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

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.

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).

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).

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.

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.

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 $(p,q)$.

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).