On the Lattice Potential KP Equation
aa r X i v : . [ n li n . S I] J u l On the Lattice Potential KP Equation
Cewen Cao , Xiaoxue Xu , Da-jun Zhang ∗ School of Mathematics and Statistics, Zhengzhou University, Zhengzhou Henan 450001, P.R. China Department of Mathematics, Shanghai University, Shanghai 200444, P.R. ChinaE-mail: [email protected], [email protected], djzhang@staff.shu.edu.cn
July 10, 2020
Abstract
The paper presents an approach to derive finite genus solutions to the lattice potentialKadomtsev-Petviashvili (lpKP) equation introduced by F.W. Nijhoff, et al. This equationis rederived from compatible conditions of three replicas of the discrete ZS-AKNS spectralproblem, which is a Darboux transformation of the continuous ZS-AKNS spectral problem.With the help of these links and by means of the so called nonlinearization technique andLiouville platform, finite genus solutions of the lpKP equation are derived. Semi-discretepotential KP equations with one and two discrete arguments, respectively, are also discussed.
Keywords: lattice potential Kadomtsev-Petviashvili equation, finite genus solutions, non-linearization, ZS-AKNS spectral problem, Liouville platform
PACS:
Discrete integrable systems and the problem of integrable discretization of given soliton equationshave attracted more and more attention in recent years [15, 16, 25]. The main purpose of thispaper is to investigate the lattice potential Kadomtsev-Petviashvili (lpKP) equationΞ lpKP( β ,β ,β ) ≡ ( β − ˜ W )( β − β + ˆ˜ W − ¯˜ W ) + ( β − ¯ W )( β − β + ˜¯ W − ˆ¯ W )+ ( β − ˆ W )( β − β + ¯ˆ W − ˜ˆ W ) = 0 , (1.1)and present an approach to construct finite genus solutions to 3D integrable lattice equations.This equation is first discovered by Nijhoff, Capel, Wiersma and Quispel by using the B¨acklundtransformation approach, and later derived through an analysis of the Cauchy matrix [16, 24].To build the integrability of the lpKP equation (1.1) and calculate its finite genus solutions,we will introduce Lax triads from the ZS-AKNS spectral problem. Compatibility of these triads,respectively, give rise to the lattice potential KP equations with 3, 2 and 1 discrete arguments, ∗ Corresponding author. E-mail: djzhang@staff.shu.edu.cn
1s (see Section 2 for the derivation)Ξ (0 , ≡
12 [( ˜ W − ¯ W ) ¯˜ W + ( ¯ W − ˆ W ) ˆ¯ W + ( ˆ W − ˜ W ) ˜ˆ W ]+ γ ( ¯˜ W − ˆ˜ W − ¯ W + ˆ W ) + γ ( ˆ¯ W − ˜¯ W − ˆ W + ˜ W )+ γ ( ˜ˆ W − ¯ˆ W − ˜ W + ¯ W ) = 0 , (1.2)Ξ (1 , ≡ ( ˜ W − ¯ W ) x − [ 12 ( ˜ W − ¯ W ) + γ − γ ]( ¯˜ W − ˜ W − ¯ W + W ) = 0 , (1.3)Ξ (2 , ≡ ( ˜ W − W ) y − [( ˜ W + W ) x + 2 γ ( ˜ W − W ) + 12 ( ˜ W − W ) ] x = 0 . (1.4)Note that (1.2) is equivalent to the lpKP equation (1.1) with β k = − γ k , k = 1 , ,
3, asΞ lpKP( − γ , − γ , − γ ) = 2Ξ (0 , . It also turns out that all these equations have the same potentialKP (pKP) equation, Ξ (3 , ≡ W xt −
14 ( W xxx + 3 W x ) x − W yy = 0 , (1.5)as their continuum limits (see Proposition 2.4).The method of finite-gap integration originated in solving the periodic initial problem of theKorteweg-de Vries (KdV) equation (cf. [21] and the references therein). Recently, an approachto deriving finite genus solutions for 2D discrete integrable systems, the lattice potential KdVequation [9] and the lattice nonlinear Schr¨odinger (lNLS) model [10] were developed. In thispaper, just as in the 2D case, explicit analytic solutions of the lattice pKP equations (1.2,1.3,1.4),together with the pKP equation (1.5), will be calculated by means of the finite-dimensionalintegrable flows of continuous and discrete types, i.e. Hamiltonian phase flows and integrablesymplectic maps. These flows are constructed through nonlinearization of the continuous anddiscrete spectral problems (see Section 3,4). It is surprising that they share same Liouvilleintegrals, same Lax matrix L ( λ ; p, q ) and same algebraic curve R . Thus the calculations can bedone on the same Liouville platform. The Abel-Jacobi variable ~φ in the Jacobian variety J ( R )straightens out both the H j - and the S γ k -flow with the velocities ~ Ω j and ~ Ω γ k , respectively. Asa result, we have a clear evolution picture for the lattice pKP equations as well as for the pKPequation, as the following,Ξ (0 , : ~φ ≡ ~φ + m ~ Ω γ + m ~ Ω γ + m ~ Ω γ , (mod T ) , Ξ (1 , : ~φ ≡ ~φ + x~ Ω + m ~ Ω γ + m ~ Ω γ , (mod T ) , Ξ (2 , : ~φ ≡ ~φ + x~ Ω + y~ Ω + m ~ Ω γ , (mod T ) , Ξ (3 , : ~φ ≡ ~φ + x~ Ω + y~ Ω + t~ Ω , (mod T ) , which will provide the basic part of the explicit analytic solutions (see Section 5,6).The paper is organized as follows. Section 2 shows that how the lattice pKP equations (1.2),(1.3) and (1.4) arise from their Lax triads. Continuum limits of these lattice pKP equationsgive rise to the same pKP equation. In Section 3, a finite-dimensional integrable Hamiltoniansystem related to the ZS-AKNS spectral problem is introduced to provide integrals, spectralcurve and Abel-Jacobi variables. In Section 4 we construct an integrable symplectic map S γ in tilde direction, develop an algebro-difference analogue of the Burchnall-Chaundy’s theory on2ommuting differential operators by which we express the potential functions in terms of thetafunction. This allows us to derive finite genus solutions for the lpKP equation in Section 5 andfor other two semi-discrete and one continuous pKP equations in Section 6. Finally, concludingremarks are given in the last Section. In order to find the suitable discrete spectral problems for (1.1), let us first recall the usualcontinuous KP equation, w t = 14 ( w xx + 3 w ) x + 34 ∂ − x w yy . (2.1)It is well-known that the KP equation has a close relation with the ZS-AKNS spectral problem( U ) [8, 19], ∂ x χ = U χ = λ/ uv − λ/ ! χ. (2.2)In fact, there is a hierarchy of isospectral equations ( X k ) related to (2.2), ∂ τ k ( u, v ) = X k , ( k = 2 , , · · · ) , (2.3)in which the first two nonlinear members, ( y = τ , t = τ ), the NLS equation and the modifiedKdV (mKdV) equation, respectively, are ∂ y ( u, v ) = X = ( u xx − u v, − v xx + 2 uv ) , (2.4a) ∂ t ( u, v ) = X = ( u xxx − uvu x , v xxx − uvv x ) . (2.4b)Corresponding to the hierarchy (2.3), there exist a series of linear spectral problems ( U k ), ∂ τ k χ = U k χ, ( k = 1 , , · · · ) , (2.5)where, apart from equation (2.2) | x = τ , we also have (with y = τ , t = τ ) ∂ y χ = U χ = λ / − uv λu + u x λv − v x − λ / uv ! χ, (2.6a) ∂ t χ = U χ = λ / − λuv − u x v + uv x λ u + λu x + u xx − u vλ v − λv x + v xx − uv − λ / λuv + u x v − uv x ! χ. (2.6b)The Lax pair for ( X k ) is composed of ( U ) and ( U k ). It is found that if ( u, v ) is a compatiblesolution of ( X ) and ( X ), then w = − uv solves the KP equation (2.1) [8, 19]. Thus the com-patible conditions of ( U ), ( U ) and ( U ) implies the KP equation. In other words, ( U , U , U )is the Lax triad for the KP equation and hence for the pKP equation via w = W x . The above facts of the KP equation lead us to consider discretization of the ZS-AKNS problem(2.2), by which we hope to find the Lax representation for the lpKP equation (1.1). One3iscretization of (2.2) is known as the Ablowitz-Ladik spectral problem [1, 2], which leads toa spatially discretized NLS equation. In this paper, we employ the following linear problem,( D ( γ ) ), adopted in [10],˜ χ = D ( γ ) χ, D ( γ ) ( λ, a, b ) = λ − γ + ab ab ! , (2.7)which provides a second discretzation for (2.2) [22] but is different from Ablowitz-Ladik’s spectralproblem (cf. [11]). Note that (2.7) is also known as a Darboux transformation of the ZS-AKNSspectral problems (2.5) [4]. Here, for the above notation, let T be shift operator along the m direction, defined for any function f : Z → R as( T f )( m , m , m ) = ˜ f ( m , m , m ) = f ( m + 1 , m , m ) . Similarly, T f = ¯ f , T f = ˆ f are shifts along the m and m direction, respectively.Two basic relations, ( a, b ) = ( u, ˜ v ) , ( u ˜ v ) x = ˜ u ˜ v − uv, (2.8)are derived from the compatibility condition of equations (2.2) and (2.7) (see [10]). The formerbridges their potential functions, while the latter suggests the setting of difference relation˜ W − W = − u ˜ v as W x = w = − uv . These facts lead to the consideration of three replicas ofequation (2.7) with distinct non-zero parameters γ = γ , γ , γ , T χ ≡ ˜ χ = D ( γ ) ( λ, u, ˜ v ) χ, (2.9a) T χ ≡ ¯ χ = D ( γ ) ( λ, u, ¯ v ) χ, (2.9b) T χ ≡ ˆ χ = D ( γ ) ( λ, u, ˆ v ) χ, (2.9c)which are denoted by ( D ( γ k ) ), k = 1 , ,
3, respectively, for short. Besides, auxiliary equationswill be assigned for each special occasion from the following list,˜ W − W = − u ˜ v, (2.10a)¯ W − W = − u ¯ v, (2.10b)ˆ W − W = − u ˆ v, (2.10c) ∂ x W = − uv. (2.10d)At first glance, these equations seem fairly hard to deal with. Here we remark that on theplatform of Liouville integrability, a pair of functions ( u, v ) of discrete arguments m , m , m can be constructed, which are finite genus potential for each of the discrete spectral problems(2.9a,b,c); and W can be solved with the help of the theta function and meromorphic differentialson the associated Riemann surface. This will lead to explicit analytic solutions to the lpKPequation (see Section 5). The approach can also be extended to the semi-discrete and purelycontinuous cases (see Section 6).To derive the discrete pKP equations (1.2), (1.3) and (1.4), we replace ( U j ) in the Lax triad( U , U , U ) successively by ( D ( γ k ) ), and then we come to the following new Lax triads,( D ( γ ) , D ( γ ) , D ( γ ) ) , ( U , D ( γ ) , D ( γ ) ) , ( U , U , D ( γ ) ) . (2.11)With the auxiliary relations (2.9), the compatibility of these triads lead to the discrete pKPequations (1.2), (1.3) and (1.4). We present the procedure of derivation via the following lemmasand propositions. 4 emma 2.1. Let ( u, v ) : Z → R be a pair of functions such that ( i ) equations (2.9a,b) havecompatible solution χ for one value of the spectral parameter λ ; ( ii ) the system (2.10a,b) has asolution W . Then ( u, v ) solve the lNLS equation [10, 18] Ξ (0 , ≡ (˜ u − ¯ u )( u ¯˜ v + 1) + ( γ − γ ) u = 0 , (2.12a)Ξ (0 , ≡ (˜ v − ¯ v )( u ¯˜ v + 1) − ( γ − γ )¯˜ v = 0 , (2.12b) and W, u, v satisfy the relation Y ( γ , γ ) ≡ (cid:16) ˜ v Ξ (0 , ( γ , γ ) + ¯ u Ξ (0 , ( γ , γ ) (cid:17) = h
12 ( ˜ W − ¯ W ) + γ − γ i ( ¯˜ W − ˜ W − ¯ W + W ) + 2(˜ u ˜ v − ¯ u ¯ v ) , (2.13) which is equal to zero due to equations (2.12).Proof. By direct calculations we see that the cross action ( T T − T T ) χ is equal to( ˜ D ( γ ) D ( γ ) − ¯ D ( γ ) D ( γ ) ) χ = Υ Ξ (0 , Ξ (0 , ! (cid:18) χ (1) χ (2) (cid:19) , (2.14)where Υ = λγ − γ [(˜ v − ¯ v )Ξ (0 , − (˜ u − ¯ u )Ξ (0 , ]+ 1 γ − γ [( γ ¯ v − γ ˜ v )Ξ (0 , − ( γ ¯ u − γ ˜ u )Ξ (0 , ] . With (2.9a,b) one can rewrite ( T T − T T ) χ in the form1 γ − γ ( γ − λ ) ¯ χ (2) − ( γ − λ ) ˜ χ (2) [( γ − λ )˜ u − ( γ − λ )¯ u ] χ (1) γ − γ ) χ (1) ! (cid:18) Ξ (0 , Ξ (0 , (cid:19) . (2.15)Usually the coefficient determinant is not zero. Since ( T T − T T ) χ = 0, we have Ξ (0 ,i ) = 0.Further, in light of equations (2.10a,b), the left-hand side of equation (2.12) can be written asΞ (0 , = (˜ u − ¯ u ) + [ 12 ( ˜ W − ¯ W ) + γ − γ ] u, Ξ (0 , = (˜ v − ¯ v ) − [ 12 ( ˜ W − ¯ W ) + γ − γ ]¯˜ v, which imply equation (2.13) by direct calculations. Proposition 2.1.
Let ( u, v ) : Z → R be a pair of functions such that ( i ) equations (2.9a,b,c)have compatible solution χ for one value of λ ; ( ii ) the system (2.10a,b,c) has a solution W .Then W solves equation (1.2), i.e. Ξ (0 , = 0 .Proof. Consider three replicas of (2.13) with parameters ( γ , γ ) , ( γ , γ ) , ( γ , γ ), respectively.Adding them together we haveΞ (0 , = Y ( γ , γ ) + Y ( γ , γ ) + Y ( γ , γ ) , (2.16)where the terms containing u, v are canceled. This yields equation (1.2).5 roposition 2.2. Let ( u, v ) : R × Z → R be a pair of functions such that ( i ) equations (2.9a,b)have compatible solution χ for one value of λ ; ( ii ) the system of equations (2.10a,b) and (2.10d)has a solution W . Then W solves equation (1.3), i.e. Ξ (1 , = 0 .Proof. In light of (2.10d), the last term in (2.13) is equal to − ( ˜ W − ¯ W ) x . Thus the proof iscompleted since Ξ (1 , = − Y ( γ , γ ). Proposition 2.3.
Let ( u, v ) : R × Z → R be a pair of functions such that ( i ) equations (2.2),(2.6a) and (2.9a) have compatible solution χ for one value of λ ; ( ii ) the system of equations(2.10a,d) has a solution W . Then W solves equation (1.4), i.e. Ξ (2 , = 0 .Proof. The compatibility condition ∂ y ∂ x χ = ∂ x ∂ y χ gives rise to the NLS equations (2.4a),rewritten as Ξ (2 , ≡ u y − u xx + 2 u v = 0 , (2.17a)Ξ (2 , ≡ v y + v xx − uv = 0 . (2.17b)In fact, ( ∂ y ∂ x − ∂ x ∂ y ) χ = ( U ,y − U ,x + [ U , U ]) χ = (2 . Ξ (2 . ! (cid:18) χ (1) χ (2) (cid:19) = χ (2) χ (1) ! (cid:18) Ξ (2 , Ξ (2 , (cid:19) . (2.18)Further, the compatibility condition ∂ x T χ = T ∂ x χ yields the semi-discrete NLS equationsΞ (1 , ≡ u x − ˜ u − γ u + u ˜ v = 0 , (2.19a)Ξ (1 , ≡ ˜ v x + v + γ ˜ v − u ˜ v = 0 , (2.19b)since the cross action leads to( ∂ x T − T ∂ x ) χ = ( D ( γ ) x − ˜ U D ( γ ) + D ( γ ) U ) χ = κ Ξ (1 , Ξ (1 , ! (cid:18) χ (1) χ (2) (cid:19) = ˜ χ (2) uχ (1) χ (1) ! (cid:18) Ξ (1 , Ξ (1 , (cid:19) , (2.20)where κ = ( u ˜ v ) x − ˜ u ˜ v + uv = ˜ v Ξ (1 , + u Ξ (1 , , and the relation ˜ χ (2) = ˜ vχ (1) + χ (2) has beenused. By calculation we have˜ v Ξ (2 , + u ˜Ξ (2 , = ( u ˜ v ) y − ( u x ˜ v − u ˜ v x ) x − u ˜ v (˜ u ˜ v − uv ) , (˜ v Ξ (1 , − u Ξ (1 , ) x = ( u x ˜ v − u ˜ v x ) x − (˜ u ˜ v + uv + 2 γ u ˜ v − u ˜ v ) x . Adding them together we arrive atΞ (2 , = − v Ξ (2 , + u ˜Ξ (2 , ) − v Ξ (1 , − u Ξ (1 , ) x , (2.21)where the term ( u x ˜ v − u ˜ v x ) x is canceled and the variable W is introduced by equations (2.10a,d).Thus Ξ (2 , = 0. 6et us back to the equations (1.2), (1.3) and (1.4). We have seen that (1.2) is nothing butthe lpKP equation (1.1). Besides, equations (1.3) and (1.4) have close relations with the (N-2)and (N-3) models that were discovered by Date, Jimbo and Miwa [12], which areΞ N2 ≡ ( ˜ V − ¯ V ) x − ( e ¯˜ V − e ˜ V − e ¯ V + e V ) = 0 , (2.22)Ξ N3 ≡ ∆( V y + 2 h V x − V V x ) − (∆ + 2) V xx = 0 , (2.23)where ∆ f = ˜ f − f for arbitrary function f . In fact, for (1.3), introducing V = ln[( ˜ W − ¯ W ) / γ − γ ] , (2.24)and then using (1.3), one finds V x = ( ˜ W − ¯ W ) x /
2( ˜ W − ¯ W ) / γ − γ = 12 ( ¯˜ W − ˜ W − ¯ W + W ) . It then follows that ( ˜ V − ¯ V ) x is equal to the second part in equation (2.22). Hence Ξ N2 = 0.Thus, for any solution W of the equation (1.3), V defined by (2.24) provides a special solutionfor (2.22). For the equation (1.4), if W is a solution, then V = ( ˜ W − W ) / γ = 1 /h )12 Ξ (2 , = ( V y + 2 h V x − V V x ) − ( V + W ) xx , which implies Ξ N3 = ∆Ξ (2 , / γ k = − /ε k , ε k = c k ε , ( k = 1 , , c , c , c are arbitrary distinct non-zero constants.For any smooth function W ( x, y, t ), define T k W = W ( x + c k ε, y − c k ε / , t + c k ε / , ( k = 1 , , . (2.26)Denote T W = ˜ W , T W = ¯ W , T W = ˆ W for short. By straightforward calculations we havethe following. Proposition 2.4.
Under the Ansatz (2.26), in the neighborhood of ε ∼ , the following Taylorexpansions hold for any smooth function W ( x, y, t ) , Ξ (2 , = Ξ (3 , c ε + O ( ε ) , (2.27a)Ξ (1 , = Ξ (3 , c c ( c − c )3 ε + O ( ε ) , (2.27b)Ξ (0 , = Ξ (3 ,
13 [ c c ( c − c ) + c c ( c − c ) + c c ( c − c )] ε + O ( ε ) . (2.27c)Thus, all the continuum limits of the lattice pKP equations (1.2), (1.3) and (1.4) give rise tothe same pKP equation (1.5). The Ansatz (2.26) is crucial, which is proposed based on comparingthe velocities of the Abel-Jacobi variable ~φ along the discrete S γ k -flow and the continuous H j -flow (see Appendix A). 7 The integrable Hamiltonian system ( H ) In [8, 10] an integrable Hamiltonian system is constructed from the ZS-AKNS spectral problem, ∂ x (cid:18) p j q j (cid:19) = (cid:18) − ∂H /∂q j ∂H /∂p j (cid:19) = α j / − < p, p >< q, q > − α j / ! (cid:18) p j q j (cid:19) , (3.1a) H ( p, q ) = − < Ap, q > + 12 < p, p >< q, q > . (3.1b)where A = diag( α , · · · , α N ), < ξ, η > = P Nj =1 ξ j η j . It can be regarded as N replicas of equation(2.2) with eigenvalues α , · · · , α N , respectively, under the constraint( u, v ) = f U ( p, q ) = ( − < p, p >, < q, q > ) . (3.2)The integrability requires enough number of involutive integrals. In deriving them, we use theLax equation ∂ x L ( λ ) = [ U ( λ ) , L ( λ )] , (3.3)which has a solution, the Lax matrix [8, 10] L ( λ ; p, q ) = / Q λ ( p, q ) − Q λ ( p, p ) Q λ ( q, q ) − / − Q λ ( p, q ) ! , (3.4)where Q λ ( ξ, η ) = < ( λI − A ) − ξ, η > . By equation (3.3), F ( λ ) = det L ( λ ) is independent of theargument x . Three sets of integrals are derived from the expansions F ( λ ) = −
14 + N X k =1 E k λ − α k = −
14 + ∞ X j =0 F j λ − j − , (3.5a) H ( λ ) = p − F ( λ ) = 12 − ∞ X k =0 H k λ − k − , (3.5b)with F = − < p, q > , H = − < p, q > / H exactly the same as in equation (3.1b), and E k = − p k q k + X ≤ j ≤ N ; j = k ( p j q k − p k q j ) α k − α j , (3.6a) F k = − < A k p, q > + X i + j = k − i, j ≥ ( < A i p, p >< A j q, q > − < A i p, q >< A j p, q > ) , (3.6b) H k = 12 F k + 2 X i + j = k − i, j ≥ H i H j . (3.6c)The functions { E k } are called confocal polynomials, satisfying N X k =1 α jk E k = F j , N X k =1 E k = F = − < p, q > . (3.7)Further, we have the Lax equation along the F ( λ )-flow,dd t λ L ( µ ) = { L ( µ ) , F ( λ ) } = 2 λ − µ [ L ( λ ) , L ( µ )] , (3.8)8hich can be verified directly. It implies { F ( µ ) , F ( λ ) } = ∂ t λ det L ( µ ) = 0. Here { A, B } is theusual Poisson bracket defined as { A, B } = N X k =1 (cid:18) ∂A∂q k ∂B∂p k − ∂A∂p k ∂B∂q k (cid:19) . As a corollary we have
Lemma 3.1.
The members in the set { E j , F k , H l } are involutive in pairs. By [8], there is an inner relation between the integral H k and ( X k ), the AKNS equation(2.3). The involutivity { H , H k } = 0 implies the commutativity of the Hamiltonian phase flows g xH , g τ k H k . This yields a compatible solution for ( H ) , ( H k ), and hence a solution to equation( X k ), respectively, as( p ( x, τ k ) , q ( x, τ k )) = g xH g τ k H k ( p , q ) , (3.9a)( u ( x, τ k ) , v ( x, τ k )) = f U ( p, q ) = ( − < p, p >, < q, q > ) . (3.9b)Let α ( λ ) = Π Nk =1 ( λ − α k ). By [8, 10], a curve R : ξ = R ( λ ), with genus g = N −
1, isconstructed by the factorization of F ( λ ) = − Λ( λ ) / [4 α ( λ )], with R ( λ ) = Λ( λ ) α ( λ ). For non-branching λ , there are two points p ( λ ), τ p ( λ ) on R , with τ : R → R the map of changingsheets. Consider two objects on the curve, the canonical basis a , · · · , a g , b , · · · , b g of homologygroup of contours, and the basis of holomorphic differentials, written in the vector form as ~ω ′ = ( ω ′ , · · · , ω ′ g ) T , ω ′ j = λ g − j d λ/ (2 ξ ). It is normalized into ~ω = C~ω ′ , where C = ( a jk ) − g × g , with a jk the integral of ω ′ j along a k . Near the infinities, the local expansions have simple relation as ~ω = ( +( ~ Ω + ~ Ω z + ~ Ω z + · · · )d z, near ∞ + , − ( ~ Ω + ~ Ω z + ~ Ω z + · · · )d z, near ∞ − . (3.10)Periodic vectors ~δ k and ~B k are defined as integrals of ~ω along a k and b k , respectively. They spana lattice T , which defines the Jacobian variety J ( R ) = C g / T . The Abel map A ( p ) is given asthe integral of ~ω from the fixed point p to p . The matrix B , with ~B k as columns, is used toconstruct the theta function θ ( ~z, B ).The elliptic variables µ j , ν j are given by the roots of the off-diagonal entries of the Laxmatrix, L ( λ ) = − < p, p > m ( λ ) α ( λ ) , m ( λ ) = Π gj =1 ( λ − µ j ) , (3.11a) L ( λ ) = < q, q > n ( λ ) α ( λ ) , n ( λ ) = Π gj =1 ( λ − ν j ) . (3.11b)They define the quasi-Abel-Jacobi and Abel-Jacobi variables, respectively, as ~ψ ′ = g X k =1 Z p ( µ k ) p ~ω ′ , ~ψ = C ~ψ ′ = A ( g X k =1 p ( µ k )) , (3.12a) ~φ ′ = g X k =1 Z p ( ν k ) p ~ω ′ , ~φ = C ~φ ′ = A ( g X k =1 p ( ν k )) . (3.12b)9he evolution of these two variables along the F ( λ )-flow is obtained by equation (3.8). Actually,in the component equation for L ( µ ), by letting µ → ν k , we calculate12 p R ( ν k ) d ν k d t λ = − n ( λ ) α ( λ )( λ − ν k ) n ′ ( ν k ) , (3.13a) { φ ′ s , F ( λ ) } = d φ ′ s d t λ = g X k =1 ν g − sk p R ( ν k ) d ν k d t λ = − λ g − s α ( λ ) , (3.13b)where ~φ ′ = ( φ ′ , · · · , φ ′ g ). By the partial fraction expansion (3.5a), we get { φ ′ s , E k } = − α g − sk /α ′ ( α k ) , ( k = 1 , · · · , N ) , (3.14a) { φ ′ s , E + · · · + E N } = { φ ′ s , F } = 0 . (3.14b) Proposition 3.1.
Each Hamiltonian system ( H k ) , k = 1 , , · · · , is integrable in Liouville sense,sharing the same integrals E , · · · , E N , which are involutive in pairs and functionally indepen-dent in R N − { } .Proof. According to Lemma 3.1, it only needs to prove the functional independence of theconfocal polynomials. Suppose P Nk =1 c k d E k = 0. Then P Nk =1 c k { φ ′ s , E k } = 0. By equation(3.14b), we have g X k =1 ( c k − c N ) { φ ′ s , E k } = 0 , (1 ≤ s ≤ g ) . The coefficient matrix is non-degenerate since by equation (3.14a) it is of Vandermonde type.Hence we have c k − c N = 0 and c N P Nk =1 d E k = 0. This implies c N = 0 since N X k =1 d E k = − d < p, q > = − N X j =1 ( q j d p j + p j d q j ) = 0 . Lemma 3.2.
The Abel-Jacobi variables straightens out the H ( λ ) -flow as { ~φ, H ( λ ) d λ } = 2 ~ω. (3.15) Proof.
Since F ( λ ) = − H ( λ ), equation (3.13b) is transformed into the following formula, which,by multiplied the matrix C , leads to equation (3.15), { φ ′ s , H ( λ )d λ } = λ g − s d λ H ( λ ) α ( λ ) = λ g − s d λ p R ( λ ) = 2 ω ′ s . Proposition 3.2.
The Abel-Jacobi variables straighten out the H k -flow, as { ~φ, H } = 0 andd ~φ d τ k = { ~φ, H k } = ~ Ω k , ( k = 1 , , · · · ) , (3.16a) ~φ ( τ k ) ≡ ~φ (0) + τ k ~ Ω k , (mod T ) . (3.16b)10 roof. By the equations (3.5b) and (3.10), we have an expansion of equation (3.15) near ∞ + .Equation (3.16a) is then obtained as its coefficient.For another Abel-Jacobi variable, by the equation (3.25) in [10], we have ~ψ + ~η + ≡ ~φ + ~η − , (mod T ) , (3.17a) ~η ± = Z p ∞ ± ~ω, ~ Ω D = ~η + − ~η − = Z ∞ − ∞ + ~ω, (3.17b) ~ψ ( τ k ) ≡ ~φ ( τ k ) − ~ Ω D ≡ ~ψ (0) + τ k ~ Ω k , (mod T ) . (3.17c) S γ In [10], an integrable symplectic map S γ : R N → R N , ( p, q ) (˜ p, ˜ q ), is constructed with thehelp of N replicas of discrete ZS-AKNS equation (2.7), (cid:18) ˜ p j ˜ q j (cid:19) = ( α j − γ ) − / D ( γ ) ( α j ; a, b ) (cid:18) p j q j (cid:19) , (1 ≤ j ≤ N ) , (4.1)under the discrete constraint ( a, b ) = f γ ( p, q ), a = − < p, p >, b = 1 Q γ ( p, p ) (cid:16) − − Q γ ( p, q ) ± p R ( γ )2 α ( γ ) (cid:17) . (4.2)It can be derived from the continuous constraint (3.2) through the relation a = u, b = ˜ v in (2.8).In fact, ˜ v − b = < ˜ q, ˜ q > − b = < ( A − γI ) − ( bp + q ) , bp + q > − b = b L ( γ ) − bL ( γ ) − L ( γ ) ≡ P ( γ ) ( b ) . Thus P ( γ ) ( b ) = 0, whose roots lead to equation (4.2). The factor ( α j − γ ) − in equation (4.1)is introduced so that the coefficient determinant equals to unity, which is necessary for makingthe resulting map S γ symplectic.As in the continuous case, the Liouville integrability of the map S γ requires enough numberof involutive integrals. Similarly, the discrete Lax equation, given as follows, plays a central role, L ( λ ; ˜ p, ˜ q ) D ( γ ) ( λ ; a, b ) = D ( γ ) ( λ ; a, b ) L ( λ ; p, q ) . (4.3)By [10], under the constraint (4.2), it has the same Lax matrix, given by equation (3.4), as itssolution. Immediately we have F ( λ ; ˜ p, ˜ q ) = F ( λ ; p, q ) by taking the determinant of (4.3). Thus F ( λ ), together with H ( λ ), E j , F k , H l , are all invariant under the action of the map S γ . Proposition 4.1. [10] The map S γ is symplectic and integrable, possessing F ( λ ) , { F j } , { H l } and the confocal polynomials E , · · · , E N , as its integrals. Construct a discrete flow ( p ( m ) , q ( m )) = S mγ ( p , q ) (4.4)by iteration. It generates the finite genus potential functions for equation (2.7),( a m , b m ) = ( u m , v m +1 ) = ( − < p, p >, < ˜ q, ˜ q > ) . (4.5)11efine L m ( λ ) = L ( λ ; p ( m ) , q ( m )), D ( γ ) m ( λ ) = D ( γ ) ( λ ; u m , v m +1 ). Rewrite equation (4.3) as L m +1 ( λ ) D ( γ ) m ( λ ) = D ( γ ) m ( λ ) L m ( λ ) . (4.6)Consider the discrete ZS-AKNS problem (2.7) with finite genus potential functions as h ( m + 1 , λ ) = D ( γ ) m ( λ ) h ( m, λ ) . (4.7)The solution space E λ is invariant under the action of L m ( λ ) due to the commutativity relation(4.6). The linear operator L m ( λ ) has eigenvalues ± H ( λ ), with associated eigenvectors h ± in E λ ,satisfying L m ( λ ) h ± ( m, λ ) = ± H ( λ ) h ± ( m, λ ) , (4.8a) h ± ( m + 1 , λ ) = D ( γ ) m ( λ ) h ± ( m, λ ) . (4.8b)Roughly speaking, the situation can be regarded as an algebro-difference analogue of theBurchnall-Chaundy’s theory on commuting differential operators [6, 7]. Actually, let L ( λ ) =2 α ( λ ) L ( λ ). Then det L ( λ ) = − R ( λ ) is a polynomial rather than a rational function. The com-mutativity relation (4.6) is rewritten as L m +1 D ( γ ) m = D ( γ ) m L m . The algebraic spectral problem(4.8a) is revised as L m h ± = ξh ± , with ξ = ± p R ( λ ). The algebraic problem and the differ-ence problem share common eigenvectors h ± , with eigenvalues satisfying the algebraic relation, ξ = R ( λ ), exactly the same as the affine equation of the algebraic curve R .Let M ( m, λ ) be fundamental solution matrix of equation (4.7). Under the normalizationcondition h (2) ± (0 , λ ) = 1, the eigenvectors are determined uniquely as h ± ( m, λ ) = (cid:18) h (1) ± ( m, λ ) h (2) ± ( m, λ ) (cid:19) = M ( m, λ ) (cid:18) c ± λ (cid:19) , (4.9a) c ± λ = L ( λ ) ± H ( λ ) L ( λ ) = − L ( λ ) L ( λ ) ∓ H ( λ ) . (4.9b)Two meromorphic functions, the Baker functions, h ( κ ) ( m, p ), p ∈ R , κ = 1 ,
2, are defined as h ( κ ) ( m, p ( λ )) = h ( κ )+ ( m, λ ) , h ( κ ) ( m, τ p ( λ )) = h ( κ ) − ( m, λ ) . (4.10)The commutativity relation (4.6) implies formulas of Dubrovin-Novikov’s type [10]. They areapplied to calculate the divisors of the Baker functions. This leads to the straightening out ofthe flow S mγ on the Jacobian variety as [8] ~ψ ( m ) ≡ ~φ (0) + m~ Ω γ − ~ Ω D , (mod T ) , (4.11a) ~φ ( m ) ≡ ~φ (0) + m~ Ω γ , (mod T ) , (4.11b) ~ Ω γ = Z ∞ + p ( γ ) ~ω, ~ Ω D = Z ∞ − ∞ + ~ω, (4.11c)where the Abel-Jacobi variables are given by (3.12) as ~ψ ( m ) = A (cid:0) g X j =1 p ( µ j ( m )) (cid:1) , ~φ ( m ) = A (cid:0) g X j =1 p ( ν j ( m )) (cid:1) . q , r ∈ R , there exists a dipole ω [ q , r ], an Abel differential of the thirdkind, with residues 1 and − q , r , respectively, satisfying [26] Z a j ω [ q , r ] = 0 , Z b j ω [ q , r ] = Z qr ω j , ( j = 1 , · · · , g ) . (4.12)With the help of these dipoles, the Baker functions can be reconstructed as [10] h (1) ( m, p ) = d (1) m θ [ − A ( p ) + ~ψ ( m ) + ~K ] θ [ − A ( p ) + ~φ (0) + ~K ] e R pp mω [ p ( γ ) , ∞ + ]+ ω [ ∞ − , ∞ + ] , (4.13a) h (2) ( m, p ) = d (2) m θ [ − A ( p ) + ~φ ( m ) + ~K ] θ [ − A ( p ) + ~φ (0) + ~K ] e R pp mω [ p ( γ ) , ∞ + ] , (4.13b)where d (1) m , d (2) m and ~K are constants, independent of p ∈ R .With these results in hand, we start to derive an explicit formula for the function u ˜ v . Tothis end we consider the local expression of the dipole near ∞ + , ( z = λ − ), ω [ p ( γ ) , ∞ + ] = [ − z − + ϕ ( z )]d z, (4.14)with ϕ ( z ) holomorphic near z ∼
0. A simple calculation yields ∂ z log( z exp Z pp ω [ p ( γ ) , ∞ + ]) = ϕ ( z ) . (4.15)Recalling equation (3.10), we have − A ( p ) = ~η + − ~ Ω z + O ( z ) , ~η + = Z p ∞ + ~ω. (4.16)Then, from (4.13a) we get z ˜ h (1)+ h (1)+ = d m +1 d m θ [ − ~ Ω z + O ( z ) + ~η + + ~ψ ( m + 1) + ~K ] θ [ − ~ Ω z + O ( z ) + ~η + + ~ψ ( m ) + ~K ] · ze R pp ω [ p ( γ ) , ∞ + ] , (4.17a) ∂ z log z ˜ h (1)+ h (1)+ = ∂ z log θ [ − ~ Ω z + O ( z ) + ~η + + ~ψ ( m + 1) + ~K ] θ [ − ~ Ω z + O ( z ) + ~η + + ~ψ ( m ) + ~K ] + ϕ ( z ) . (4.17b)On the other hand, since h ± = ( h (1) ± , h (2) ± ) T satisfies equation (4.8b), we have z ˜ h (1)+ h (1)+ = 1 + ( u ˜ v − γ ) z + uh (2)+ h (1)+ z = 1 + ( u ˜ v − γ ) z + O ( z ) , (4.18)where the following estimation is used, uh (2)+ h (1)+ = L ( λ ) − H ( λ ) − L ( λ ) = < q, q > λ − [1 + O ( λ − )] = O ( z ) . Now, taking derivative of the equation (4.18) with respect to z and comparing it with (4.17b) at z = 0, with the help of the relation ~ψ + ~η + ≡ ~φ + ~η − in equation (3.17a), we obtain the following. Proposition 4.2.
Let ( a, b ) = ( u, ˜ v ) be finite genus potential functions of equation (2.7), definedby equation (4.5). Then we have u ˜ v = − ∂ z | z =0 log θ [ ~ Ω z + ~φ ( m + 1) + ~η − + ~K ] θ [ ~ Ω z + ~φ ( m ) + ~η − + ~K ] + [ γ + ϕ (0)] . (4.19)13 Finite genus solutions to the lpKP
Let γ = γ , γ , γ be distinct and non-zero. We can apply the same theory we developed in Sec-tion 4 to the three corresponding cases, respectively. The resulting integrable maps S γ , S γ , S γ commute in pairs since they share the same integrals E , · · · , E N (see Appendix in [10]). By iter-ation we have discrete flows S m γ , S m γ , S m γ , and hence well-defined functions from any startingpoint ( p ( m , m , m ) , q ( m , m , m )) = S m γ S m γ S m γ ( p , q ) , (5.1a)( u ( m , m , m ) , v ( m , m , m )) = ( − < p, p >, < q, q > ) | ( m ,m ,m ) . (5.1b)Define a = u , and let b take ˜ v = T v, ¯ v = T v, ˆ v = T v , respectively. By the commutativity ofthe flows, one can present the functions given by equation (5.1a) in three ways, respectively as( p ( m k ) , q ( m k )) = S m k γ k ( p ( k )0 , q ( k )0 ) , ( k = 1 , , . (5.2)Thus, from equation (4.1) in the three special cases, the j -th component satisfies three equationssimultaneously with λ = α j , T k (cid:18) p j q j (cid:19) = ( α j − γ k ) − / D ( γ k ) ( α j ; u, T k v ) (cid:18) p j q j (cid:19) , ( k = 1 , , . (5.3)Introducing χ = ( α j − γ ) m / ( α j − γ ) m / ( α j − γ ) m / (cid:18) p j q j (cid:19) , (5.4)we then have T k χ = D ( γ k ) ( α j ; u, T k v ) χ, ( k = 1 , , . (5.5)In other words, the overdetermined system of equations (2.9a-c) has a compatible solution χ forthe parameter λ = α j . Now, for the lpKP equation (1.2), recalling Proposition 2.1, we arrive atthe following. Proposition 5.1.
The lpKP equation (1.2), Ξ (0 , = 0 , has a special solution W ( m , m , m ) =2 ∂ z | z =0 log θ [ z~ Ω + ~φ ( m , m , m ) + ~η − + ~K ] θ [ z~ Ω + ~φ (0 , ,
0) + ~η − + ~K ] − X s =1 m s [ γ s + ϕ s (0)] + W (0 , , , (5.6) where ~φ ( m , m , m ) = X s =1 m s ~ Ω γ s + ~φ (0 , , , (5.7) and ϕ s ( z ) is defined by equation (4.14) in the case of γ = γ s .Proof. We have ( k = 1 , , T k W − W = 2 ∂ z | z =0 log θ [ z~ Ω + T k ~φ ( m , m , m ) + ~η − + ~K ] θ [ z~ Ω + ~φ ( m , m , m ) + ~η − + ~K ] − γ k + ϕ k (0)] . It is equal to − u ( T k v ) by equation (4.19). Thus W solves (2.10a-c) simultaneously. Accordingto Proposition 2.1, W solves equation (1.2). 14 Solutions of other equations
In solving pKP equation Ξ ( j,k ) = 0 that contains at least one continuous argument x , we willderive an explicit analytic expression for uv , which is similar to (4.19) and also meets theauxiliary equation (2.10d). This can be done on the Liouville integrable platform as well, likein the discrete case. We list the main steps as follows.Consider ( p ( x ) , q ( x )) = g xH ( p , q ). Hence ( u ( x ) , v ( x )) = ( − < p, p >, < q, q > ) provide thefinite genus potential functions. For the ZS-AKNS equation (2.2) with these potential functions,the solution space E λ is invariant under the action of L ( λ ) due to the commutativity relation(3.3). The linear operator L ( λ ) has eigenvalues ± H ( λ ), with associated eigenvectors h ± in E λ ,satisfying L ( λ ) h ± ( x, λ ) = ± H ( λ ) h ± ( x, λ ) , (6.1a) ∂ x h ± ( x, λ ) = U ( λ ; u ( x ) , v ( x )) h ± ( x, λ ) . (6.1b)Let M ( x, λ ) be basic solution matrix of equation (6.1b). The eigenvectors are uniquely deter-mined under the normalized condition h (2) ± (0 , λ ) = 1 and can be expressed as h ± ( x, λ ) = (cid:18) h (1) ± ( x, λ ) h (2) ± ( x, λ ) (cid:19) = M ( x, λ ) (cid:18) c ± λ (cid:19) , (6.2a) c ± λ = L (0 , λ ) ± H ( λ ) L (0 , λ ) . (6.2b)Two meromorphic functions h ( κ ) ( x, p ), κ = 1 ,
2, are defined in
R − {∞ + , ∞ − } by h ( κ ) ( x, p ( λ )) = h ( κ )+ ( x, λ ) , h ( κ ) ( x, τ p ( λ )) = h ( κ ) − ( x, λ ) . A formula of Dubrovin-Novikov’s type is derived from the commutativity relation (3.3). Itis used to calculate the divisor of h (2) ( x, p ), which is equal to P gj =1 [ p ( ν j ( x )) − p ( ν j (0))]. Byequations (3.12b) and (3.16b), we have ~φ ( x ) = A (cid:16) g X j =1 p ( ν j ( x )) (cid:17) ≡ x~ Ω + ~φ (0) , (mod T ) . (6.3)On the two-sheeted Riemann surface R , an Abel differential, ω (1) [ ∞ − , ∞ + ], of the third kind isconstructed, having only poles at ∞ − , ∞ + with ω (1) [ ∞ − , ∞ + ] = ( [ − z − − a (1) ( z )]d z, near ∞ + , [+ z − + a (1) ( z )]d z, near ∞ − , (6.4)where a (1) ( z ) is holomorphic near z ∼
0. Without loss of generality, it can be arranged to satisfythe condition Z a j ω (1) = 0 , Z b j ω (1) = − π iΩ j , (1 ≤ j ≤ g ) , (6.5)where ~ Ω = (Ω , · · · , Ω g ) T . Actually, by adding a linear combination of holomorphic differentials ω , · · · , ω g to ω (1) , we can make the former formula in equation (6.5) valid. The latter isa corollary of the former, which can be verified by using the canonical representation of the15iemann surface R [13, 26]. The form of local expressions (6.4) is invariant with adjusted a (1) ( z ). We adopt the same symbol, for short. Through a usual analysis we reconstruct [5, 26] h (2) ( x, p ) = c (2) ( x ) θ [ − A ( p ) + ~φ ( x ) + ~K ] θ [ − A ( p ) + ~φ (0) + ~K ] · exp (cid:16) x Z pp ω (1) [ ∞ − , ∞ + ] (cid:17) , (6.6)where c (2) is independent of p ∈ R . Equations (6.5) are used to cancel the extra factors causedby the uncertain linear combination of the contours a , · · · , a g , b , · · · , b g in the integration routefrom the point p to p , both in A ( p ) and in the integral of ω (1) .By equation (3.10), near ∞ − we have ( z = λ − ∼ − A ( p ) = ~η − + ~ Ω z + O ( z ) , ~η − = Z p ∞ − ~ω. Exerting action ∂ z ∂ x log on equation (6.6), we obtain ∂ z ∂ x log h (2) − , which is equal to ∂ z ∂ x log θ [ ~ Ω z + O ( z ) + ~φ ( x ) + ~η − + ~K ] + 12 [ z − + a (1) ( z )] . (6.7)On the other hand, by equation (6.1a) we estimate v h (1) − h (2) − = v L ( λ ) − H ( λ ) L ( λ ) = − uvλ − + O ( λ − ) , (6.8)where the following estimations are employed, L ( λ ) = 12 + < p, q > z + < Ap, q > z + O ( z ) ,L ( λ ) = < q, q > z + O ( z ) ,H ( λ ) = 12 − H z − H z + O ( z ) . From equation (6.1b) and the estimation (6.8), we have ∂ x log h (2) − = − λ v h (1) − h (2) − = − z − − uvz + O ( z ) , (6.9a) ∂ z ∂ x log h (2) − = z − / − uv + O ( z ) . (6.9b)Then, equating equation (6.7) with (6.9b) to cancel the singular term z − /
2, we obtain thefollowing.
Proposition 6.1.
Let ( u, v ) be finite genus potential functions for equation (2.2). Then − uv = 2 ∂ z | z =0 ∂ x log θ [ ~ Ω z + ~φ ( x ) + ~η − + ~K ] + a (1) (0) . (6.10)Next, we can recover W for the pKP equations with continuous arguments. In order to solveΞ (1 , = 0, we consider the integrable maps S γ , S γ and g xH , which commute in pairs since theyshare the same integrals { E j } (cf. [10]). Well-defined functions are constructed as( p ( x, m , m ) , q ( x, m , m )) = g xH S m γ S m γ ( p , q ) , (6.11a)( u ( x, m , m ) , v ( x, m , m )) = ( − < p, p >, < q, q > ) | ( x,m ,m ) . (6.11b)16y the commutativity of the flows, the functions in equation (6.11a) can be presented in threeways, respectively, as ( p ( x ) , q ( x )) = g xH ( p ′ , q ′ ) , (6.12a)( p ( m k ) , q ( m k )) = S m k γ k ( p ( k )0 , q ( k )0 ) , ( k = 1 , . (6.12b)Thus the j -th component satisfies three equations simultaneously with λ = α j , ∂ x (cid:18) p j q j (cid:19) = U ( α j ; u, v ) (cid:18) p j q j , (cid:19) , (6.13a) T k (cid:18) p j q j (cid:19) = ( α j − γ k ) − / D ( γ k ) ( α j ; u, T k v ) (cid:18) p j q j (cid:19) , ( k = 1 , . (6.13b)Introducing χ = ( α j − γ ) m / ( α j − γ ) m / ( p j q j ) T , we have ∂ x χ = U ( α j ; u, v ) χ, (6.14a) T k χ = D ( γ k ) ( α j ; u, T k v ) χ, ( k = 1 , . (6.14b)Thus equations (2.2) and (2.9a,b) have a compatible solution χ for the parameter λ = α j . Proposition 6.2.
The semi-discrete pKP equation (1.3), i.e. Ξ (1 , = 0 , has a solution W ( x, m , m ) =2 ∂ z | z =0 log θ [ z~ Ω + ~φ ( x, m , m ) + ~η − + ~K ] θ [ z~ Ω + ~φ (0 , ,
0) + ~η − + ~K ] − X s =1 m s [ γ s + ϕ s (0)] + a (1) (0) x + W (0 , , , (6.15) where ~φ ( x, m , m ) = x~ Ω + P s =1 m s ~ Ω γ s + ~φ (0 , , , ϕ s ( z ) defined by equation (4.14) with γ = γ s , and a (1) ( z ) given by equation (6.4).Proof. From (6.15) we have ( k = 1 , ∂ x W = 2 ∂ z | z =0 ∂ x log θ [ z~ Ω + ~φ ( x, m , m ) + ~η − + ~K ] + a (1) (0) ,T k W − W = 2 ∂ z | z =0 log θ [ z~ Ω + T k ~φ ( x, m , m ) + ~η − + ~K ] θ [ z~ Ω + ~φ ( x, m , m ) + ~η − + ~K ] − γ k + ϕ k (0)] , which are equal to − uv and − u ( T k v ) according to equations (6.10) and (4.19), respectively.Recalling Proposition 2.2, W solves equation (1.3).By similar analysis we have the following. Proposition 6.3.
The semi-discrete pKP equation (1.4), i.e. Ξ (2 , = 0 , has a solution W ( x, y, m ) =2 ∂ z | z =0 log θ [ z~ Ω + ~φ ( x, y, m ) + ~η − + ~K ] θ [ z~ Ω + ~φ (0 , ,
0) + ~η − + ~K ] − m [ γ + ϕ (0)] + a (1) (0) x + W (0 , , , (6.16) where ~φ ( x, y, m ) = x~ Ω + y~ Ω + m ~ Ω γ + ~φ (0 , , . roposition 6.4. The pKP equation (1.5), i.e. Ξ (3 , = 0 , is solved by W ( x, y, t ) =2 ∂ z | z =0 log θ [ z~ Ω + ~φ ( x, y, t ) + ~η − + ~K ] θ [ z~ Ω + ~φ (0 , ,
0) + ~η − + ~K ]+ a (1) (0) x + W (0 , , , (6.17) where ~φ ( x, y, t ) = x~ Ω + y~ Ω + t~ Ω + ~φ (0 , , . In this paper we have shown that the lpKP equation, semi-discrete pKP equations and continuouspKP equation can be derived as compatibilities of Lax triads that originate from the ZS-AKNSspectral problems. The approach to constructing finite genus solutions for 2D lattice equations[9, 10] was extended to 3D cases. As a result, we obtained finite genus solutions for the discrete,semi-discrete and continuous pKP equations. Note that these solutions are different from theelliptic solitons that are genus-one solutions obtained by Nijhoff, et al. in [23, 27].In deriving those pKP equations, we employed the auxiliary relations (2.10). We note thatusually W in (2.10) can not be exactly solved out for all arbitrarily given ( u, v ); therefore it ishard to say when Lax triads provide strict integrability for 3D equations in some cases (cf. [20]).However, as for the case of finite genus solutions, since the finite-dimensional integrable flows g τ j H j and S m k γ k share same Liouville integrals, same Lax matrix and same algebraic curve, it enablesus to treat (2.10) on the same Liouville platform and obtain explicit expressions for W from(2.10) by algebro-geometric integration.The lpKP equation is one of the five octahedron-type equations with 4D consistency [3]. Webelieve our approach can be extended to other octahedron-type integrable equations. This willbe a part of our future work. Acknowledgments
The authors are grateful to the referee for the invaluable comments. This work is supported bythe National Natural Science Foundation of China (grant nos 10971200, 11501521, 11426206,11631007 and 11875040 for the three authors, respectively).
AppendicesA An heuristic deduction of Ansatz (2.26)
The Abel-Jacobi variable ~φ in the Jacobian variety J ( R ) provides a favorable window to observethe evolution of the discrete symplectic flow S m k γ k as well as the Hamiltonian flow g τ j H j . Thediscrete velocity ~ Ω γ k of ~φ is given by equation (4.11c), while the continuous velocity ~ Ω j iscalculated by equation (3.10) and (3.16a). They are bridged by the normalized basis ~ω ofholomorphic differentials. Let the parameter γ = γ k tend to be infinity in the way as γ k = − /ε k ,with ε k = c k ε , ε →
0. By the local expression of ~ω near ∞ + given by equation (3.10), we have ~ Ω γ k = Z ∞ + p ( γ k ) ~ω = ε k ~ Ω − ε k ~ Ω + ε k ~ Ω + O ( ε ) . ~φ − ~φ = X k =1 m k (cid:0) ε k ~ Ω − ε k ~ Ω + ε k ~ Ω (cid:1) + O ( ε ) . On the other hand, by equation (6.17), the 3D continuous evolution of ~φ reads ~φ − ~φ = ( x − x ) ~ Ω + ( y − y ) ~ Ω + ( t − t ) ~ Ω . Thus, up to O ( ε ), we have x − x = X s =1 m k ε k , y − y = − X s =1 m k ε k , t − t = X s =1 m k ε k ,T k x = x + ε k , T k y = y − ε k , T k t = t + ε k . By substituting them into T k W = W ( T k x, T k y, T k t ), we obtain Ansatz (2.26). B Continuum limit of the lNLS
The lNLS equation (2.12), i.e. Ξ (0 , = 0, is first obtained by Konopelchenko [18]. It is solvedin [10]. At first glance, its relation with the NLS equation (2.17), Ξ (2 , = 0, is not clear. Itturns out that there is a transformation of Nijhoff’s type, u = ( − γ ) m ( − γ ) m u ′ , v = ( − γ ) − m ( − γ ) − m v ′ , which reduces the lNLS equation into an equation of ( u ′ , v ′ ),(Ξ ′ ) (0 , ≡ ( γ γ ) − ( γ ˜ u ′ − γ ¯ u ′ ) u ′ ¯˜ v ′ + γ (˜ u ′ − u ′ ) − γ (¯ u ′ − u ′ ) = 0 , (Ξ ′ ) (0 , ≡ ( γ γ ) − ( γ ˜ v ′ − γ ¯ v ′ ) u ′ ¯˜ v ′ + γ (˜¯ v ′ − ¯ v ′ ) − γ (¯˜ v ′ − ˜ v ′ ) = 0 . Let − γ − k = ε k = c k ε , ( k = 1 , c , c are distinct non-zero constants. For any smoothfunctions u ′ ( x, y ), v ′ ( x, y ), define˜ u ′ = u ′ ( x + ε , y − ε / , ¯ u ′ = u ′ ( x + ε , y − ε / , ¯˜ u ′ = u ′ ( x + ε + ε , y − ε / − ε / , and similar expressions for ˜ v ′ , ¯ v ′ , ¯˜ v ′ . Then, as ε ∼
0, we have the following Taylor expansion whichconfirms that the continuum limit of the lNLS is NLS up to a Nijhoff’s type transformation,(Ξ ′ ) (0 , = (cid:18) u ′ y − u ′ xx + 2( u ′ ) v ′ v ′ y + v ′ xx − u ′ ( v ′ ) (cid:19) c − c ε + O ( ε ) . Similarly, the semi-discrete NLS equation (2.19), Ξ (1 , = 0, is transformed as(Ξ ′ ) (1 , ≡ (cid:18) u ′ x + γ (˜ u ′ − u ′ ) − γ − ( u ′ ) ˜ v ′ ˜ v ′ x + γ (˜ v ′ − v ′ ) + γ − u ′ (˜ v ′ ) (cid:19) = 0 ,
19y the transformation of Nijhoff’s type, u = ( − γ ) m u ′ , v = ( − γ ) − m v ′ . Let˜ u ′ = u ′ ( x + ε , y − ε / , ˜ v ′ = v ′ ( x + ε , y − ε / . Then, as ε ∼
0, we have the following Taylor expansion, which confirms that the continuumlimit of the time-discrete NLS equation (2.19) is the NLS equation up to a Nijhoff’s type trans-formation, (Ξ ′ ) (1 , = (cid:18) u ′ y − u ′ xx + 2( u ′ ) v ′ v ′ y + v ′ xx − u ′ ( v ′ ) (cid:19) c ε + O ( ε ) . References [1] Ablowitz M J and Ladik J 1975 Nonlinear differential-difference equations
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