Symmetric discrete AKP and BKP equations
Shangshuai Li, Frank W. Nijhoff, Ying-ying Sun, Da-jun Zhang
aa r X i v : . [ n li n . S I] S e p Symmetric discrete AKP and BKP equations
Shangshuai Li , Frank W. Nijhoff , Ying-ying Sun , Da-jun Zhang ∗ Department of Mathematics, Shanghai University, Shanghai 200444, China School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom Department of Mathematics, University of Shanghai for Science and Technology, Shanghai 200093, China
September 10, 2020
Abstract
We show that when KP (Kadomtsev-Petviashvili) τ functions allow special symmetries,the discrete BKP equation can be expressed as a linear combination of the discrete AKPequation and its reflected symmetric forms. Thus the discrete AKP and BKP equationscan share the same τ functions with these symmetries. Such a connection is extended to4 dimensional (i.e. higher order) discrete AKP and BKP equations in the correspondingdiscrete hierarchies. Various explicit forms of such τ functions, including Hirota’s form,Gramian, Casoratian and polynomial, are given. Symmetric τ functions of Cauchy matrixform that are composed of Weierstrass σ functions are investigated. As a result we obtaina discrete BKP equation with elliptic coefficients. PACS numbers:
Keywords: discrete AKP, discrete BKP, symmetric τ function, solution, elliptic function The Kadomtsev-Petviashvili (KP) equation is one of the most famous (2+1)-dimensional in-tegrable systems. In discrete case, the discrete AKP (dAKP) equation and BKP (dBKP)equation are two master equations in the KP family. With parameterised coefficients, they are,respectively,
A . = ( a − b ) be τ τ + ( b − c ) b τ e τ + ( c − a ) e τ b τ = 0 , (1)and B . =( a − b )( b − c )( c − a ) be τ τ + ( a − b )( a + c )( b + c ) be τ τ + ( b − c )( b + a )( c + a ) b τ e τ + ( c − a )( c + b )( a + b ) e τ b τ = 0 . (2)Here a, b, c are spacing parameters and tilde, hat, bar serve as notations of shifts in differentdirections (see (12)). An alternative form of Eq.(1) is c ( a − b ) be τ τ + a ( b − c ) b τ e τ + b ( c − a ) e τ b τ = 0 , (3)which is connected with (1) by changing( a, b, c ) −→ (1 /a, /b, /c ) . (4)Note that Eq.(2) does not change under the above replacement. The coefficients in both thedAKP and dBKP equation can be arbitrary nonzero numbers if we do not require τ = 1 is asolution. In fact, all the coefficients z i of the following dAKP equation z b τ e τ + z e τ b τ + z be τ τ = 0 , (5) ∗ Corresponding author. Email: djzhang@staff.shu.edu.cn τ → z − ml z − nl z − nm τ ′ (cf. [16, 17]), and for the dBKP equation z b τ e τ + z e τ b τ + z be τ τ + z be τ τ = 0 , (6)the transformation is (cf. [18]) τ → (cid:18) z z z z (cid:19) ml (cid:18) z z z z (cid:19) nl (cid:18) z z z z (cid:19) nm τ ′ . (7)Eq.(1) originated from Hirota’s discrete analogue of the generalized Toda equation (DAGTE)[9] that was parameterised later by Miwa [12] for the sake of expression of N -soliton solutions.It is also known as the discrete KP equation, the Hirota equation, or the Hirota-Miwa equation.Note that the DAGTE and the dAKP equation are equivalent in the sense that there exist a setof parameter transformations to transform them to each other [11]. Equation (2) was first givenby Miwa [12] and now bears his name. It also appears as a nonlinear superposition formula ofthe (2+1) dimensional sine-Gordon system [18]. Both equations have Lax triads [16–18], andboth equations are 4D consistent [1, 2].Both the dAKP and dBKP equations have N -soliton solutions, which are possible to bewritten out from those of the continuous AKP and BKP equation (cf. [5]) by means of Miwa’stransformation [12]. Let us present these solutions by the following uniform formula, τ = X µ =0 , exp N X j =1 µ j η j + N X ≤ i Theorem 1. If τ satisfies the dAKP equation (1) and has symmetries (16) , then τ satisfiesthe dBKP equation (2) .Proof. For the dAKP τ function, when it has symmetries (16), τ satisfies (1) and (17) simulta-neously. Direct calculation gives rise to( c − a )( a − b ) × A + ( a − b )( b − c ) × A + ( b − c )( c − a ) × A + ( a + b + c + 3 ab + 3 bc + 3 ca ) × A = 3 × B. (18) Remark 1. Among the dAKP equation (1) and the reflected dAKP equations (17) , A, A , A , A are not linearly independent. Apart from the relation A + A + A = A, any element in { A, A , A , A } can be a linear combination of any two elements of the sameset. For example, A = a − ca + b A + b − ca + b A , A = − b + ca + b A − a + ca + b A . This indicates that there are alternative expressions of (18) in terms of only two elements of { A, A , A , A } . For example, we have B = ( a + b )( a + c ) A − ( a − b )( a − c ) A = ( a − c )( b + c ) A − ( b − c )( a + c ) A , (19) which is simpler than (18) . Remark 2. Note that the symmetries can be extended to τ (( n, a ) , m, l ) = γ A n B m C l τ (( − n, − a ) , m, l ) , (20a) τ ( n, ( m, b ) , l ) = γ A n B m C l τ ( n, ( − m, − b ) , l ) , (20b) τ ( n, m, ( l, c )) = γ A n B m C l τ ( n, m, ( − l, − c )) , (20c) where A i , B i , C i , γ i are nonzero constants. Due to the gauge property of discrete bilinear equa-tions (e.g. [8]), Theorem 1 and (17) are still valid if replacing (16) with (20) . s sss s ss ✧✧✧✧✧✧ ✧✧✧✧✧✧✧✧✧✧✧✧✟✟✟✟ ✟✟ s s ss s sss ˙ τ ˙ e τ e ττ ˙ b τ b τ ˙ be τ be τ ˙ be τ be τ ˙ b τ b τ ˙ e τ e τ ˙ τ τ Figure 1: 4D hypercube The 4D dAKP equation is given in [19] via a compact form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a e τ ˙ b τb b b τ ˙ e τc c τ ˙ be τd d τ be τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (21)which has an explicit expression( b − c )( c − d )( b − d ) e τ ˙ b τ − ( a − c )( c − d )( a − d ) b τ ˙ e τ + ( a − b )( b − d )( a − d ) τ ˙ be τ − ( a − b )( b − c )( a − c ) ˙ τ be τ = 0 . (22)The dAKP equation (1) is 4D consistent [2]. We note that the above 4D dAKP equation is aresult of 4D consistency of (1). In fact, embedding eight copies of the dAKP equations on ahypercube (see Fig.1) and then calculating a triplly shifted τ with proper initial points, (e.g.calculating ˙ be τ with initials e τ , b τ , τ , ˙ τ , e τ , ˙ e τ , ˙ e τ , ˙ b τ ), one gets the 4D dAKP equation (22). In additionto (22), the dAKP equation defined on three elementary cubes can yield another 4D latticeequation (cf. Eq.(29) in [2]) A . = ( a − b )( c − d ) be τ ˙ τ − ( a − c )( b − d ) ˙ b τ e τ + ( a − d )( b − c ) ˙ e τ b τ . (23)With regard to solutions, by virtue of 4D consistency, if we consistently extend the dAKPplane wave factor in (9) to the 4th dimension, then the resulted τ function will be a solution tothe 4D equations (22) and (23) as well. The 4D dBKP equation is (see Eq.(2.4) in [23])( a − b )( a − c )( a − d )( b − c )( b − d )( c − d ) ˙ be τ τ − ( a − b )( a + c )( a + d )( b + c )( b + d )( c − d ) be τ ˙ τ + ( a + b )( a − c )( a + d )( b + c )( b − d )( c + d ) e τ ˙ b τ − ( a + b )( a + c )( a − d )( b − c )( b + d )( c + d ) b τ ˙ e τ = 0 . (24)5here are at least three ways by which the above equation is connected with known lat-tice equations. First, this equation is a result of the 4D consistency of the dBKP equation(2) [1]. Second, a more precise relation between the 4D dBKP and 3D dBKP can be given.Denoting the 3D dBKP equation (2) by dBKP( n, m, l ) = 0 and the 4D dBKP equation (24) by4DdBKP( n, m, l, h ) = 0, then we have (cf. [24]) − ˙ τ × n, m, l, h )( a + b )( a + c )( a − d ) ˙ e τ × dBKP( h, m, l ) + ( a + b )( b + c )( b − d ) ˙ b τ × dBKP( n, h, l )+ ( a + c )( b + c )( c − d ) ˙ τ × dBKP( n, m, h ) + ( a − d )( b − d )( c − d ) τ × E h dBKP( n, m, l ) . The third way is the connection with the 4D dAKP equation (22) when τ allows symmetries(16) and τ ( n, m, l, ( h, d )) = τ ( n, m, l, ( − h, − d )) . (25)With these symmetries the 4D dAKP equation (22) yields A . = ( b − c )( c − d )( b − d ) τ ˙ be τ − ( a + c )( c − d )( a + d ) be τ ˙ τ + ( a + b )( b − d )( a + d ) e τ ˙ b τ − ( a + b )( b − c )( c + a ) ˙ e τ b τ = 0 , A . = ( a − c )( c − d )( a − d ) τ ˙ be τ − ( b + c )( c − d )( d + b ) be τ ˙ τ + ( a + b )( b + d )( a − d ) b τ ˙ e τ − ( a + b )( b + c )( a − c ) ˙ b τ e τ = 0 , A . = ( a − b )( b − d )( a − d ) τ ˙ be τ + ( a + c )( c + d )( a − d ) b τ ˙ e τ − ( b + c )( c + d )( b − d ) e τ ˙ b τ − ( a − b )( b + c )( c + a ) ˙ τ be τ = 0 , A . = ( a − b )( b − c )( a − c ) τ ˙ be τ + ( a − c )( c + d )( d + a ) ˙ b τ e τ − ( a − b )( b + d )( d + a ) ˙ τ be τ − ( b − c )( c + d )( d + b ) ˙ e τ b τ = 0 . Together with (23), on can find4 × n, m, l, h )= ( a − b )( a − c )( a − d ) A + ( a − b )( b − c )( b − d ) A + ( a − c )( b − c )( c − d ) A + ( a − d )( b − d )( c − d ) A + [( a − b + c − d ) − a + b )( b + c )( a + d )( c + d )] A . Then, for solutions of the 4D dBKP equation (24), we have the following. Theorem 2. Once we have a dAKP τ function, we consistently extend its plane wave factorto the 4th dimension and impose symmetries (16) and (25) . Then the τ function is a solutionto the 4D dBKP equation (24) . τ functions with symmetries We construct τ function that possesses symmetries (16) or (20). Obviously, the τ function inHirota’s form (8) with (9) | q i = − p i , i.e. η i = (cid:18) a + p i a − p i (cid:19) n (cid:18) b + p i b − p i (cid:19) m (cid:18) c + p i c − p i (cid:19) l , A ij = ( p i − p j ) ( p i + p j ) , (26)agrees with the symmetries (16). Such a τ function provides soliton solutions for the dAKPequation (1) as well as for the dBKP equation (2) in light of Theorem 1. It is remarkable that(26) cannot be obtained from the dBKP plane wave factor and phase factor (10) by imposingconstraints on ( p j , q j ). In the following we will go through more forms of τ functions with thedesired symmetries. 6 .1 Gramian form via Cauchy matrix approach In order to derive a more general τ function, we construct it by means of Cauchy matrixapproach (cf. [7, 15, 28]). Consider the Sylvester equation KM + M K = rc T , (27)where K is a given N × N constant matrix and K and − K do not share eigenvalues, M ∈ C N × N [ n, m, l, · · · ], r = ( r , r , · · · , r N ) T and r i are functions r i : Z ∞ C , and c = ( c , c , · · · , c N ) T with c i ∈ C . Dispersion relations are introduced through e r = ( a − K ) − ( a + K ) r , (28a) b r = ( b − K ) − ( b + K ) r , (28b) r = ( c − K ) − ( c + K ) r . (28c)Since K and − K do not have common eigenvalues, M is uniquely determined by (27) withgive ( K , r , c ) [22]. Here for the spacing parameters a, b, c we skip the unit matrix I without anyconfusion in notations. Then, it can be proved that M obeys the following shift evolutions [28] f M ( a + K ) − ( a + K ) M = e rc T , (29a)( a − K ) f M − M ( a − K ) = rc T , (29b)and the parallel relations for ( m, b ) and ( l, c ). Then we introduce scalar functions S ( i,j ) = c T K j ( I + M ) − K i r , (30a) S ( α, j ] = c T K j ( I + M ) − ( α + K ) − r , (30b) S [ i, β ) = c T ( β + K ) − ( I + M ) − K i r , (30c)with i, j ∈ Z , α, β ∈ C , and define τ function as τ = | I + M | . (31)One can derive evolutions of these functions. S = ( S ( i,j ) ) ∞×∞ is a symmetric matrix [28], i.e. S ( i,j ) = S ( j,i ) . Evolutions for S ( i,j ) can be described as (cf. [15, 28]) a e S ( i,j ) − e S ( i,j +1) = aS ( i,j ) + S ( i +1 ,j ) − S (0 ,j ) e S ( i, , (32a) aS ( i,j ) + S ( i,j +1) = a e S ( i,j ) + e S ( i +1 ,j ) − e S (0 ,j ) S ( i, , (32b)and parallel relations for ( m, b ) and ( l, c ). One more symmetric relation is S ( α, j ] = S [ j, α ) dueto S ( i,j ) = S ( j,i ) and the expansion S ( α, j ] = ∞ X i =0 ( − i a i +1 S ( i,j ) , S [ j, α ) = ∞ X i =0 ( − i a i +1 S ( j,i ) . With the above expansion and (32) one can obtain evolutions of S ( α, j ] as a e S ( α, j ] − e S ( α, j + 1] = ( a − α ) S ( α, j ] + S (0 ,j ) (1 − e S ( α, , (33a) b b S ( α, j ] − b S ( α, j + 1] = ( b − α ) S ( α, j ] + S (0 ,j ) (1 − b S ( α, , (33b) cS ( α, j ] − S ( α, j + 1] = ( c − α ) S ( α, j ] + S (0 ,j ) (1 − S ( α, . (33c)Next, let us derive some relations on τ and the functions in (30) (cf. [7]).7 emma 1. τ function (31) obeys evolutions e τ /τ = 1 − S [0 , − a ) , (34a) τ / e τ = 1 − e S [0 , a ) . (34b) and the parallel relations for ( m, b ) and ( l, c ) .Proof. First, | a − K | e τ = | a − K || I + f M | = | a − K + M ( a − K ) + rc T | , i.e., e τ = | I + M + rc T ( a − K ) − | = | I + M || I + rc T ( a − K ) − ( I + M ) − | . Then, by means of Weinstein-Aronszajn identity | I + rc T | = 1 + c T r , one has e τ /τ = 1 − c T ( K − a ) − ( I + M ) − r = 1 − S [0 , − a ) . (34b) can be proved as the following. | a + K | τ = | a + K || I + M | = | a + K + f M ( a + K ) − e rc T | , and τ = | I + f M − e rc T ( a + K ) − | = | I + f M || I − e rc T ( a + K ) − ( I + f M ) − | . It then follows from the Weinstein-Aronszajn identity that one gets (34b). Lemma 2. Setting u = S (0 , , we have the following relations ( a − b + b u − e u ) = ( a − b ) be τ τ e τ b τ , (35a)( b − c + u − b u ) = ( b − c ) b τ τ b τ τ , (35b)( c − a + e u − u ) = ( c − a ) e τ ττ e τ . (35c) Proof. The proof has been give in [15] for the case K is diagonal. Here let us extract out mainsteps and extend them to the case of arbitrary K .Considering the evolutions of S ( α, j ] given in (33a) and (33b) where we take α = a, j = 0,we get a e S ( a, − e S ( a, 1] = u (1 − e S ( a, , (36a) b b S ( a, − b S ( a, 1] = ( b − a ) S ( a, 0] + u (1 − be S ( a, . (36b)Eliminating S ( a, 1] from the above gives rise to( a − b + b u − e u )(1 − be S ( a, − ( a − b )(1 − e S ( a, . (37)which indicates the relation (35a) using (34b). (35b) and (35c) can be obtained in a similarway.Thus, combining the three equations in (35) together, we get the dAKP equation. Theorem 3. The τ function defined in (31) satisfies the dAKP equation (1) . M to the Sylvester equation (27) and r to the dispersion relation (28) can be writtenout in terms of the canonical forms of K . For a given K , the dispersion relation (28) indicatesthe symmetry for r : r (( n i , a i )) = r (( − n i , − a i )), so is for M , i.e. M (( n i , a i )) = M (( − n i , − a i )),and so is for τ , i.e. τ (( n i , a i )) = τ (( − n i , − a i )).Canonical form of K is composed of a diagonal matrix and different Jordan blocks. When K is a diagonal matrix K = diag { p , p , · · · , p N } , r consists of r i = η i where η i is given in (26),and M = ( m ij ) N × N consists of m ij = r i c j p i + p j . (38)Then, τ = | I + M | is a Gramian which is a special case of the solution obtained in [19]. When K is a Jordan block and a more general form, one can refer to Sec.4 of [28] for explicit formsof r and M . The deformed dAKP equation (17b) also appeared as a member in the bilinear forms of H3equation in the Adler-Bobenko-Suris (ABS) list [1], (see Eq.(5.20a) in [8]). It allows a Casoratiansolution [8] τ ( ψ ) = | ψ ( n, m, l ) , ψ ( n, m, l + 1) , ψ ( n, m, l + 2) , · · · , ψ ( n, m, l + N − | , (39a)where ψ = ( ψ , ψ , · · · , ψ N ) T and ψ i = γ + i ( a + p i ) n ( b + p i ) m ( c + p i ) l + γ − i ( a − p i ) n ( b − p i ) m ( c − p i ) l , γ ± i ∈ C . (39b)At the first glance, τ ( ψ ) does not have symmetries (16). However, by means of the gaugeproperty of Hirota’s discrete bilinear equations, τ ( ψ ) does satisfy the dAKP and dBKP equationssimultaneously. Theorem 4. The τ function τ ( ψ ) defined in (39) is a solution of the dAKP equation (1) aswell as the dBKP equation (2) .Proof. In addition to ψ , we introduce N -th order column vectors ϕ, ω and θ that are composedof, respectively, (cf. [8]) ϕ i = γ + i ( a − p i ) − n ( b + p i ) m ( c + p i ) l + γ − i ( a + p i ) − n ( b − p i ) m ( c − p i ) l , (40a) ω i = γ + i ( a + p i ) n ( b − p i ) − m ( c + p i ) l + γ − i ( a − p i ) n ( b + p i ) − m ( c − p i ) l , (40b) θ i = γ + i ( a + p i ) n ( b + p i ) m ( c − p i ) − l + γ − i ( a − p i ) n ( b − p i ) m ( c + p i ) − l . (40c)One can prove that (cf. [8]) τ ( ψ (( n, a ) , ( m, b ) , l )) = A n τ ( ϕ (( n, a ) , m, l )) = ( − N × n A n τ ( ψ (( − n, − a ) , m, l ))= B m τ ( ω ( n, ( m, b ) , l ) = ( − N × m B m τ ( ψ ( n, ( − m, − b ) , l )) , where A = Q Ni =1 ( a − p i ) and B = Q Ni =1 ( b − p i ). This means that τ ( ψ ) satisfies the extendedsymmetries (20) in n and m -direction. Meanwhile, note that due to the relation e ψ − ψ = ( a − c ) ψ ,the τ function (39a) can be equivalently constructed in terms of shifts of n , i.e. τ ( ψ ) = | ψ ( n, m, l ) , ψ ( n + 1 , m, l ) , ψ ( n + 2 , m, l ) , · · · , ψ ( n + N − , m, l ) | , (41)(see Eq.(2.24) in [8]). With this notation, τ ( ψ ( n, m, ( l, c ))) = C l τ ( θ ( n, m, ( l, c )) = ( − N × l C l τ ( ψ ( n, m, ( − l, − c ))) , where C = Q Ni =1 ( c − p i ). This gives the extended symmetries (20c). Thus, due to the gaugeproperty of Hirota’s discrete bilinear equations, (39) allows symmetries (20) and consequentlyprovides a solution for both the dAKP and dBKP equations.9ultiple pole solutions of the deformed dAKP equation (17b) is given by τ ( ψ ) in the Caso-ratian form (39a), but where ψ = (39 b ) | i =1 , ψ j = 1( j − ∂ j − p ψ , ( j = 2 , , · · · ) . (42)This can be found in Theorem 1 in [21]. We claim that τ ( ψ ) with (42) satisfies the extendedsymmetries (20) and then it solves the dAKP and the dBKP as well. To elaborate this, weintroduce lower triangular Toeplitz matrix (LTTM) T = t · · · t t · · · t N − t N − · · · t t N − t N − · · · t t (43)and note that such a matrix can be generated by some function f ( p ) via taking t j = 1 j ! ∂ jp f ( p ) , j = 0 , , · · · . (44)For convenience, by T [ f ( p )] we denote a LTTM generated by the function f ( p ) via (44). Wealso introduce new auxiliary vectors ϕ, ω and θ by ϕ = (40 a ) | i =1 , ϕ j = 1( j − ∂ j − p ϕ , (45a) ω = (40 b ) | i =1 , ϕ j = 1( j − ∂ j − p ω , (45b) θ = (40 a ) | i =1 , θ j = 1( j − ∂ j − p θ , (45c)for j = 2 , , · · · . Note that these vectors are connected to ψ composed of (42) by ψ = T [ a − p ] ϕ = T [ b − p ] ω = T [ c − p ] θ. (46)Then we have τ ( ψ (( n, a ) , ( m, b ) , ( l, c ))) = T [ a − p ] τ ( ϕ (( n, a ) , m, l )) = ( p − a ) n τ ( ψ (( − n, − a ) , m, l ))= T [ b − p ] τ ( ω ( n, ( m, b ) , l ) = ( p − b ) m τ ( ψ ( n, ( − m, − b ) , l ))= T [ c − p ] τ ( θ ( n, m, ( l, c )) = ( p − c ) l τ ( ψ ( n, m, ( − l, − c ))) , which are in the form of the extended symmetries (20).Let us sum up the above discussion by the following theorem. Theorem 5. The function τ ( ψ ) composed of (42) provides a multiple pole solution to the dAKPequation (1) as well as the dBKP equation (2) . The dAKP equation has polynomial solutions (cf. [16]). Explicit form of these solutions can bedescribed as the following. Lemma 3. [29] Let ψ +0 = ̺ (1 + p/a ) n (1 + p/b ) m (1 + p/c ) l (1 + p ) s , (47)10 here ̺ = 12 e − P ∞ j =1 ( − p ) jj γ j , γ j ∈ C . Then, ψ +0 = 12 ∞ X j =0 φ j p j , ( φ j = 2 j ! ∂ jp ψ +0 | p =0 )= 12 exp (cid:20) − ∞ X j =1 ( − p ) j j ˇ x j (cid:21) , where ˇ x j = x j + s, x j = na − j + mb − j + lc − j + γ j . (48) φ j = φ j ( n, m, l, s ) can be expressed in terms of x j by φ j = ( − j X || µ || = j ( − | µ | ˇ x µ µ ! (49) where µ = ( µ , µ , · · · ) , µ j ∈ { , , , · · · } , || µ || = ∞ X j =1 jµ j , | µ | = ∞ X j =1 µ j , µ ! = µ ! · µ ! · · · , ˇ x µ = (cid:18) ˇ x (cid:19) µ (cid:18) ˇ x (cid:19) µ · · · . The first few φ j are φ = 1 , φ = ˇ x , φ = 12 (ˇ x − ˇ x ) , φ = 16 (ˇ x − x ˇ x + 2ˇ x ) ,φ = 124 (ˇ x − x ˇ x + 8ˇ x ˇ x + 3ˇ x − x ) . Define φ = ( φ , φ , φ , · · · , φ N − ) T . (50) The Casoratian τ N ( φ ) = | φ ( n, m, l, , φ ( n, m, l, , φ ( n, m, l, , · · · , φ ( n, m, l, N − | (51) is a solution of the deformed dAKP equation (17c) (i.e. Eq.(3.15) in [29]). Note that τ N ( φ ) satisfies the superposition formula [27] τ N − ( E n i τ N +1 ) − τ N +1 ( E n i τ N − ) = 1 a i τ N ( E n i τ N ) , (52)and provides a discrete analogue of the remarkable Burchnall-Chaundy polynomials (cf. [25]).Let us look at symmetries of τ N ( φ ) The first three are τ ( φ ) = x , τ ( φ ) = x − x , τ ( φ ) = 145 x − x x + 15 x x − x , which only depend on { x i +1 } . For general N , it has been proved that (in Appendix C of [27]) Lemma 4. τ N ( φ ) depends only on { x , x , · · · , x N − } . Thus, from the definition (48) of x j , we immediately find that τ N ( φ ) has symmetries (16).This then leads to polynomial solutions of the dBKP equation. Theorem 6. The τ function τ ( φ ) defined by (51) provides polynomial solutions for the dAKPequation (1) as well as the dBKP equation (2) . Elliptic case Eq.(2.51) in [26] is a version of dAKP equation ready for elliptic solitons. It is written as E . = Φ − ba be τ τ + Φ − cb b τ e τ + Φ − ac e τ b τ = 0 , (53)where Φ a ( b ) = Φ ba = σ ( a + b ) σ ( a ) σ ( b ) . (54)Here and below, σ ( z ), ζ ( z ) and ℘ ( z ) are the Weierstrass functions. Eq.(53) can also be writtenas σ ( c ) σ ( a − b ) be τ τ + σ ( a ) σ ( b − c ) b τ e τ + σ ( b ) σ ( c − a ) e τ b τ = 0 . (55)Note that this is similar to (3), not to (1).The following τ function is given as an elliptic soliton solution of the dAKP equation (53), [26] τ = σ ( ξ ) | I + M| (56)where ξ = an + bm + cl + ξ , ξ ∈ C , (57) M = ( M ij ) N × N , M ij = ρ i M ij ν j , (58a) ρ i = (Φ a ( − κ i )) n (Φ b ( − κ i )) m (Φ c ( − κ i )) l e ζ ( ξ ) κ i ρ ( κ i ) , (58b) ν j = (Φ a ( κ ′ j )) − n (Φ b ( κ ′ j )) − m (Φ c ( κ ′ j )) − l e ζ ( ξ ) κ ′ j ν ( κ ′ j ) , (58c) M ij = Φ ξ ( κ i + κ ′ j )e − ζ ( ξ )( κ i + κ ′ j ) , (58d)and κ i , κ ′ j ∈ C for i, j = 1 , , · · · , N . τ function To get a τ function that allows symmetries (16), we take κ ′ j = κ j , ρ ( κ j ) = ν ( κ j ) , ( j = 1 , , · · · , N ) . (59)Obviously, ξ ( − n, − a ) = ξ ( n, a ). In addition, with (59) we have M ij = S i Φ ξ ( κ i + κ j ) T j , (60a)where S i = (Φ a ( − κ i )) n (Φ b ( − κ i )) m (Φ c ( − κ i )) l ρ ( κ i ) , ( i = 1 , , · · · , N ) , (60b) T j = (Φ a ( κ j )) − n (Φ b ( κ j )) − m (Φ c ( κ j )) − l ρ ( κ j ) , ( j = 1 , , · · · , N ) . (60c)Noticing that S i ( − n, − a ) = A ni S i ( n, a ) , T i ( − n, − a ) = A − ni T i ( n, a ) , where A i = − σ ( a ) σ ( κ i ) σ ( κ i + a ) σ ( κ i − a ) = 1 ℘ ( κ i ) − ℘ ( a ) , we then have τ ( − n, − a ) = σ ( ξ ) | I + M| ( − n, − a ) = σ ( ξ )Det[Diag( A n , A n , · · · , A nN )( I + M ) ( n,a ) Diag( A − n , A − n , · · · , A − nN )]= σ ( ξ ) | I + M| ( n,a ) = τ ( n, a ) . Since the symmetries w.r.t. m and l can be proved similarly, one can conclude that12 emma 5. The function τ = σ ( ξ ) | I + M| , where M ij are defined in (60) , allows symmetries (16) . Now that the dAKP τ function defined in Lemma 5 allows symmetries (16), as the counterpartsof (17), from (53) we have deformations E . = Φ − cb be τ τ + Φ ac be τ τ − Φ ba e τ b τ = 0 , (61a) E . = Φ − ac be τ τ + Φ ba b τ e τ − Φ cb be τ τ = 0 , (61b) E . = Φ − ba be τ τ + Φ cb e τ b τ − Φ ac b τ e τ = 0 , (61c)where we have made use of Φ ba = Φ ab = − Φ − b − a . To derive a dBKP equation with ellipticcoefficients, multiplying Φ − ac Φ − ba , Φ − ba Φ − cb , Φ − cb Φ − ac to the three equations in (61), respectively,and summing them together, we getΦ − ac Φ − ba × E + Φ − ba Φ − cb × E + Φ − cb Φ − ac × E = 3Φ − ba Φ − cb Φ − ac be τ τ + Φ − ba (cid:0) Φ ac Φ − ac − Φ cb Φ − cb (cid:1) be τ τ + Φ − cb (cid:16) Φ ba Φ − ba − Φ ac Φ − ac (cid:17) b τ e τ + Φ − ac (cid:16) Φ cb Φ − cb − Φ ba Φ − ba (cid:17) e τ b τ = 0 , (62)i.e. 3Φ − ba Φ − cb Φ − ac be τ τ + Φ − ba (2 ℘ ( c ) − ℘ ( a ) − ℘ ( b )) be τ τ + Φ − cb (2 ℘ ( a ) − ℘ ( b ) − ℘ ( c )) b τ e τ + Φ − ac (2 ℘ ( b ) − ℘ ( a ) − ℘ ( c )) e τ b τ = 0 , (63)where we have made use of Φ yx Φ − yx = ℘ ( x ) − ℘ ( y ) . (64)Finally, multiplying ( ℘ ( a ) + ℘ ( b ) + ℘ ( c )) to the dAKP (53) and then adding it to the aboveequation, we get F . = Φ − ba Φ − cb Φ − ac be τ τ + Φ − ba ℘ ( c ) be τ τ + Φ − cb ℘ ( a ) b τ e τ + Φ − ac ℘ ( b ) e τ b τ = 0 , (65)which is a dBKP equation with elliptic coefficients. Noting that3 F = Φ − ac Φ − ba × E + Φ − ba Φ − cb × E + Φ − cb Φ − ac × E + ( ℘ ( a ) + ℘ ( b ) + ℘ ( c )) × E , (66)we immediately have the following. Theorem 7. Both the dAKP equation (53) and dBKP equation (65) with elliptic coefficientsallow a solution τ defined in Lemma 5. By means of the transformation like (7), we can transform (65) to be of arbitrary nonzerocoefficients. We have the following. Theorem 8. Substituting τ = A ml B nl C nm × f (67a) into (65) , where A = ℘ ( b ) ℘ ( c ) ℘ ( a )(Φ cb ) , B = ℘ ( c ) ℘ ( a ) ℘ ( b )(Φ ac ) , C = ℘ ( a ) ℘ ( b ) ℘ ( c )(Φ ba ) , (67b) one can transform Eq. (65) to the following dBKP equation, Φ − ba Φ − cb Φ − ac be f f + Φ − ba Φ cb Φ ac be f f + Φ − cb Φ ba Φ ac b f e f + Φ − ac Φ ba Φ cb e f b f = 0 . (68)13 lliptic solutions of this dBKP equation are given by f = A − ml B − nl C − nm σ ( ξ ) | I + M| , (69) where M ij are defined in (60) and A, B, C are given in (67b) . Remark 3. E i ( i = 0 , , , can be expressed as linear combinations of any two elements of {E , E , E , E } , for example, Φ ba E = − Φ − ac E + Φ − cb E , Φ ba E = − Φ cb E − Φ ac E . (70) This also means there are alternative expressions of (66) , e.g. Φ ba F = − Φ − ac ℘ ( b ) E + Φ − cb ℘ ( a ) E , F = ℘ ( a ) E + Φ − ac Φ − ba E . Note that to obtain (70) we need to make use of a special case ( v = 0) of the well-known identity σ ( x + y ) σ ( x − y ) σ ( u + v ) σ ( u − v )= σ ( x + u ) σ ( x − u ) σ ( y + v ) σ ( y − v ) − σ ( x + v ) σ ( x − v ) σ ( y + u ) σ ( y − u ) . (71)Finally, let us back to the equation dAKP (55) and dBKP (65). Both of them have thesimplest solution τ = σ ( ξ ) where ξ is given in (57). In this case (55) is related to the identity(71) by (cf. [14, 26]) x = 12 ( a − b + c ) , y = 12 ( c − a + b ) , u = ξ + 12 ( a + b + c ) , v = 12 ( a + b − c ) , or, equivalently, a = x + v, b = y + v, c = x + y, ξ = u − x − y − v. (72)When τ = σ ( ξ ), since the three equations in (61) that are used to derive (65) are essentiallythe identity (71) with reparameters of x, y, u and v , we can substitute (72) into the dBKP (65)where τ = σ ( ξ ), and we arrive at the the following equality σ ( x − y ) σ ( v − x ) σ ( y − v ) σ ( u + x + y + v ) σ ( u − x − y − v )+ ℘ ( x + v ) σ ( v + x ) σ ( y + v ) σ ( x + y ) σ ( v − x ) σ ( u − y ) σ ( u + y )+ ℘ ( y + v ) σ ( y + v ) σ ( v + x ) σ ( x + y ) σ ( y − v ) σ ( u − x ) σ ( u + x )+ ℘ ( x + y ) σ ( x + y ) σ ( v + x ) σ ( y + v ) σ ( x − y ) σ ( u − v ) σ ( u + v )= 0 . (73)The latter elliptic identity is a consequence of the derivation of the dBKP equation, relyingon the fact that τ = σ ( ξ ) is a solution, and obtained by substituting (72) and a − b = x − y, b − c = v − x, c − a = y − v,a + b = v + ( x + y + v ) , a + c = x + ( x + y + v ) , b + c = y + ( x + y + v ) ,τ = σ ( ξ ) = σ ( u − ( x + y + v )) , be τ = σ ( u + ( x + y + v )) , e τ = σ ( u − y ) , b τ = σ ( u − x ) , τ = σ ( u − v ) , b τ = σ ( u + y ) , e τ = σ ( u + x ) , be τ = σ ( u + v ) . The identity (73) demonstrates that at a basic level, the discrete equations can be interpretedas addition formulae, albeit of a special type, for the relevant functions; in the present case forthe Weierstrass σ and ℘ functions. 14 Concluding remarks We have shown that under special (i.e. reflection) symmetries (16) the dBKP equation (2) canbe expressed as a linear combination of the dAKP equation (1) and their reflected symmetricforms (17). This leads to a common subset of solution spaces of the dAKP equation (1) and thedBKP equation (2), which is different from the Pfaffian type link (11) by coordinates reduction.As argued earlier, in a sense such solutions are reducible, as they obey simultaneously two dif-ferent partial difference equations, each of which allow for in principle different solution classes.Nonetheless, we conjecture that these solutions give an insight into the elliptic parametrisationof the dBKP equation which was unkown hitherto.We also checked the case of 4D equations, comprising the higher-order equations in therelevant dKP hierarchies. Since the 4D dAKP (22) and 4D dBKP (24) are direct results ofthe 4D consistency of the dAKP and dBKP, respectively, the two 4D lattice equations allowsymmetric τ function solutions as well. In addition, the 4D dBKP is connected to the 4D dAKPas a linear combination of the latter and its symmetric deformations.It is also remarkable that the plane wave factors entering in the symmetric τ functions inSec.3 (e.g. (39b)) coincide with the plane wave factors, and their corresponding multidimen-sional extensions, of the ABS list of lattice equations, [1], in the parametrisation that allowstheir uniform treatment of soliton solutions, cf. [15, 28]. This explains why the dAKP and itsreflected symmetric forms frequently (sometimes simultaneously) appear in the bilinearisationsof the ABS equations, e.g. (4.7) in [3], (5.20a,b) in [8], and (3.15) in [29]. This also implies apossible yet uncovered link between the dBKP equation and ABS lattice equations. Finally, inSec.4 we explored the elliptic version of the dBKP equation and we obtained a parametrisationof the dBKP equation (65) with elliptic coefficients and its gauge equivalent form (68). Espe-cially in this elliptic case it would be interesting to establish whether there are non-symmetricsolutions of the KP equation that would obey a relation of the type (11). This will be a subjectfor future investigations. Acknowledgments This project is supported by the NSF of China (Nos.11875040 and 11631007) and ShanghaiSailing Program (No. 20YF1433000). References [1] V.E. Adler, A.I. Bobenko, Yu.B. Suris, Classification of integrable equations on quad-graphs. The consistency approach, Commun. Math. Phys., 233 (2003) 513-543.[2] V.E. Adler, A.I. Bobenko, Yu.B. Suris, Classification of integrable discrete equations ofoctahedron type, Int. Math. Res. Not., 2012 (2012) 1822-1889.[3] J. Atkinson, J. Hietarinta, F.W. Nijhoff, Soliton solutions for Q3, J. Phys. A: Math. Theor.,41 (2008) 142001 (11pp).[4] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equationsIV. A new hierarchy of soliton equations of KP-type, Phys. D, 4 (1982) 343-365.[5] E. Date, M. Kashiwara, T. Miwa, Vertex pperators and τ functions. Transformation groupsfor soliton equations. II, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981) 387-393.[6] W. Fu, F.W. Nijhoff, Direct linearizing transform for three-dimensional discrete inte-grable systems: the lattice AKP, BKP and CKP equations, Proc. R. Soc. A, 473 (2017)No.20160915 (22pp). 157] J. Hietarinta, N. Joshi, F.W. Nijhoff, Discrete Systems and Integrablity, Camb. Univ. Press,Cambridge, 2016.[8] J. Hietarinta, D.J. Zhang, Soliton solutions for ABS lattice equations. II. Casoratians andbilinearization, J. Phys. A: Math. Theor., 42 (2009) 404006 (30pp).[9] R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan, 50 (1981)3785-3791.[10] R. Hirota, Soliton solutions to the BKP equation. I. the Pfaffian technique, J. Phys. Soc.Japan, 58 (1989) 2285-2296.[11] R. Hirota, Solutions to discrete soliton equations, RIMS Kˆokyˆuroku Bessatsu, B47 (2014)97-115.[12] T. Miwa, On Hirota’s difference equations, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982)9-12.[13] T. Miwa, M. Jimbo, E. Date, Solitons: Differential equations, symmetries and infinitedimensional algebras, Camb. Univ. Press, Cambridge, 2000.[14] F.W. Nijhoff, J. Atkinson, Elliptic N -soliton solutions of ABS lattice equations, Int. Math.Res. Not., 2010 (2010) 3837-3895.[15] F.W. Nijhoff, J. Atkinson, J. Hietarinta, Soliton solutions for ABS lattice equations. I.Cauchy matrix approach, J. Phys. A: Math. Theor., 42 (2009) 404005 (34pp).[16] J.J.C. Nimmo, Darboux transformations and the discrete KP equation, J. Phys. A: Math.Gen., 30 (1997) 8693-8704.[17] J.J.C. Nimmo, Darboux transformations for discrete systems, Chaos, Solitons and Fractals,11 (2000) 115-120.[18] J.J.C. Nimmo, W.K. Schief, Non-linear superposition principles associated with themoutard transformation: An integrable discretization of a (2+1) dimensional sine-Gordonsystem, Proc. R. Soc. London A, 453 (1997) 255-279.[19] Y. Ohta, R. Hirota, S. Tsujimoto, T. Imai, Casorati and discrete Gram type determinantrepresentations of solutions to the discrete KP hierarchy, J. Phys. Soc. Japan, 62 (1993)1872-1886.[20] M. Sato, Soliton equations as dynamical systems on an infinite dimensional Grassmannmanifolds, RIMS Kokyuroku Kyoto Univ., 439 (1981) 30-46,[21] Y. Shi, D.J. Zhang, Rational solutions of the H3 and Q1 models in the ABS lattice list,Symmetry, Integrability Geom.: Meth. Appl., 7 (2011) No.046 (11pp).[22] J. Sylvester, Sur l’equation en matrices px = xq, C. R. Acad. Sci. Paris., 99 (1884) 67-71,115-116.[23] S. Tsujimoto, R. Hirota, Pfaffian representation of solutions to the discrete BKP hhierarchyin bilinear form, J. Phys. Soc. Japan, 65 (1996) 2797-2806.[24] V.E. Vekslerchik, Solitons of the (2+2)-dimensional Toda lattice, J. Phys. A: Math. Theor.,52 (2019) 045202 (11pp).[25] A.P. Veselov, R. Willox, Burchnall-Chaundy polynomials and the Laurent phenomenon, J.Phys. A: Math. Theor., 48 (2015) 205201 (15pp).1626] S. Yoo-Kong, F.W. Nijhoff, Elliptic ( N, N ′ )-soliton solutions of the lattice Kadomtsev-Petviashvili equation, J. Math. Phys., 54 (2013) 043511 (20pp).[27] D.D. Zhang, D.J. Zhang, Rational solutions to the ABS list: Transformation approach,Symmetry Integrability Geom.: Meth. Appl., 13 (2017) 078 (24pp).[28] D.J. Zhang, S.L. Zhao, Solutions to ABS lattice equations via generalized Cauchy matrixapproach, Stud. Appl. Math., 131 (2013) 72-103.[29] S.L. Zhao, D.J. Zhang, Rational solutions to Q3 δδ