)\psi(P),~{}\psi_{y}(P)=~{}\widehat{V}_{4}(z)\psi(P),\\\
&\psi_{t_{p}}(P)=~{}\widehat{V}_{2p+2}(z)\psi(P),~{}~{}p>2,\\\
&\psi_{1}(P,x_{0},y_{0},t_{p,0})=1,~{}~{}(P,x_{0},y_{0},t_{p,0})\in
X\times\mathbb{R}^{3}.\end{split}$ (4.1)

To find its explicit form, we need some preparations.

######  Lemma 4.1.

Suppose $q,r\in C^{\infty}(\mathbb{R}^{2+1})$ and $z\in\mathbb{C}$. Then
$\mathscr{G}_{2n+2},\mathscr{F}_{2n+1},\mathscr{H}_{2n+1}$ and basic
fundamental solutions $(\psi_{1}^{\pm},\psi_{2}^{\pm})$ of linear system (4.1)
have the following algebraic relation:

$\displaystyle(\psi_{1}^{+}(z)\psi_{2}^{-}(z)-\psi_{1}^{-}(z)\psi_{2}^{+}(z))^{2}=4\mathscr{R}_{4n+4}^{2}(z).$
(4.2)

If we take
$\psi_{1}^{+}(z)\psi_{2}^{-}(z)-\psi_{1}^{-}(z)\psi_{2}^{+}(z)=2\sqrt{\mathscr{R}_{4n+4}(z)},$
then

$\displaystyle\psi_{1}^{+}(z)\psi_{2}^{-}(z)=\mathscr{G}_{2n+2}(z)+\sqrt{\mathscr{R}_{4n+4}(z)},$
(4.3)
$\displaystyle\psi_{1}^{-}(z)\psi_{2}^{+}(z)=\mathscr{G}_{2n+2}(z)-\sqrt{\mathscr{R}_{4n+4}(z)},$
(4.4)
$\displaystyle\frac{\psi_{2}^{\pm}(z)}{\psi_{1}^{\pm}(z)}=\frac{\mathscr{H}_{2n+1}(z)}{\mathscr{G}_{2n+2}(z)\pm\sqrt{\mathscr{R}_{4n+4}(z)}}=\frac{\mathscr{G}_{2n+2}\mp\sqrt{\mathscr{R}_{4n+4}(z)}}{\mathscr{F}_{2n+1}}.$
(4.5)

###### Proof.

Expressions (4.2)-(4.5) can easily be verified by (2), (3.2). ∎

######  Lemma 4.2.

Suppose $q,r\in C^{\infty}(\mathbb{R}^{2+1})$. Then we have the following
relations

$\displaystyle\Big{[}\frac{q(x,y,t_{p})z}{\mathscr{F}_{2n+1}(z,x,y,t_{p})}\Big{]}_{t_{j}}=\Big{[}\frac{{\widehat{F}_{2j+1}(z,x,y,t_{p})}}{\mathscr{F}_{2n+1}(z,x,y,t_{p})}\Big{]}_{x},$
(4.6)
$\displaystyle\Big{[}\frac{r(x,y,t_{p})z}{\mathscr{H}_{2n+1}(z,x,y,t_{p})}\Big{]}_{t_{j}}=-\Big{[}\frac{\widehat{H}_{2j+1}(z,x,y,t_{p})}{\mathscr{F}_{2n+1}(z,x,y,t_{p})}\Big{]}_{x},~{}~{}j=2,p,$
(4.7)
$\displaystyle\Big{[}\frac{{\widehat{F}_{3}(z,x,y,t_{p})}}{\mathscr{F}_{2n+1}(z,x,y,t_{p})}\Big{]}_{t_{p}}=\Big{[}\frac{{\widehat{F}_{2p+1}(z,x,y,t_{p})}}{\mathscr{F}_{2n+1}(z,x,y,t_{p})}\Big{]}_{y},$
(4.8)
$\displaystyle\Big{[}\frac{{\widehat{H}_{3}(z,x,y,t_{p})}}{\mathscr{F}_{2n+1}(z,x,y,t_{p})}\Big{]}_{t_{p}}=\Big{[}\frac{{\widehat{H}_{2p+1}(z,x,y,t_{p})}}{\mathscr{F}_{2n+1}(z,x,y,t_{p})}\Big{]}_{y}.$
(4.9)

###### Proof.

The proof is straightforward. By (2.115), (2.116), (2.117) and

$q_{t_{j}}-F_{2j+1,x}+2zqG_{2j+2}+(-2z^{2}+\frac{1}{2}qr)F_{2j+1}=0,~{}~{}j=2,p,$
(4.10)

we have

$\begin{split}\Big{[}\frac{qz}{\mathscr{F}_{2n+1}}\Big{]}_{t_{j}}=&~{}\frac{q_{t_{j}}z}{\mathscr{F}_{2n+1}}-\frac{qz\mathscr{F}_{2n+1,t_{j}}}{\mathscr{F}_{2n+1}^{2}}\\\
=&~{}\frac{\widehat{F}_{2j+1,x}+(2z^{2}-\frac{1}{2}qr)\widehat{F}_{2j+1}-2qz\widehat{G}_{2j+2}}{\mathscr{F}_{2n+1}}-\frac{qz(2\widehat{F}_{2j+1}\mathscr{G}_{2n+2}-2\widehat{G}_{2j+2}\mathscr{F}_{2n+1})}{\mathscr{F}_{2n+1}^{2}}\\\
=&\frac{\widehat{F}_{2j+1,x}+\frac{(\mathscr{F}_{2n+1,x}-2qzG_{2n+2})}{\mathscr{F}_{2n+1}}\widehat{F}_{2j+1}}{\mathscr{F}_{2n+1}}-\frac{2qz\widehat{F}_{2j+1}\mathscr{G}_{2n+2}}{\mathscr{F}_{2n+1}^{2}}\\\
=&\frac{\widehat{F}_{2j+1,x}\mathscr{F}_{2n+1}-\widehat{F}_{2j+1}\mathscr{F}_{2n+1,x}}{\mathscr{F}_{2n+1}^{2}}=\Big{[}\frac{{\widehat{F}_{2j+1}}}{\mathscr{F}_{2n+1}}\Big{]}_{x},~{}~{}j=2,p.\\\
\end{split}$

Moreover, (4.7)-(4.9) follows by the relations

$r_{t_{n}}z+\widehat{H}_{2n+1,x}-2zr\widehat{G}_{2n+2}+(-2z^{2}+\frac{1}{2}qr)\widehat{H}_{2n+1}=0,$
(4.11)

(2.134), (2.118), (2.119) and (2.120). ∎

Now we lift the fundamental solution $(\psi_{1}^{\pm}(z),\psi_{2}^{\pm}(z))$,
$z\in\mathbb{C},$ to Riemann surface $X$. Define the Baker-Akhiezer function
by

$\displaystyle\psi_{j}(P,x,y,t_{p})=\psi_{j}^{+}(z,x,y,t_{p}),~{}~{}j=1,2,~{}~{}y(P)=\frac{1}{2}\sqrt{\mathscr{R}_{4n+4}(z)},~{}~{}\textrm{for}~{}~{}P\in\Pi_{+},$
$\displaystyle\psi_{j}(P,x,y,t_{p})=\psi_{j}^{-}(z,x,y,t_{p}),~{}~{}j=1,2,~{}~{}y(P)=-\frac{1}{2}\sqrt{\mathscr{R}_{4n+4}(z)},~{}~{}\textrm{for}~{}~{}P\in\Pi_{-},$
$\displaystyle\lim_{P\rightarrow P_{0}}\psi_{j}(P,x,t_{p})=\lim_{P\rightarrow
P_{0}}\psi_{j}(P^{*},x,t_{p}),~{}~{}\textrm{for}~{}~{}j=1,2,~{}~{}P\in\Pi_{\pm},~{}~{}P_{0}\in\mathcal{C},$

where we choose the branches of $\sqrt{\mathscr{R}_{2n+2}(\cdot)}$ satisfying

$\lim_{z\rightarrow\infty}\frac{\mathscr{G}_{n+1}}{\sqrt{\mathscr{R}_{2n+2}(z)}}=1.$

Moreover, let

$\displaystyle\hat{\mu}_{j}(x,y,t_{p})=$
$\displaystyle\Big{(}\mu_{j}(x,y,t_{p}),-2\sqrt{\mathscr{R}_{4n+4}(\mu_{j}(x,y,t_{p}))}\Big{)}\in\Pi_{-},$
$\displaystyle\hat{\nu}_{j}(x,y,t_{p})=$
$\displaystyle\Big{(}\nu_{j}(x,y,t_{p}),2\sqrt{\mathscr{R}_{4n+4}(\mu_{j}(x,y,t_{p}))}\Big{)}\in\Pi_{+},~{}~{}j=1,\ldots,n.$

Based on above preparations, we shall study explicit forms of Baker-Akhiezer
function $\psi_{j}(P),j=1,2$.

######  Theorem 4.3.

Suppose $q,r\in C^{\infty}(\mathbb{R}^{2+1})$. Then Baker-Akhiezer function
satisfying condition (4.1) can be expressed as

$\displaystyle\psi_{1}(P,x,y,t_{p})=$
$\displaystyle~{}\sqrt{\frac{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}}\exp\Big{(}-2\int_{x_{0}}^{x}\frac{q(x^{\prime},y,t_{p})\sqrt{\eta}y(P)}{\mathscr{F}_{2n+1}(\sqrt{\eta},x^{\prime},y,t_{p})}dx^{\prime}$
$\displaystyle-2\int_{y_{0}}^{y}\frac{y(P)\widehat{F}_{3}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}dy^{\prime}-2\int_{t_{p,0}}^{t_{p}}\frac{y(P)\widehat{F}_{2p+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}dt^{\prime}\Big{)},$
(4.12) $\displaystyle\psi_{2}(P,x,y,t_{p})=$
$\displaystyle~{}\sqrt{\frac{\mathscr{H}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}}\exp\Big{(}2\int_{x_{0}}^{x}\frac{r(x^{\prime},y,t_{p})\sqrt{\eta}y(P)}{\mathscr{F}_{2n+1}(\sqrt{\eta},x^{\prime},y,t_{p})}dx^{\prime}$
$\displaystyle-2\int_{y_{0}}^{y}\frac{y(P)\widehat{H}_{3}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}dy^{\prime}-2\int_{t_{p,0}}^{t_{p}}\frac{y(P)\widehat{H}_{2p+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}dt^{\prime}\Big{)},$
(4.13)

where $\eta=z^{2}$ denotes the projection of $P$ to complex sphere.

###### Proof.

By (2.111), (2.115)-(2.120), (4.5), one obtains

$\displaystyle\psi_{1,x}^{\pm}(z)=$
$\displaystyle-\left(z^{2}-\frac{1}{4}qr\right)\psi_{1}^{\pm}(z)+qz\psi_{2}^{\pm}(z)$
$\displaystyle=$
$\displaystyle\left[\left(-\frac{qz\mathscr{G}_{2n+2}(z)}{\mathscr{F}_{2n+1}(z)}+\frac{1}{2}\frac{\mathscr{F}_{2n+1,x}(z)}{\mathscr{F}_{2n+1}(z)}\right)+qz\frac{\psi_{2}^{\pm}(z)}{\psi_{1}^{\pm}(z)}\right]\psi_{1}^{\pm}(z)$
$\displaystyle=$
$\displaystyle\left[\left(-\frac{qz\mathscr{G}_{2n+2}(z)}{\mathscr{F}_{2n+1}(z)}+\frac{1}{2}\frac{\mathscr{F}_{2n+1,x}(z)}{\mathscr{F}_{2n+1}(z)}\right)+qz\frac{\mathscr{H}_{2n+1}(z)}{\mathscr{G}_{2n+2}(z)\pm\sqrt{\mathscr{R}_{4n+4}(z)}}\right]\psi_{1}^{\pm}(z)$
$\displaystyle=$ $\displaystyle\frac{\mp
qz\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{F}_{2n+1,x}(z)}{\mathscr{F}_{2n+1}(z)}\psi_{1}^{\pm}(z),$
(4.14) $\displaystyle\psi_{1,t_{j}}^{\pm}(z)=$
$\displaystyle-\widehat{G}_{2j+2}(z)\psi_{1}^{\pm}(z)+\widehat{F}_{2j+1}(z)\psi_{2}^{\pm}(z)$
$\displaystyle=$
$\displaystyle\left[\left(-\frac{\widehat{F}_{2j+1}(z)\mathscr{G}_{2n+2}(z)}{\mathscr{F}_{2n+1}(z)}+\frac{1}{2}\frac{\mathscr{F}_{2n+1,t_{j}}(z)}{\mathscr{F}_{2n+1}(z)}\right)+\widehat{F}_{2j+1}(z)\frac{\psi_{2}^{\pm}(z)}{\psi_{1}^{\pm}(z)}\right]\psi_{1}^{\pm}(z)$
$\displaystyle=$
$\displaystyle\left[\left(-\frac{\widehat{F}_{2j+1}(z)\mathscr{G}_{2n+2}(z)}{\mathscr{F}_{2n+1}(z)}+\frac{1}{2}\frac{\mathscr{F}_{2n+1,t_{j}(z)}}{\mathscr{F}_{2n+1}(z)}\right)+\widehat{F}_{2j+1}(z)\frac{\mathscr{H}_{2n+1}(z)}{\mathscr{G}_{2n+2}(z)\pm\sqrt{\mathscr{R}_{4n+4}(z)}}\right]\psi_{1}^{\pm}(z)$
$\displaystyle=$
$\displaystyle\frac{\mp\widehat{F}_{2j+1}(z)\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{F}_{2n+1,t_{j}}(z)}{\mathscr{F}_{2n+1}(z)}\psi_{1}^{\pm}(z),~{}~{}j=2,p,$
(4.15)

and similarly

$\displaystyle\psi_{2,x}^{\pm}(z)=$
$\displaystyle~{}rz\psi_{1}^{\pm}(z)+\left(z^{2}-\frac{1}{4}qr\right)\psi_{2}^{\pm}(z)$
$\displaystyle=$
$\displaystyle~{}\left[rz\frac{\psi_{1}^{\pm}(z)}{\psi_{2}^{\pm}(z)}+\left(\frac{\mathscr{H}_{2n+1,x}(z)}{2\mathscr{H}_{2n+1}(z)}-\frac{rz\mathscr{G}_{2n+2}(z)}{\mathscr{H}_{2n+1}(z)}\right)\right]\psi_{2}^{\pm}(z)$
$\displaystyle=$
$\displaystyle~{}\left[rz\frac{\mathscr{G}_{2n+2}(z)\pm\sqrt{\mathscr{R}_{4n+4}(z)}}{\mathscr{H}_{2n+1}(z)}+\left(\frac{\mathscr{H}_{2n+1,x}(z)}{2\mathscr{H}_{2n+1}(z)}-\frac{rz\mathscr{G}_{2n+2}(z)}{\mathscr{H}_{2n+1}(z)}\right)\right]\psi_{2}^{\pm}(z)$
$\displaystyle=$ $\displaystyle~{}\frac{\pm
rz\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{H}_{2n+1,x}(z)}{\mathscr{H}_{2n+1}(z)}\psi_{2}^{\pm}(z),$
(4.16) $\displaystyle\psi_{2,t_{j}}^{\pm}(z)=$
$\displaystyle~{}-\widehat{H}_{2j+1}(z)\psi_{1}^{\pm}(z)+\widehat{G}_{2j+2}(z)\psi_{2}^{\pm}(z)$
$\displaystyle=$
$\displaystyle~{}\left[-\widehat{H}_{2j+1}(z)\frac{\psi_{1}^{\pm}(z)}{\psi_{2}^{\pm}(z)}+\left(\frac{\mathscr{H}_{2n+1,t_{j}}(z)}{2\mathscr{H}_{2n+1}(z)}+\widehat{H}_{2j+1}(z)\frac{\mathscr{G}_{2n+2}(z)}{\mathscr{H}_{2n+1}(z)}\right)\right]\psi_{2}^{\pm}(z)$
$\displaystyle=$
$\displaystyle~{}\left[-\widehat{H}_{2j+1}(z)\frac{\mathscr{G}_{2n+2}(z)\pm\sqrt{\mathscr{R}_{4n+4}(z)}}{\mathscr{H}_{2n+1}(z)}+\left(\frac{\mathscr{H}_{2n+1,t_{j}}(z)}{2\mathscr{H}_{2n+1}(z)}+\widehat{H}_{2j+1}(z)\frac{\mathscr{G}_{2n+2}(z)}{\mathscr{H}_{2n+1}(z)}\right)\right]\psi_{2}^{\pm}(z)$
$\displaystyle=$
$\displaystyle~{}\frac{\mp\widehat{H}_{2j+1}(z)\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{H}_{2n+1,t_{j}}(z)}{\mathscr{H}_{2n+1}(z)}\psi_{2}^{\pm}(z),~{}~{}j=2,p.$
(4.17)

Thus, we have

$\displaystyle d\ln(\psi_{1}^{\pm}(z,x,y,t_{p}))=$
$\displaystyle~{}\left(\frac{\mp
qz\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{F}_{2n+1,x}(z)}{\mathscr{F}_{2n+1}(z)}\right)dx$
$\displaystyle~{}+\left(\frac{\mp\widehat{F}_{3}(z)\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{F}_{2n+1,y}(z)}{\mathscr{F}_{2n+1}(z)}\right)dy$
$\displaystyle~{}+\left(\frac{\mp\widehat{F}_{2p+1}(z)\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{F}_{2n+1,t_{p}}(z)}{\mathscr{F}_{2n+1}(z)}\right)dt_{p}$
(4.18)

and

$\displaystyle d\ln(\psi_{2}^{\pm}(z,x,y,t_{p}))=$
$\displaystyle~{}\left(\frac{\pm
rz\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{H}_{2n+1,x}(z)}{\mathscr{H}_{2n+1}(z)}\right)dx$
$\displaystyle~{}+\left(\frac{\mp\widehat{H}_{3}(z)\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{H}_{2n+1,y}(z)}{\mathscr{H}_{2n+1}(z)}\right)dy$
$\displaystyle~{}+\left(\frac{\mp\widehat{H}_{2p+1}(z)\sqrt{\mathscr{R}_{4n+4}(z)}+\frac{1}{2}\mathscr{H}_{2n+1,t_{p}}(z)}{\mathscr{H}_{2n+1}(z)}\right)dt_{p}.$
(4.19)

According to relations (4.7)-(4.9), it follows that the integrals

$\int_{(x_{0},y_{0},t_{p,0})}^{(x,y,t_{p})}d\ln\big{(}\frac{\psi_{j}^{\pm}(z,x,y,t_{p})}{\sqrt{\mathscr{F}_{2n+1}(z,x,y,t_{p})}}\big{)},~{}~{}j=1,2$
(4.20)

is independent of the path. Therefore taking into account the normalization
condition $\psi_{1}(P,x_{0},y_{0},t_{p,0})=1$, and choosing a special path

$(x_{0},y_{0},t_{p,0})\rightarrow(x_{0},y_{0},t_{p})\rightarrow(x_{0},y,t_{p})\rightarrow(x,y,t_{p})$

in (4.20), we finally obtain (4.12) and (4.13). ∎

Basic properties of Baker-Akhiezer functions $\psi_{j}(P),j=1,2$ are
summarized in the following result.

######  Lemma 4.4.

Suppose $q,r\in C^{\infty}(\mathbb{R}^{2+1})$ and $P\in
X\backslash\\{P_{\infty\pm}\\}$. Then the Baker-Akhiezer function derived in
(4.12), (4.13) satisfy

$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{1}(P^{*},x,y,t_{p})=\frac{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})},$
(4.21)
$\displaystyle\psi_{2}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})=\frac{\mathscr{H}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})},$
(4.22)
$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})=\frac{\mathscr{G}_{2n+2}(\sqrt{\eta},x,y,t_{p})+\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})},$
(4.23)
$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})+\psi_{1}(P^{*},x,y,t_{p})\psi_{2}(P,x,y,t_{p})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{2\mathscr{G}_{2n+2}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})},$
(4.24)
$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})-\psi_{1}(P^{*},x,y,t_{p})\psi_{2}(P,x,y,t_{p})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{2\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}.$
(4.25)

###### Proof.

It can be easily seen that (4.21) and (4.22) hold by (4.12) and (4.13),
respectively. Then from (4.21), (4.22) and the fact that

$\left(\frac{1}{2}(\psi_{1}(P)\psi_{2}(P^{*})+\psi_{1}(P^{*})\psi_{2}(P)),\psi_{1}(P)\psi_{1}(P^{*}),\psi_{2}(P)\psi_{2}(P^{*})\right)$

and

$\left(\frac{\mathscr{G}_{2n+2}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})},\frac{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})},\frac{\mathscr{H}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}\right)$

are both solutions of linear system (2.112)-(2.120) satisfying the same
initial condition, one gets (4.24). Using (4.21), (4.22) and (4.24), we have

$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})-\psi_{1}(P^{*},x,y,t_{p})\psi_{2}(P,x,y,t_{p})$
$\displaystyle=$
$\displaystyle\sqrt{(\psi_{1}(P)\psi_{2}(P^{*})+\psi_{1}(P^{*})\psi_{2}(P))^{2}-4\psi_{1}(P)\psi_{1}(P^{*})\psi_{2}(P)\psi_{2}(P^{*})}$
$\displaystyle=$
$\displaystyle\frac{2\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}.$
(4.26)

Finally, (4.23) is the direct result of (4.24), (4.25). ∎

Next we shall study the analytic property and asymptotic behavior of
$\psi_{1}(P)\psi_{2}(P^{*})$, and $\psi_{j}(P),j=1,2.$

######  Lemma 4.5.

The function $\psi_{1}(P)\psi_{2}(P^{*})$ is a meromorphic function on $X$
with divisor

$(\psi_{1}(P)\psi_{2}(P^{*}))=\mathcal{D}_{\underline{\hat{\nu}}^{*}(x,y,t_{p})\underline{\hat{\mu}}(x,y,t_{p})P_{\infty+}}-\mathcal{D}_{\underline{\hat{\mu}}^{*}(x_{0},y_{0},t_{0})\underline{\hat{\mu}}(x_{0},y_{0},t_{0})P_{\infty-}},$
(4.27)

where we abbreviate
$\underline{\hat{\mu}}=(\hat{\mu}_{1},\ldots,\hat{\mu}_{n}),\underline{\hat{\nu}}=(\hat{\nu}_{1},\ldots,\hat{\nu}_{n}).$
Moreover, using the local coordinate $\zeta=z^{-1}$ near $P_{\infty\pm},$ we
have

$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})\overset{\zeta\rightarrow
0}{=}\begin{cases}\frac{2}{q(x_{0},y_{0},t_{p,0})}\zeta^{-1}+O(1),&\textrm{as}~{}~{}P\rightarrow
P_{\infty+},\cr-\frac{q(x,y,t_{p})r(x,y,t_{p})}{2q(x_{0},y_{0},t_{p,0})}\zeta+O(\zeta^{2}),&\textrm{as}~{}~{}P\rightarrow
P_{\infty-}.\cr\end{cases}$ (4.28)

###### Proof.

Noticing (3.2), (3.4), (4.23) and

$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})=$
$\displaystyle~{}\frac{\mathscr{G}_{2n+2}(\sqrt{\eta},x,y,t_{p})+\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}$
$\displaystyle=$
$\displaystyle~{}\frac{\mathscr{G}_{2n+2}(\sqrt{\eta},x,y,t_{p})+\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}}{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}\frac{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}$
$\displaystyle=$
$\displaystyle~{}\frac{\mathscr{H}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{G}_{2n+2}(\sqrt{\eta},x,y,t_{p})-\sqrt{\mathscr{R}_{4n+4}(\sqrt{\eta})}}\frac{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})},$

one easily proves (4.27) and (4.28). ∎

Now we turn to study the analytic structure of $\psi_{j}(P,x,y,t_{p})$ on
$X\backslash\\{P_{\infty\pm}\\}.$

######  Theorem 4.6.

Assume auxiliary spectrum points $\mu_{j}(x,y,t_{p})$, $\nu_{j}(x,y,t_{p}),$
$j=1,\ldots,n,$ are mutually distinct and finite for all
$(x,y,t_{p})\in\Omega,$ where $\Omega\in\mathbb{R}^{3}$ is an open interval.
Moreover, let $P\in X\backslash\\{P_{\infty\pm}\\}.$ Then

  * I.

$\psi_{1}(P,x,y,t_{p})$ and $\psi_{2}(P,x,y,t_{p})$ are meromorphic on $P\in
X\backslash\\{P_{\infty\pm}\\}.$ Their divisor of poles coincides with
$\mathcal{D}_{\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})}.$

  * II.

The divisor of zeros for $\psi_{1}(P,x,y,t_{p})$ and $\psi_{2}(P,x,y,t_{p})$
coincides with $\mathcal{D}_{\underline{\hat{\mu}}(x,y,t_{r+1})}$ and
$\mathcal{D}_{\underline{\hat{\nu}}(x,y,t_{p})},$ respectively.

  * III.

As $P\rightarrow P_{\infty\pm},$ the asymptotic behavior of
$(\psi_{1}(P,x,y,t_{p}),\psi_{2}(P,x,y,t_{p}))^{T}$ is given by the equations,

$\displaystyle\left(\begin{array}[]{c}\psi_{1}(P,x,y,t_{p})\\\
\psi_{2}(P,x,y,t_{p})\\\ \end{array}\right)=$
$\displaystyle~{}\left(\begin{array}[]{c}\frac{q(x,y,t_{p})}{q(x_{0},y_{0},t_{p,0})}+O(\zeta)\\\
-\frac{q(x,y,t_{p})r(x,y,t_{p})}{2q(x_{0},y_{0},t_{p,0})}\zeta+O(\zeta^{2})\\\
\end{array}\right)\times\exp\Big{(}-(x-x_{0})\zeta^{-1}$ (4.33)
$\displaystyle-(y-y_{0})\zeta^{-2}-(t_{p}-t_{p,0})\zeta^{-(p+1)}$
$\displaystyle+\Big{(}-\frac{1}{4}\int_{x_{0}}^{x}q(x^{\prime},y,t_{p})r(x^{\prime},y,t_{p})dx^{\prime}+\frac{1}{2}\int_{y_{0}}^{y}g_{4}(x_{0},y^{\prime},t_{p})dy^{\prime}$
$\displaystyle+\frac{1}{2}\int_{t_{p,0}}^{t_{p}}g_{2p+2}(x_{0},y_{0},t^{\prime})dt^{\prime}\Big{)}\Big{)},~{}~{}\textrm{at}~{}~{}P\rightarrow
P_{\infty+},$ (4.34)

and

$\displaystyle\left(\begin{array}[]{c}\psi_{1}(P,x,y,t_{p})\\\
\psi_{2}(P,x,y,t_{p})\\\ \end{array}\right)=$
$\displaystyle~{}\left(\begin{array}[]{c}1+O(\zeta)\\\
\frac{2}{q(x,y,t_{p})}\zeta^{-1}+O(1)\\\
\end{array}\right)\times\exp\Big{(}(x-x_{0})\zeta^{-1}$ (4.39)
$\displaystyle+(y-y_{0})\zeta^{-2}+(t-t_{p,0})\zeta^{-(p+1)}$
$\displaystyle+\Big{(}\frac{1}{4}\int_{x_{0}}^{x}q(x^{\prime},y,t_{p})r(x^{\prime},y,t_{p})dx^{\prime}-\frac{1}{2}\int_{y_{0}}^{y}g_{4}(x_{0},y^{\prime},t_{p})dy^{\prime}$
$\displaystyle-\frac{1}{2}\int_{t_{p,0}}^{t_{p}}g_{2p+2}(x_{0},y_{0},t^{\prime})dt^{\prime}\Big{)}\Big{)},~{}~{}\textrm{at}~{}~{}P\rightarrow
P_{\infty-}.$ (4.40)

###### Proof.

First we study the function $\psi_{1}$. By (3.10), (3.11), (3.12) it follows

$\displaystyle-2\frac{q(x^{\prime},y,t_{p})\sqrt{\eta}y(P)}{\mathscr{F}_{2n+1}(\sqrt{\eta},x^{\prime},y,t_{p})}$
$\displaystyle\overset{P\rightarrow\hat{\mu}_{j}(x^{\prime},y,t_{p})}{=}\partial_{x^{\prime}}\ln\sqrt{\eta-\mu_{j}(x^{\prime},y,t_{p})},$
(4.41)
$\displaystyle-\frac{2y(P)\widehat{F}_{3}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}{\mathscr{F}_{2n+1}(\eta,x_{0},y^{\prime},t_{p})}$
$\displaystyle\overset{P\rightarrow\hat{\mu}_{j}(x_{0},y^{\prime},t_{p})}{=}\partial_{y^{\prime}}\ln\sqrt{\eta-\mu_{j}(x_{0},y^{\prime},t_{p})},$
(4.42)
$\displaystyle-\frac{2y(P)\widehat{F}_{2p+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}$
$\displaystyle\overset{P\rightarrow\hat{\mu}_{j}(x_{0},y_{0},t^{\prime})}{=}~{}~{}\partial_{t^{\prime}}\ln\sqrt{\eta-\mu_{j}(x_{0},y_{0},t^{\prime})},$
(4.43)

and hence one obtains

$\displaystyle\exp\Big{(}-2\int_{x_{0}}^{x}\frac{q(x^{\prime},y,t_{p})\sqrt{\eta}y(P)}{\mathscr{F}_{2n+1}(\sqrt{\eta},x^{\prime},y,t_{p})}dx^{\prime}-2\int_{y_{0}}^{y}\frac{y(P)\widehat{F}_{3}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}dy^{\prime}$
$\displaystyle-2\int_{t_{p,0}}^{t_{p}}\frac{y(P)\widehat{F}_{2p+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}dt^{\prime}\Big{)}$
$\displaystyle=$
$\displaystyle\exp\Big{(}\int_{x_{0}}^{x}\partial_{x^{\prime}}\ln\sqrt{\eta-\mu_{j}(x^{\prime},y,t_{p})}dx^{\prime}+\int_{y_{0}}^{y}\partial_{y^{\prime}}\ln\sqrt{\eta-\mu_{j}(x_{0},y^{\prime},t_{p})}dy^{\prime}$
$\displaystyle+\int_{t_{p,0}}^{t_{p}}\partial_{t^{\prime}}\ln\sqrt{\eta-\mu_{j}(x_{0},y_{0},t^{\prime})}dt^{\prime}+O(1)\Big{)}$
$\displaystyle=$
$\displaystyle\begin{cases}\sqrt{\eta-\mu_{j}(x_{0},y_{0},t_{p,0})}^{-1},&P\rightarrow\hat{\mu}_{j}(x_{0},y_{0},t_{p,0}),\cr\sqrt{\eta-\mu_{j}(x,y,t_{p})},&P\rightarrow\hat{\mu}_{j}(x,y,t_{p})\neq\hat{\mu}_{j}(x_{0},y_{0},t_{p,0}),\cr
O(1),&P\rightarrow\hat{\mu}_{j}(x,y,t_{p})=\hat{\mu}_{j}(x_{0},y_{0},t_{p,0}),\cr
O(1),&P\rightarrow\textrm{other
points}\neq\hat{\mu}_{j}(x,y,t_{p}),~{}\hat{\mu}_{j}(x_{0},y_{0},t_{p,0}).\end{cases}$
(4.44)

Then taking into account (4.12), one proves I. and II. for $\psi_{1}$. Next we
study the asymptotic behaviour of $\psi_{1}$ near $P_{\infty\pm}.$ Again using
(4.12) and local coordinate $\eta={z}^{-1}$ near $P_{\infty\pm}$, one infers

$\displaystyle\psi_{1}(P,x,y,t_{p})=$
$\displaystyle~{}\sqrt{\frac{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t_{p,0})}}\exp\Big{(}-2\int_{x_{0}}^{x}\frac{q(x^{\prime},y,t_{p})y(P)}{\mathscr{F}_{2n+1}(\sqrt{\eta},x^{\prime},y,t_{p})}dx^{\prime}$
$\displaystyle-2\int_{y_{0}}^{y}\frac{y(P)\widehat{F}_{3}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}dy^{\prime}-2\int_{t_{p,0}}^{t_{p}}\frac{y(P)\widehat{F}_{2p+1}(z,x_{0},y_{0},t^{\prime})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}dt^{\prime}\Big{)}$
$\displaystyle=$
$\displaystyle\exp\Big{(}\int_{x_{0}}^{x}\left(-2\frac{q(x^{\prime},y,t_{p})y(P)}{\mathscr{F}_{2n+1}(\sqrt{\eta},x^{\prime},y,t_{p})}+\frac{\mathscr{F}_{2n+1,x^{\prime}}(\sqrt{\eta},x^{\prime},y,t_{p})}{2\mathscr{F}_{2n+1}(\sqrt{\eta},x^{\prime},y,t_{p})}\right)dx^{\prime}$
$\displaystyle+\int_{y_{0}}^{y}\left(-2\frac{y(P)\widehat{F}_{3}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}+\frac{\mathscr{F}_{2n+1,y^{\prime}}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}{2\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y^{\prime},t_{p})}\right)dy^{\prime}$
$\displaystyle+\int_{t_{p,0}}^{t_{p}}\left(-2\frac{y(P)\widehat{F}_{2p+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}{\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}+\frac{\mathscr{F}_{2n+1,t^{\prime}}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}{2\mathscr{F}_{2n+1}(\sqrt{\eta},x_{0},y_{0},t^{\prime})}\right)dt^{\prime}\Big{)}$
$\displaystyle=$
$\displaystyle\begin{cases}\exp\Big{(}\int_{x_{0}}^{x}\left(\frac{-2q(x^{\prime},y,t_{p})}{\sum_{j=0}^{\infty}\hat{f}_{2j+1}(x^{\prime},y,t_{p})\zeta^{j+1}}+\frac{q_{x^{\prime}}(x^{\prime},y,t_{p})}{2q(x^{\prime},y,t_{p})}\right)dx^{\prime}\\\
~{}~{}~{}~{}+\int_{y_{0}}^{y}\left(\frac{-2\widehat{F}_{3}(x_{0},y^{\prime},t_{p})}{\sum_{j=0}^{\infty}\hat{f}_{2j+1}(x_{0},y^{\prime},t_{p})\zeta^{j+1}}+\frac{q_{y^{\prime}}(x_{0},y^{\prime},t_{p})}{2q(x_{0},y^{\prime},t_{p})}\right)dy^{\prime}\\\
~{}~{}~{}~{}+\int_{t_{p,0}}^{t_{p}}\left(\frac{-2\widehat{F}_{2p+1}(x_{0},y_{0},t^{\prime})}{\sum_{j=0}^{\infty}\hat{f}_{2j+1}(x_{0},y_{0},t^{\prime})\zeta^{j+1}}+\frac{q_{t^{\prime}}(x_{0},y_{0},t^{\prime})}{2q(x_{0},y_{0},t^{\prime})}\right)dt^{\prime}\Big{)},&\textrm{as}~{}~{}P\rightarrow
P_{\infty-},\cr\exp\Big{(}\int_{x_{0}}^{x}\left(\frac{2q(x^{\prime},y,t_{p})}{\sum_{j=0}^{\infty}\hat{f}_{2j+1}(x^{\prime},y,t_{p})\zeta^{j+1}}+\frac{q_{x^{\prime}}(x^{\prime},y,t_{p})}{2q(x^{\prime},y,t_{p})}\right)dx^{\prime}\\\
~{}~{}~{}~{}+\int_{y_{0}}^{y}\left(\frac{2\widehat{F}_{3}(x_{0},y^{\prime},t_{p})}{\sum_{j=0}^{\infty}\hat{f}_{2j+1}(x_{0},y^{\prime},t_{p})\zeta^{j+1}}+\frac{q_{y^{\prime}}(x_{0},y^{\prime},t_{p})}{2q(x_{0},y^{\prime},t_{p})}\right)dy^{\prime}\\\
~{}~{}~{}~{}+\int_{t_{p,0}}^{t_{p}}\left(\frac{2\widehat{F}_{2p+1}(x_{0},y_{0},t^{\prime})}{\sum_{j=0}^{\infty}\hat{f}_{2j+1}(x_{0},y_{0},t^{\prime})\zeta^{j+1}}+\frac{q_{t^{\prime}}(x_{0},y_{0},t^{\prime})}{2q(x_{0},y_{0},t^{\prime})}\right)dt^{\prime}\Big{)},&\textrm{as}~{}~{}P\rightarrow
P_{\infty+},\end{cases}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{q(x,y,t_{p})}{q(x_{0},y_{0},t_{p,0})}\exp\Big{(}-(x-x_{0})\zeta^{-1}-(y-y_{0})\zeta^{-2}-(t_{p}-t_{p,0})\zeta^{-(p+1)}\\\
+(-\frac{1}{4}\int_{x_{0}}q(x^{\prime},y,t_{p})r(x^{\prime},y,t_{p})dx^{\prime}+\frac{1}{2}\int_{y_{0}}^{y}g_{4}(x_{0},y^{\prime},t_{p})dy^{\prime}\\\
+\frac{1}{2}\int_{t_{p,0}}^{t_{p}}g_{2p+2}(x_{0},y_{0},t^{\prime})dt^{\prime})+O(\zeta)\Big{)},~{}~{}~{}~{}\textrm{as}~{}~{}P\rightarrow
P_{\infty-},\cr\exp\Big{(}(x-x_{0})\zeta^{-1}+(y-y_{0})\zeta^{-2}+(t_{p}-t_{p,0})\zeta^{-(p+1)}\\\
+(\frac{1}{4}\int_{x_{0}}q(x^{\prime},y,t_{p})r(x^{\prime},y,t_{p})dx^{\prime}-\frac{1}{2}\int_{y_{0}}^{y}g_{4}(x_{0},y^{\prime},t_{p})dy^{\prime}\\\
-\frac{1}{2}\int_{t_{p,0}}^{t_{p}}g_{2p+2}(x_{0},y_{0},t^{\prime})dt^{\prime})+O(\zeta)\Big{)},~{}~{}~{}~{}\textrm{as}~{}~{}P\rightarrow
P_{\infty+}.\end{cases}$

Here we have used asymptotic spectral expansion

$\frac{\mathscr{F}_{2n+1}(\sqrt{\eta},x,y,t_{p})}{y(P)}=\pm\sum_{j=0}^{\infty}\hat{f}_{2j+1}(x,y,t_{p})\zeta^{j+\frac{3}{2}},~{}~{}\textrm{as}~{}~{}P\rightarrow
P_{\infty\pm},$ (4.46)

in the last equality of (LABEL:4.33dp), and the expression (4.46) can be
obtained by induction. Finally, the statements for $\psi_{2}$ follows by Lemma
4.5 and (LABEL:4.33dp). ∎

##  5 Algebro-Geometric Solutions

In this section, we shall study the theta function representation of Baker-
Akhiezer functions $\psi_{1}(P),\psi_{2}(P)$ and algebro-geometric solutions
of the whole mKP hierarachy. For more details about Riemann surfaces and theta
functions we recommend the reference [13].

First, choosing a convenient base point $Q_{0}\in
X\setminus\\{P_{\infty\pm}\\}$, the Abel maps $\underline{A}_{Q_{0}}(\cdot)$
and $\underline{\alpha}_{Q_{0}}(\cdot)$ are defined by

$\displaystyle\underline{A}_{Q_{0}}:X\rightarrow J(X)$ $\displaystyle=$
$\displaystyle\mathbb{C}^{n}/L_{n},$ $\displaystyle
P\mapsto\underline{A}_{Q_{0}}(P)$ $\displaystyle=$
$\displaystyle(A_{Q_{0},1}(P),\ldots,A_{Q_{0},n}(P))$ $\displaystyle=$
$\displaystyle\left(\int_{Q_{0}}^{P}\omega_{1},\ldots,\int_{Q_{0}}^{P}\omega_{n}\right)(\mathrm{mod}~{}L_{n}),$

and

$\displaystyle\underline{\alpha}_{Q_{0}}:\mathrm{Div}(X)\rightarrow J(X),$
$\displaystyle~{}~{}~{}~{}~{}\qquad\mathcal{D}\mapsto\underline{\alpha}_{Q_{0}}(\mathcal{D})=\sum_{P\in\mathcal{K}_{n}}\mathcal{D}(P)\underline{A}_{Q_{0}}(P),$

where
$L_{n}=\\{\underline{z}\in\mathbb{C}^{n}|~{}\underline{z}=\underline{N}+\Gamma\underline{M},~{}\underline{N},~{}\underline{M}\in\mathbb{Z}^{n}\\}$,
and $\Gamma$, $\underline{\Xi}_{Q_{0}}$ are the Riemann matrix and the vector
of Riemann constants, respectively. Moreover, we choose a homology basis
$\\{a_{j},b_{j}\\}_{j=1}^{n}$ on $X$ in such a way that the intersection
matrix of the cycles satisfies

$a_{j}\circ b_{k}=\delta_{j,k},\quad a_{j}\circ a_{k}=0,\quad b_{j}\circ
b_{k}=0,\quad j,~{}k=1,\ldots,n.$ (5.1)

For brevity, define the function
$\underline{z}:X\times\sigma^{n}X\rightarrow\mathbb{C}^{n}$
by555$\sigma^{n}X$= $\underbrace{X\times\ldots\times X}_{n}.$

$\displaystyle\underline{z}(P,\underline{Q})$
$\displaystyle=~{}\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{Q}}),$
$\displaystyle P\in\mathcal{K}_{n},\,~{}\underline{Q}$
$\displaystyle=~{}(Q_{1},\ldots,Q_{n})\in\sigma^{n}\mathcal{K}_{n},$ (5.2)

here $\underline{z}(\cdot,\underline{Q})$ is independent of the choice of base
point $Q_{0}$. The Riemann theta function $\theta(\underline{z})$ associated
with $X$ and the homology is defined by

$\theta(\underline{z})=\sum_{\underline{n}\in\mathbb{Z}}\exp\left(2\pi
i<\underline{n},\underline{z}>+\pi
i<\underline{n},\underline{n}\Gamma>\right),\quad\underline{z}\in\mathbb{C}^{n},$

where
$<\underline{B},\underline{C}>=\overline{\underline{B}}\cdot\underline{C}^{t}=\sum_{j=1}^{N}\overline{B}_{j}C_{j}$
denotes the scalar product in $\mathbb{C}^{n-1}$.

Let $\omega_{P_{\infty\pm},q}^{(2)}$ be the normalized differentials of the
second kind with a unique pole at $P_{\infty\pm}$, respectively, and principal
parts

$\omega_{P_{\infty\pm},q}^{(2)}\underset{\zeta\rightarrow
0}{=}\left(\zeta^{-2-q}+O(1)\right)d\zeta,~{}~{}P\rightarrow
P_{\infty\pm},~{}~{}\zeta=z^{-1},~{}~{}q\in\mathbb{N}_{0}$ (5.3)

with vanishing $a$-periods,

$\int_{a_{j}}\omega_{P_{\infty\pm},q}^{(2)}=0,~{}~{}j=1,\ldots,n,$

and $\omega_{P_{\infty+}P_{\infty-}}$ be normalized differential of third kind
satisfying

$\begin{split}\int_{Q_{0}}^{P}\omega_{P_{\infty-}P_{\infty+}}=\zeta^{-1}+c_{\infty+}+O(\zeta),~{}~{}P\rightarrow
P_{\infty+},\zeta=z^{-1},c_{\infty+}\in\mathbb{C},\end{split}$

and

$\begin{split}\int_{a_{j}}\omega_{P_{\infty+}P_{\infty-}}=0~{}~{}j=1,\ldots,n.\end{split}$

Moreover, we introduce the notation

$\Omega_{r}^{(2)}=\frac{1}{2}(r+1)(\omega_{P_{\infty+},r}^{(2)}-\omega_{P_{\infty-},r}^{(2)}),~{}~{}r\in\mathbb{N}_{0}.$
(5.4)

and

$\omega_{r}^{(2)}=\lim_{P\rightarrow
P_{\infty+}}\int_{Q_{0}}^{P}\Omega_{r}^{(2)}.$ (5.5)

Next we study the theta function representation for
$\psi_{1}(P)\psi_{2}(P^{*})$.

######  Lemma 5.1.

Assume spectral curve defined in (3.4) is nonsingular and $q,r$ satisfy
(2.122). Moreover, let $P\in X\backslash\\{P_{\infty\pm}\\}$ and suppose the
divisors $\mathcal{D}_{\underline{\hat{\nu}}(x,y,t_{p})}$,
$\mathcal{D}_{\underline{\hat{\mu}}(x,y,t_{p})}$ are nonspecial. Then

$\displaystyle\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})$ $\displaystyle=$
$\displaystyle\frac{2}{q(x_{0},y_{0},t_{p,0})}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x,y,t_{p})))\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}$
$\displaystyle\times\frac{\theta(\underline{z}(P^{*},\underline{\hat{\nu}}(x,y,t_{p})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P^{*},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle\times\exp\Big{(}\int_{Q_{0}}^{P}\omega_{P_{\infty-}P_{\infty+}}-c_{\infty+}\Big{)}.$
(5.6)

###### Proof.

Let

$\displaystyle\circledR(P,x,t_{p})=$
$\displaystyle~{}\frac{2}{q(x_{0},y_{0},t_{r+1,0})}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x,y,t_{p})))\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}$
$\displaystyle\times\frac{\theta(\underline{z}(P^{*},\underline{\hat{\nu}}(x,y,t_{p})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P^{*},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle\times\exp\Big{(}\int_{Q_{0}}^{P}\omega_{P_{\infty-}P_{\infty+}}-c_{\infty+}\Big{)}.$
(5.7)

Then it is not difficult to know divisors of
$\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})$ coincides with that of
$\circledR(P,x,t_{p})$ on $X$. Therefore, the function
$\frac{\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})}{\circledR(P,x,y,t_{p})}$
is a constant independent of $P$ and we denote it by $C(x,y,t_{p})$. Taking
$P\rightarrow P_{\infty+}$ and using Lemma 4.5, one finally obtains

$C(x,y,t_{p})=\lim_{P\rightarrow
P_{\infty+}}\frac{\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})}{\circledR(P,x,y,t_{p})}=1,~{}~{}$

and the relation

$\psi_{1}(P,x,y,t_{p})\psi_{2}(P^{*},x,y,t_{p})=\circledR(P,x,y,t_{p})$

for any $P\in X.$ ∎

Given these preparations, one finally derives the following theta function
representation for Baker-Akhiezer functions $\psi_{j}(P,x,y,t_{p}),j=1,2,$ and
algebro-geometric solutions $q(x,y,t_{p}),r(x,y,t_{p}).$

######  Theorem 5.2.

Assume spectral curve defined in (3.4) is nonsingular and $q,p$ satisfy
(2.122). Moreover, let $P\in X\backslash\\{P_{\infty\pm}\\}$ and suppose the
divisors $\mathcal{D}_{\underline{\hat{\nu}}(x,y,t_{p})}$,
$\mathcal{D}_{\underline{\hat{\mu}}(x,y,t_{p})}$ are nonspecial. Then
functions $\psi_{1},\psi_{2},q,r$ have the following theta function
representations

$\displaystyle\psi_{1}(P,x,y,t_{p})=$
$\displaystyle~{}C_{1}(x,y,t_{p})\frac{\theta(\underline{z}(P,\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}\exp\Big{(}-(x-x_{0})\int_{Q_{0}}^{P}\Omega^{(2)}_{0}$
$\displaystyle-(y-y_{0})\int_{Q_{0}}^{P}\Omega^{(2)}_{1}-(t_{p}-t_{p,0})\int_{Q_{0}}^{P}\Omega^{(2)}_{p}\Big{)},$
(5.8) $\displaystyle\psi_{2}(P,x,y,t_{p})=$
$\displaystyle~{}C_{2}(x,y,t_{p})\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x,y,t_{p})))}{\theta(\underline{z}(P,\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}\exp\Big{(}-(x-x_{0})\int_{Q_{0}}^{P}\Omega^{(2)}_{0}$
$\displaystyle-(y-y_{0})\int_{Q_{0}}^{P}\Omega^{(2)}_{1}-(t_{p}-t_{p,0})\int_{Q_{0}}^{P}\Omega^{(2)}_{p}+\sqrt{\int_{Q_{0}}^{P}\omega_{P_{\infty+}P_{\infty-}}}\Big{)},$
(5.9)

and

$\displaystyle q(x,y,t_{p})=$
$\displaystyle~{}q(x_{0},y_{0},t_{p,0})\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle\times\exp\Big{(}c_{0}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)},$
(5.10) $\displaystyle r(x,y,t_{p})=$
$\displaystyle~{}-q(x_{0},y_{0},t_{p,0})\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x,y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle~{}\times\exp\Big{(}-c_{0}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)}.$
(5.11) $\displaystyle q(x,y,t_{p})$ $\displaystyle
r(x,y,t_{p})=-2c_{0}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x,y,t_{p})))}.$
(5.12)

where $c_{0}=2e^{c_{\infty-}-c_{\infty+}}$ and

$\displaystyle C_{1}(x,y,t_{p})=$
$\displaystyle~{}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}\times\exp\Big{(}\frac{c_{0}}{2}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}$
$\displaystyle~{}\times\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)},$
(5.13) $\displaystyle C_{2}(x,y,t_{p})=$
$\displaystyle~{}~{}~{}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}\times\exp\Big{(}-\frac{c_{0}}{2}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}$
$\displaystyle~{}\times\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)}.$
(5.14)

Here we emphasize that the initial values $q(x_{0},y_{0},t_{p,0})$ and
$r(x_{0},y_{0},t_{p,0})$ should satisfy

$\displaystyle r(x_{0},y_{0},t_{p,0})=$
$\displaystyle~{}-q(x_{0},y_{0},t_{p,0})\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle~{}\times\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}.$
(5.15)

Moreover, the Abel map linearizes the auxiliary divisors
$\mathcal{D}_{\hat{\underline{\mu}}(x,y,t_{p})},\mathcal{D}_{\hat{\underline{\nu}}(x,y,t_{p})}$
in the sense that

$\displaystyle\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x,y,t_{p})})=$
$\displaystyle~{}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})})+\underline{U}_{0}^{(2)}(x-x_{0})+\underline{U}_{1}^{(2)}(y-y_{0})$
$\displaystyle+\underline{U}_{r}^{(2)}(t_{p}-t_{p,0}),$ (5.16)
$\displaystyle\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x,y,t_{p})})=$
$\displaystyle~{}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})})+\underline{U}_{0}^{(2)}(x-x_{0})+\underline{U}_{1}^{(2)}(y-y_{0})$
$\displaystyle+\underline{U}_{r}^{(2)}(t_{p}-t_{p,0})+\frac{1}{2}\Big{(}A_{Q_{0}}(P_{\infty-})-A_{Q_{0}}(P_{\infty+})\Big{)}.$
(5.17)

###### Proof.

First, we prove expression (5.8). Denote the right hand of (5.8) by
$\tilde{\psi}_{1}$, and then one finds $\tilde{\psi}_{1}$ is meromorphic on
$X\backslash\\{P_{\infty\pm}\\}$ with simple zeros at
$\hat{\mu}_{j}(x,y,t_{p}),j=1,\ldots,n,$ and simple poles at
$\hat{\mu}_{j}(x_{0},y_{0},t_{p,0}),j=1,\ldots,n,$ by Riemann vanishing
theorem. Comparing (4.34), (4.39), (5.8) for $\tilde{\psi}_{1}$, and taking
into account (5.3), (5.4), one infers $\psi_{1}$ and $\tilde{\psi}_{1}$ have
identical exponential behavior up to order $O(1)$ near $P_{\infty\pm}.$ Thus,
$\psi_{1}$ and $\tilde{\psi}_{1}$ share the same singularities and zeros and
the Riemann-Roch-type uniqueness result then proves that $\psi_{1}$ and
$\tilde{\psi}_{1}$ coincide up to normalization and we denote this
normalization constant by $C_{1}(x,y,t_{p}).$ Hence $\psi_{1}(P,x,y,t_{p})$
has the form of (5.8). Using (4.1), (4.39), (5.8), one obtains

$\displaystyle(\ln\psi_{1}(P,x,y,t_{p}))_{x}=$
$\displaystyle~{}-\eta+\frac{1}{4}q(x,y,t_{p})r(x,y,t_{p})$
$\displaystyle~{}+q(x,y,t_{p})\sqrt{\eta}\frac{\psi_{2}(P,x,y,t_{p})}{\psi_{1}(P,x,y,t_{p})}$
(5.18)

and

$\displaystyle-\frac{q(x,y,t_{p})r(x,y,t_{p})}{4}=$
$\displaystyle~{}\Big{(}\ln
C_{1}(x,y,t_{p})+\ln\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}\Big{)}_{x}$
as $P\rightarrow P_{\infty+}$, (5.19)
$\displaystyle\frac{q(x,y,t_{p})r(x,y,t_{p})}{4}=$ $\displaystyle~{}\Big{(}\ln
C_{2}(x,y,t_{p})+\ln\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}\Big{)}_{x}$
as $P\rightarrow P_{\infty-}$. (5.20)

Then by Theorem 4.6 and Lemma 5.1, one infers

$\displaystyle q(x,y,t_{p})r(x,y,t_{p})=$
$\displaystyle~{}-4e^{c_{\infty-}-c_{\infty+}}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x,y,t_{p})))}.$
(5.21)

Thus, from (5.19), (5.20) and (5.9), one gets

$\displaystyle C_{1}(x,y,t_{p})=$
$\displaystyle~{}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}\times\exp\Big{(}-\frac{1}{4}\int_{x_{0}}^{x}q(x^{\prime},y,t_{p})r(x^{\prime},y,t_{p})dx^{\prime}\Big{)}$
$\displaystyle=$
$\displaystyle~{}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}\times\exp\Big{(}e^{c_{\infty-}-c_{\infty+}}$
$\displaystyle~{}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)},$
(5.22) $\displaystyle C_{2}(x,y,t_{p})=$
$\displaystyle~{}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}\times\exp\Big{(}\frac{1}{4}\int_{x_{0}}^{x}q(x^{\prime},y,t_{p})r(x^{\prime},y,t_{p})dx^{\prime}\Big{)}$
$\displaystyle=$
$\displaystyle~{}~{}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}\times\exp\Big{(}-e^{c_{\infty-}-c_{\infty+}}$
$\displaystyle~{}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)}.$
(5.23)

On the other hand, taking into account the asymptotic behavior of both sides
in (4.21) and (4.22) for $P\rightarrow P_{\infty+}$, one infers

$\displaystyle q(x,y,t_{p})=$
$\displaystyle~{}q(x_{0},y_{0},t_{p,0})C_{1}(x,y,t_{p})^{2}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle\times\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle=$
$\displaystyle~{}q(x_{0},y_{0},t_{p,0})\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x,y,t_{p})))}$
$\displaystyle\times\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}\times\exp\Big{(}2e^{c_{\infty-}-c_{\infty+}}$
$\displaystyle~{}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)},$
(5.24)

and

$\displaystyle r(x,y,t_{p})=$
$\displaystyle~{}-q(x_{0},y_{0},t_{p,0})C_{2}(x,y,t_{p})^{2}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle\times\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle=$
$\displaystyle~{}-q(x_{0},y_{0},t_{p,0})\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x,y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x,y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x_{0},y_{0},t_{p,0})))}$
$\displaystyle~{}\times\exp\Big{(}-2e^{c_{\infty-}-c_{\infty+}}\int_{x_{0}}^{x}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(x^{\prime},y,t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(x^{\prime},y,t_{p})))}dx^{\prime}\Big{)}.$
(5.25)

Moreover, (5.2) follows from (5.24) and (5.25) by taking
$(x,y,t_{p})=(x_{0},y_{0},t_{p,0}).$ Finally, the linearization property of
the Abel map in (5.21) and (5.17) is a standard investigation of the
differentials

$\Omega_{i}(x,y,t_{p})=d\ln(\psi_{i}(\cdot,x,y,t_{p})),~{}i=1,2,$

or standard Langrange interpolation procedure (see [18], [42]). ∎

Thus, the results of theorem 2.1 and theorem 5.2 enable us to construct the
algebro-geometric solutions of the whole mKP hierarchy.

######  Theorem 5.3.

Assume spectral curve defined in (3.4) is nonsingular and $q,r$ satisfy
(2.122). Moreover, let $P\in X\backslash\\{P_{\infty\pm}\\}$ and the divisors
$\mathcal{D}_{\underline{\hat{\nu}}(x,y,t_{p})}$,
$\mathcal{D}_{\underline{\hat{\mu}}(x,y,t_{p})}$ are nonspecial. The algbro-
geometric solutions of $p$th mKP equation ($p\geq 2$) possess the following
theta function representation

$u(x,y,t_{p})=2^{p}c_{0}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(2x,-y,-(-2)^{p-1}t_{p})))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(2x,-y,-(-2)^{p}t_{p})))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(2x,-y,(-2)^{p-1}t_{p})))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(2x,-y,-(-2)^{p}t_{p})))}.$
(5.26)

where the constant $c_{0}\in\mathbb{C}$ is the same with that in theorem 5.2.

######  Remark 5.4.

Here we consider the special case $p=2.$
(i) In the case $p=2,$ the expression (5.26) yields algebro-geometric
solutions of the mKP equation and real algebro-geometric solutions of the mKP
equation can easily obtained from (5.10) and (5.11) by studying reduction
condition $q=\pm\bar{r}$.
(ii) The spectral curve of the mKP equation (1.1) is hyperelliptic which
compactified by two different points $P_{\infty\pm}$ at infinity. Similarly we
can discuss algebro-geometric solution of the mKP hierarchy on some specific
trigonal curves following this way. Moreover, if the solution (5.26) is
independent of $y$, then we can derive theta function representations for
algebro-geometric solutions of modified KdV equation.

## Acknowledgements

The work described in this paper was supported by grants from the National
Science Foundation of China (Project No. 10971031), and the Shanghai Shuguang
Tracking Project (Project No. 08GG01).

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