Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation

Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation K enji K ajiw ara Faculty of Mathematics, K yushu Univ ersity , 6-10- 1 Hakozaki, Fukuok a 812- 8581 , Japan Y asuhir o O ht a Departmen t of Mathematics, K obe University , Rokko, K obe 657 -8501 , Japan Abstract Casorati determinant solution to the non-autonomou s discrete KdV equation is constructed by using the bilinear formalism. W e pres ent three di ff erent bili near formulations which ha ve di ff erent origins. 1 Introd uction In this article, we consider the following partial di ff erence equation 1 a m + 1 b n + 1 ! v m n + 1 − 1 a m + 1 + 1 b n ! v m + 1 n = 1 a m − 1 b n ! 1 v m n − 1 a m + 1 − 1 b n + 1 ! 1 v m + 1 n + 1 , (1) where m , n are the discrete indepen dent variables, v m n is the dependent variable on th e lattice site ( m , n ), and a m , b n are arbitrary fu nctions of m and n , re spectiv ely . Recently eq .(1) h as b een d eriv ed by Matsuu ra[1] as the equation o f motion of discrete curves o n the centro-a ffi n e p lane. In this context, v m n is related to the curvature and it is essential that a m and b n depend on m and n , respectively . For analyzing th e mo tion of discrete cu rve, constructing solutions o f eq.(1) explicitly is an interesting subject. W e c all eq.( 1) the non- autono mous discrete KdV equ ation, for if a m and b n are constants, e.g. a m = a , b n = b , eq.(1) reduces to the discrete KdV equation [2, 3 ] 1 a + 1 b ! v m n + 1 − 1 a + 1 b ! v m + 1 n = 1 a − 1 b ! 1 v m n − 1 a − 1 b ! 1 v m + 1 n + 1 , (2) or v m n + 1 − v m + 1 n = a − b a + b       1 v m + 1 n + 1 − 1 v m n       . (3) Since th e con stants a and b co rrespond to the lattice intervals of m and n , resp ectiv ely , eq.(1) can b e also regar ded as the discrete KdV equation on inhomo geneou s lattice. The non-auton omou s v ersion of discrete i ntegrable systems on tw o-d imensional lattice ha ve not been in vestigated well, although those on three-d imensional lattice, such as the Hirota-Miwa (discrete KP) equation or the discrete two- dimensiona l T oda lattice equation , have been studied well toge ther with their solutio ns[4, 5, 6, 7, 8]. Let us explain a reason taking eq.(1) as an examp le. The auton omou s version eq.(2) can be transfo rmed into so-called the bilinear equation by suitable depen dent variable transformation . Th e bilinear equation is re gard ed as a reduction of the Hirota- Miwa equatio n[9, 10], which is well-kn own to admit various types of exact so lutions, such a s so liton solution s[3, 9], rational solutions expr essible in terms o f th e Schur fun ctions[10, 11], or per iodic solutions that are wr itten in term s of the Riemann theta fu nctions[12]. Theref ore one can obta in solutions to th e d iscrete KdV equatio n (2) by app lying the r eduction pr ocedur e to those fo r the Hirota-Miwa equ ation. Now , the Hirota- Miwa equation and its solutions can b e g eneralized to non-au tonomo us case in a straightfor ward manner . Howe ver , it is shown that o ne cann ot apply the redu ction proced ure to the non -auton omous Hirota-Miwa equatio n con sistently . Moreover , eq. (1) canno t be put into b ilinear equa tion by the proced ure similar to the auto nomo us case b ecause o f the non-au tonom ous pro perty . Therefo re it was not clear how to construct solutions to the two-dimen sional non-a utonom ous discrete integrable systems systematically . In this article, we construct the Casorati determinant solution to the non- autono mous discr ete KdV equation (1) by using the bilinear formalism. W e present three di ff erent bilinearizatio ns: T he first one can be derived by the reduction of n on-au tonomo us d iscrete KP hierarchy with a new techniqu e. The second one is the b ilinearization ob tained by introdu cing certain au xiliary τ fun ction which has a similar stru cture to the on es that appear ed in the study of R I and R I I biortho gonal rational fun ctions[13, 14]. The th ird bilineariza tion is through the use of no n-auton omou s potential discrete KdV equation. 1 This article is organized as fo llows. I n Section 2 we revie w the bilin earization o f the d iscrete KdV equ ation (2), and d iscuss briefly why th e similar calculation f ails for the non-au tonomo us ca se. In Sec tion 3 we discuss the bilinearization s of eq.(1) and construct th e Casorati determinan t solution. Finally , concluding re marks are given in Section 4. 2 Bilinearization of the discr ete KdV equation The discrete KdV equation (2) can be transformed to the bilinear equation 1 a + 1 b ! τ m n + 1 τ m + 1 n − 1 − 1 a − 1 b ! τ m n − 1 τ m + 1 n + 1 = 2 b τ m n τ m + 1 n , (4) by the depend ent variable transformation v m n = τ m n + 1 τ m + 1 n τ m n τ m + 1 n + 1 . (5) In fact, substituting eq.(5) into eq.(2) we ha ve 1 a + 1 b ! τ m n + 1 τ m + 1 n − 1 τ m + 2 n − 1 a + 1 b ! τ m + 2 n − 1 τ m n τ m + 1 n + 1 = 1 a − 1 b ! τ m n − 1 τ m + 1 n + 1 τ m + 2 n − 1 a − 1 b ! τ m + 2 n + 1 τ m n τ m + 1 n − 1 . (6) Interchan ging the second term of the left hand side and the first term of the right hand side, and di vidin g the both sides by τ m + 2 n τ m n τ m + 1 n we get  1 a + 1 b  τ m n + 1 τ m + 1 n − 1 −  1 a − 1 b  τ m n − 1 τ m + 1 n + 1 τ m n τ m + 1 n =  1 a + 1 b  τ m + 1 n + 1 τ m + 2 n − 1 −  1 a − 1 b  τ m + 1 n − 1 τ m + 2 n + 1 τ m + 1 n τ m + 2 n . (7) Equation (7) can be decoupled as 1 a + 1 b ! τ m n + 1 τ m + 1 n − 1 − 1 a − 1 b ! τ m n − 1 τ m + 1 n + 1 = α ( n ) τ m n τ m + 1 n , (8) since the right hand side of eq.(7) is obtained from the left hand side by shifting m to m + 1 . Here α ( n ) is an arbitrar y function in n , which can be ab sorbed by suitable gau ge transformation on τ m n . W e o btain eq.(4) by ch oosing α ( n ) = 2 b so that τ m n = 1 is a solution . The bilinear equation (4) can be obtained by applying the reduction to the Hirota-Miwa equation a 1 ( a 2 − a 3 ) τ ( l 1 + 1 , l 2 , l 3 ) τ ( l 1 , l 2 + 1 , l 3 + 1) + a 2 ( a 3 − a 1 ) τ ( l 1 , l 2 + 1 , l 3 ) τ ( l 1 + 1 , l 2 , l 3 + 1) + a 3 ( a 1 − a 2 ) τ ( l 1 , l 2 , l 3 + 1) τ ( l 1 + 1 , l 2 + 1 , l 3 ) = 0 , (9) where a 1 , a 2 , a 3 are arbitrary constants. In fact, imposing the condition τ ( l 1 + 1 , l 2 + 1 , l 3 ) ≎ τ ( l 1 , l 2 , l 3 ) , (10) where ≎ mean s the eq uiv alence up to gauge tran sformation , using eq.(10) to supp ress the l 1 depend ence and p utting a 1 = − a 2 , eq.(9) yields − ( a 2 − a 3 ) τ ( l 2 − 1 , l 3 ) τ ( l 2 + 1 , l 3 + 1) + ( a 2 + a 3 ) τ ( l 2 + 1 , l 3 ) τ ( l 2 − 1 , l 3 + 1) − 2 a 3 τ ( l 2 , l 3 + 1) τ ( l 2 , l 3 ) = 0 , (11) which is equiv alent to eq. ( 4) with l 2 = n , l 3 = m , a 2 = b , a 3 = a and τ ( l 2 , l 3 ) = τ m n . Now let us co nsider the non- autono mous case. W e sh ow that neither direct b ilinearization nor r eduction fro m the non-au tonomo us Hirota- Miwa equation work successfully for this case. First, substituting eq.(5) into eq.(1) and doing 2 the same calculation as above, we arrive at the following equation  1 a m + 1 b n  τ m n + 1 τ m + 1 n − 1 −  1 a m − 1 b n − 1  τ m n − 1 τ m + 1 n + 1 τ m n τ m + 1 n =  1 a m + 1 + 1 b n − 1  τ m + 1 n + 1 τ m + 2 n − 1 −  1 a m + 1 − 1 b n  τ m + 1 n − 1 τ m + 2 n + 1 τ m + 1 n τ m + 2 n , (12) which cannot be decoupled into the bilinear equation because of n dependence of the coe ffi cien ts. Theref ore naive bilinearization fails for the non-auton omou s cas e. Secondly , let us consider the reduction from the non-a utonom ous Hirota-Miwa equatio n[7, 8] a 1 ( l 1 )( a 2 ( l 2 ) − a 3 ( l 3 )) τ ( l 1 + 1 , l 2 , l 3 ) τ ( l 1 , l 2 + 1 , l 3 + 1) + a 2 ( l 2 )( a 3 ( l 3 ) − a 1 ( l 1 )) τ ( l 1 , l 2 + 1 , l 3 ) τ ( l 1 + 1 , l 2 , l 3 + 1) + a 3 ( l 3 )( a 1 ( l 1 ) − a 2 ( l 2 )) τ ( l 1 , l 2 , l 3 + 1) τ ( l 1 + 1 , l 2 + 1 , l 3 ) = 0 , (13) where a i ( l i ) ( i = 1 , 2 , 3) are arbitrary f unctions. Imposing the co ndition (10) on eq.(13) and suppressing the l 1 - depend ence, we obtain two di ff erent bilinear equations a 1 ( l 1 )( a 2 ( l 2 ) − a 3 ( l 3 )) τ ( l 2 − 1 , l 3 ) τ ( l 2 + 1 , l 3 + 1) + a 2 ( l 2 )( a 3 ( l 3 ) − a 1 ( l 1 )) τ ( l 2 + 1 , l 3 ) τ ( l 2 − 1 , l 3 + 1) + a 3 ( l 3 )( a 1 ( l 1 ) − a 2 ( l 2 )) τ ( l 2 , l 3 + 1) τ ( l 2 , l 3 ) = 0 , (14) a 1 ( l 1 − 1)( a 2 ( l 2 − 1) − a 3 ( l 3 )) τ ( l 2 − 1 , l 3 ) τ ( l 2 + 1 , l 3 + 1) + a 2 ( l 2 − 1)( a 3 ( l 3 ) − a 1 ( l 1 − 1) ) τ ( l 2 + 1 , l 3 ) τ ( l 2 − 1 , l 3 + 1) + a 3 ( l 3 )( a 1 ( l 1 − 1) − a 2 ( l 2 − 1) ) τ ( l 2 , l 3 + 1) τ ( l 2 , l 3 ) = 0 . (15) Since those two equations should be equiv alent, the coe ffi cien ts must satisfy a 1 ( l 1 )( a 2 ( l 2 ) − a 3 ( l 3 )) a 1 ( l 1 − 1)( a 2 ( l 2 − 1) − a 3 ( l 3 )) = a 2 ( l 2 )( a 3 ( l 3 ) − a 1 ( l 1 )) a 2 ( l 2 − 1)( a 3 ( l 3 ) − a 1 ( l 1 − 1)) = a 3 ( l 3 )( a 1 ( l 1 ) − a 2 ( l 2 )) a 3 ( l 3 )( a 1 ( l 1 − 1) − a 2 ( l 2 − 1)) , (16) which yields a 3 ( l 3 ) 1 a 1 ( l 1 ) a 2 ( l 2 − 1) − 1 a 1 ( l 1 − 1) a 2 ( l 2 ) ! + 1 a 1 ( l 1 − 1) − 1 a 1 ( l 1 ) ! − 1 a 2 ( l 2 − 1) − 1 a 2 ( l 2 ) ! = 0 . (17) Since th is sho uld ho ld for any a ( l 3 ) fo r all l 3 , we d educe that a ( l 1 ) an d a ( l 2 ) mu st be co nstants. T his imp lies that it is not possible to im pose the cond ition (10) on the non-a utonom ous Hirota-Miwa equation (13) consistently , unless it is reduced to the autonom ous case. 3 Bilinearizations of the non-autonomous discrete KdV equation 3.1 Reduction fr om the discrete KP hierar chy The non-a utonom ous discrete KP hierarchy in the biline ar form is expressed as[7, 3]             1 a i 1 ( l i 1 ) a i 1 ( l i 1 ) 2 · · · a i 1 ( l i 1 ) m − 2 a i 1 ( l i 1 ) m − 2 τ i 1 τ ˆ i 1 1 a i 2 ( l i 2 ) a i 2 ( l i 2 ) 2 · · · a i 2 ( l i 2 ) m − 2 a i 2 ( l i 2 ) m − 2 τ i 2 τ ˆ i 2 . . . . . . . . . . . . . . . 1 a i m ( l i m ) a i m ( l i m ) 2 · · · a i m ( l i m ) m − 2 a i m ( l i m ) m − 2 τ i m τ ˆ i m             = 0 , (18) 3 where { i 1 , . . . , i m } ⊂ { 1 , . . . , n } , τ i k and τ ˆ i k ( k = 1 , . . . , m ) are given by τ i k = T i k τ , (19) τ ˆ i k = T i 1 T i 2 · · · T i k − 1 T i k + 1 · · · T i m τ , (20) respectively , a ν ( l ν ) are arb itrary functio ns in l ν for each ν , and n and m ar e arb itrary in tegers satisfying n ≥ m ≥ 3. Here T i is the shift operator of l i defined by T i τ ( l 1 , l 2 , · · · , l n ) = τ ( l 1 , l 2 , · · · , l i − 1 , l i + 1 , l i + 1 , · · · , l n ) . (21) The simplest equation in the hierarch y ( m = 3 ) is the non-auto nomou s Hirota-Miwa equation a i ( l i )( a j ( l j ) − a k ( l k )) τ ( l i + 1 , l j , l k ) τ ( l i , l j + 1 , l k + 1) + a j ( l j )( a k ( l k ) − a i ( l i )) τ ( l i , l j + 1 , l k ) τ ( l i + 1 , l j , l k + 1) + a k ( l k )( a i ( l i ) − a j ( l j )) τ ( l i , l j , l k + 1) τ ( l i + 1 , l j + 1 , l k ) = 0 , (22) where { i , j , k } ⊂ { 1 , . . . , n } an d we suppressed other indep endent variables. The Casorati d eterminant solution to the hierarchy can be written as τ ( l 1 , · · · , l n ) =              ϕ ( s ) 1 ( l 1 , · · · , l n ) ϕ ( s + 1) 1 ( l 1 , · · · , l n ) · · · ϕ ( s + N − 1) 1 ( l 1 , · · · , l n ) ϕ ( s ) 2 ( l 1 , · · · , l n ) ϕ ( s + 1) 2 ( l 1 , · · · , l n ) · · · ϕ ( s + N − 1) 2 ( l 1 , · · · , l n ) . . . . . . · · · . . . ϕ ( s ) N ( l 1 , · · · , l n ) ϕ ( s + 1) N ( l 1 , · · · , l n ) · · · ϕ ( s + N − 1) N ( l 1 , · · · , l n )              , (23) where ϕ ( s ) r ( r = 1 , . . . , N ) satisfy the linear eq uations ϕ ( s ) r ( l 1 , · · · , l ν + 1 , · · · , l n ) − ϕ ( s ) r ( l 1 , · · · , l ν , · · · , l n ) a ν ( l ν ) = ϕ ( s + 1) r ( l 1 , · · · , l ν , · · · , l n ) , (24) for ν = 1 , . . . , n . For example, the N -soliton solutio n is obtained by choosing ϕ ( s ) r as ϕ ( s ) r ( l 1 , · · · , l n ) = α r p s r n Y ν = 1 l ν − 1 Y i = i ν (1 + a ν ( i ) p r ) + β r q s r n Y ν = 1 l ν − 1 Y i = i ν (1 + a ν ( i ) q r ) , (25) where α r , β r , p r , q r ( r = 1 , . . . , N ) are arbitr ary constants. Let u s consider the reduction to the non -autono mous discrete KdV equation . T he key idea is to specialize some of the independen t v ariables to be autonomo us by choosing the lattice intervals as constants and use them as auxiliary variables. The n one can use the autonomo us variables for the reduction procedure to get the non- autono mous discrete KdV equation. W e consider the four independen t v ariables k = l 1 , l = l 2 , m = l 3 , n = l 4 with the lattice intervals being δ = a 1 ( k ), ǫ = a 2 ( l ), a m = a 3 ( m ), b n = a 4 ( n ), respecti vely . W e note that δ and ǫ are co nstants, n amely , k and l are autonom ous v ariables. Then we have the following bilinear equation s from eq.(22): δ ( a m − b n ) τ ( k + 1 , l , m , n ) τ ( k , l , m + 1 , n + 1) + a m ( b n − δ ) τ ( k , l , m + 1 , n ) τ ( k + 1 , l , m , n + 1) + b n ( δ − a m ) τ ( k , l , m , n + 1) τ ( k + 1 , l , m + 1 , n ) = 0 , (26) ǫ ( a m − b n ) τ ( k , l + 1 , m , n ) τ ( k , l , m + 1 , n + 1) + a m ( b n − ǫ ) τ ( k , l , m + 1 , n ) τ ( k , l + 1 , m , n + 1) + b n ( ǫ − a m ) τ ( k , l , m , n + 1) τ ( k , l + 1 , m + 1 , n ) = 0 . (27) W e impose the condition τ ( k + 1 , l + 1 , m , n ) ≎ τ ( k , l , m , n ) . (28) This is achiev ed by imposing the condition on ϕ ( s ) r ( r = 1 , . . . , N ) as ϕ ( s ) r ( k + 1 , l + 1 , m , n ) ≎ ϕ ( s ) r ( k , l , m , n ) . (29) 4 For the case of soliton solutions, ϕ ( s ) r ( k , l , m , n ) is expressed as ϕ ( s ) r ( k , l , m , n ) = α r p s r (1 + δ p r ) k (1 + ǫ p r ) l m − 1 Y i = m 0 (1 + a i p r ) n − 1 Y j = n 0 (1 + b j p r ) + β r q s r (1 + δ q r ) k (1 + ǫ q r ) l m − 1 Y i = m 0 (1 + a i q r ) n − 1 Y j = n 0 (1 + b j q r ) . In order to satisfy eq.(29), one may take q r = − p r , δ = − ǫ , (30) so that ϕ ( s ) r ( k + 1 , l + 1 , m , n ) = (1 − ǫ 2 p 2 r ) ϕ ( s ) r ( k , l , m , n ) , τ ( k + 1 , l + 1 , m , n ) = N Y r = 1 (1 − ǫ 2 p 2 r ) τ ( k , l , m , n ) . (31) Then, suppressing the k -dep endenc e by using eq.(28), the bilinear equation s (26 ) and (27) are red uced to − ǫ ( a m − b n ) τ ( l , m , n ) τ ( l + 1 , m + 1 , n + 1) + a m ( b n + ǫ ) τ ( l + 1 , m + 1 , n ) τ ( l , m , n + 1) − b n ( ǫ + a m ) τ ( l + 1 , m , n + 1) τ ( l , m + 1 , n ) = 0 , ǫ ( a m − b n ) τ ( l + 1 , m , n ) τ ( l , m + 1 , n + 1) + a m ( b n − ǫ ) τ ( l , m + 1 , n ) τ ( l + 1 , m , n + 1) + b n ( ǫ − a m ) τ ( l , m , n + 1) τ ( l + 1 , m + 1 , n ) = 0 , respectively . By putting τ m n = τ ( l , m , n ) , σ m n = τ ( l + 1 , m , n ) , (32) the above bilinear equation s are re written as − ǫ ( a m − b n ) τ m n σ m + 1 n + 1 + a m ( b n + ǫ ) τ m n + 1 σ m + 1 n − b n ( ǫ + a m ) τ m + 1 n σ m n + 1 = 0 , (33) ǫ ( a m − b n ) σ m n τ m + 1 n + 1 + a m ( b n − ǫ ) τ m + 1 n σ m n + 1 + b n ( ǫ − a m ) τ m n + 1 σ m + 1 n = 0 , (34 ) respectively . Equations (33) and ( 34) can b e regarded as a b ilinearization of the n on-au tonomo us discrete KdV eq ua- tion (1). In fact, introduc ing the variables Ψ m n and v m n by Ψ m n = σ m n τ m n , (35a) v m n = τ m n + 1 τ m + 1 n τ m n τ m + 1 n + 1 , (35b) we obtain 1 b n − 1 a m ! 1 v m n Ψ m + 1 n + 1 − 1 ǫ + 1 b n ! Ψ m + 1 n + 1 ǫ + 1 a m ! Ψ m n + 1 = 0 , (36) 1 b n − 1 a m ! 1 v m n Ψ m n + 1 ǫ − 1 b n ! Ψ m n + 1 + 1 a m − 1 ǫ ! Ψ m + 1 n = 0 , (37) which are regarde d as th e au xiliary linear pro blem fo r the n on-au tonomo us discrete KdV equation . Eliminating Ψ n m by consider ing the compatibility cond ition we obtain the n on-au tonomo us discrete K dV equ ation (1). The N -soliton solution is giv en by τ m n =                ϕ ( s ) 1 ( m , n ) ϕ ( s + 1) 1 ( m , n ) · · · ϕ ( s + N − 1) 1 ( m , n ) ϕ ( s ) 2 ( m , n ) ϕ ( s + 1) 2 ( m , n ) · · · ϕ ( s + N − 1) 2 ( m , n ) . . . . . . · · · . . . ϕ ( s ) N ( m , n ) ϕ ( s + 1) N ( m , n ) · · · ϕ ( s + N − 1) N ( m , n )                , (38) 5 ϕ ( s ) r ( m , n ) = α r p s r m − 1 Y i = m 0 (1 + a i p r ) n − 1 Y j = n 0 (1 + b j p r ) + β r ( − p r ) s m − 1 Y i = m 0 (1 − a i p r ) n − 1 Y j = n 0 (1 − b j p r ) . (39) W e r emark th at we obtain the non-auto nomou s potential discrete modified Kd V eq uation fo r Ψ m n by eliminating v m n from eqs.(36) and (37). 3.2 Alternate b ilinearization There is anoth er interesting bilinearization to the non-au tonomo us discrete KdV equation (1). Let us consider the following bilinear equations, b n ( a m − 1 + a m ) κ m n + 1 τ m n − a m − 1 ( a m + b n ) τ m − 1 n + 1 τ m + 1 n + a m ( a m − 1 − b n ) τ m + 1 n + 1 τ m − 1 n = 0 , (40) b n ( a m − 1 − a m ) τ m + 1 n τ m − 1 n + 1 − a m ( a m − 1 − b n ) τ m n + 1 κ m n + a m − 1 ( a m − b n ) τ m n κ m n + 1 = 0 . (41) W e obtain eq.(1) by introdu cing v m n by eq.(35 b) and eliminating κ m n . The Casorati deter minant solution is gi ven by κ m n =                ψ ( s ) 1 ( m , n ) ψ ( s + 1) 1 ( m , n ) · · · ψ ( s + N − 1) 1 ( m , n ) ψ ( s ) 2 ( m , n ) ψ ( s + 1) 2 ( m , n ) · · · ψ ( s + N − 1) 2 ( m , n ) . . . . . . · · · . . . ψ ( s ) N ( m , n ) ψ ( s + 1) N ( m , n ) · · · ψ ( s + N − 1) N ( m , n )                , (42) where ψ ( s ) r ( m , n ) = λ r p s r (1 + a m p r ) m − 2 Y i = m 0 (1 + a i p r ) n − 1 Y j = n 0 (1 + b j p r ) + µ r ( − p r ) s (1 − a m p r ) m − 2 Y i = l 0 (1 − b i p r ) n − 1 Y j = n 0 (1 − b j p r ) , (43) λ r , µ r are arbitrary constants ( r = 1 , . . . , N ) and τ m n is giv en by eqs.(38) and (39). W e n ote that in the autonomo us case, κ m n reduces to τ m n , the bilin ear equa tion (40) yields eq.(4), an d eq .(41) becomes trivial, respecti vely . Second ly , because of the symmetry with r espect to m , n in eqs.(1) and (35b), the following bilinearization i s also po ssible: a m ( b n − 1 + b n ) θ m + 1 n τ m n − b n − 1 ( b n + a m ) τ m + 1 n − 1 τ m n + 1 + b n ( b n − 1 − a m ) τ m + 1 n + 1 τ m n − 1 = 0 , (44) a m ( b n − 1 − b n ) τ m n + 1 τ m + 1 n − 1 − b n ( b n − 1 − a m ) τ m + 1 n θ m n + b n − 1 ( b n − a m ) τ m n θ m + 1 n = 0 , (45) whose solution is expressed as θ m n =                φ ( s ) 1 ( m , n ) φ ( s + 1) 1 ( m , n ) · · · φ ( s + N − 1) 1 ( m , n ) φ ( s ) 2 ( m , n ) φ ( s + 1) 2 ( m , n ) · · · φ ( s + N − 1) 2 ( m , n ) . . . . . . · · · . . . φ ( s ) N ( m , n ) φ ( s + 1) N ( m , n ) · · · φ ( s + N − 1) N ( m , n )                , (46) φ ( s ) r ( m , n ) = λ r p s r (1 + b n p r ) m − 1 Y i = m 0 (1 + a i p r ) n − 2 Y j = n 0 (1 + b j p r ) + µ r ( − p r ) s (1 − b n p r ) m − 1 Y i = l 0 (1 − a i p r ) n − 2 Y j = n 0 (1 − b j p r ) . (47) W e a lso rem ark that th e similar structure in the above au xiliary τ functions has appeared in the study of R I and R I I biortho gonal f unctions[1 3 , 1 4]. Also, similar soliton type solutio n h as b een con structed f or the no n-auton omous discrete-time T oda lattice equatio n[15]. 6 W e can s how that τ m n and κ m n satisfy eqs.(40) and (4 1) by the techniqu e similar to that w as used in refs.[13, 14, 15]. Namely , by using the linea r relations am ong ϕ ( s ) r and ψ ( s ) r , we fir st constru ct such d i ff erence form ulas that express the determinan ts whose columns are app ropriately shifted by τ m n or κ m n . Then eqs. (41) a nd (44) are d erived from Pl ¨ ucker relations which are quadratic identities of determinan ts whose columns are shifted. From eqs.(39) and (43), we see that ϕ ( s ) r and ψ ( s ) r satisfy ϕ ( s ) r ( m + 1 , n ) − ϕ ( s ) r ( m , n ) = a m ϕ ( s + 1) r ( m , n ) , (48) ϕ ( s ) r ( m − 1 , n ) + a m ϕ ( s + 1) r ( m − 1 , n ) = ψ ( s ) r ( m , n ) , (49) ψ ( s ) r ( m , n ) − a m ψ ( s + 1) r ( m , n ) = (1 − a 2 m p 2 r ) ϕ ( s ) r ( m − 1 , n ) , (50) ϕ ( s ) r ( m , n + 1) − ϕ ( s ) r ( m , n ) = b n ϕ ( s + 1) r ( m , n ) . (51) W e introdu ce a notatio n τ m n = | 0 , 1 , · · · , N − 2 , N − 1 | , (52) where “ k ” den otes the column vector k m n =              ϕ ( s + k ) 1 ( m , n ) . . . ϕ ( s + k ) N ( m , n )              . (53) Then the fo llowing d i ff erence f ormulas are derived from eqs.(48)-(51) by the similar calcu lations to those gi ven in ref.[15]: − a m − 1 τ m − 1 n = | 0 , 1 , · · · , N − 3 , N − 2 , N − 2 m − 1 | , (54) − b n − 1 τ m n − 1 = | 0 , 1 , · · · , N − 3 , N − 2 , N − 2 n − 1 | , (55) A ( m ) − 1 a m τ m + 1 n =     0 , 1 , · · · , N − 3 , N − 2 , g N − 2 m + 1     , (56) − A ( m ) − 1 ( a m − 1 + a m ) σ m n =     0 , 1 , · · · , N − 3 , g N − 2 m + 1 , N − 2 m − 1     , (57) − ( a m − 1 − b n − 1 ) τ m − 1 n − 1 = | 0 , 1 , · · · , N − 3 , N − 2 n − 1 , N − 2 m − 1 | , (58) A ( m ) − 1 ( a m + b n − 1 ) τ m + 1 n − 1 =     0 , 1 , · · · , N − 3 , N − 2 n − 1 , g N − 2 m + 1     , (59) where e k m + 1 =              A 1 ( m ) − 1 ϕ ( s + k ) 1 ( m + 1 , n ) . . . A N ( m ) − 1 ϕ ( s + k ) N ( m + 1 , n )              , (60) A r ( m ) = 1 − a 2 m p 2 r ( r = 1 , . . . , N ) , A ( m ) = N Y r = 1 (1 − a 2 m p 2 r ) . (61) W e giv e the proof of the above formu las in the appendix. App lying the the di ff erence formulas to the Pl ¨ ucker relation 0 = | 0 , · · · , N − 3 , N − 2 , N − 2 n − 1 | ×     0 , · · · , N − 3 , N − 2 m − 1 , g N − 2 m + 1     − | 0 , · · · , N − 3 , N − 2 , N − 2 m − 1 | ×     0 , · · · , N − 3 , N − 2 n − 1 , g N − 2 m + 1     +     0 , · · · , N − 3 , N − 2 , g N − 2 m + 1     × | 0 , · · · , N − 3 , N − 2 n − 1 , N − 2 m − 1 | , (62) we obtain the bilinear equation (41). Equation (44) is derived by ap plying the di ff erence formulas (54), (55), (58) and − a m − 2 σ m − 1 n =     0 , 1 , · · · , N − 3 , N − 2 , d N − 2 m − 1     , (63) − ( a m − 2 − a m − 1 ) τ m − 2 n =     0 , 1 , · · · , N − 3 , N − 2 m − 1 , d N − 2 m − 1     , (64) − ( a m − 2 − b n − 1 ) σ m − 1 n − 1 =     0 , 1 , · · · , N − 3 , N − 2 n − 1 , d N − 2 m − 1     , (65) 7 where b k m − 1 =              ψ ( s + k ) 1 ( m − 1 , n ) . . . ψ ( s + k ) N ( m − 1 , n )              , (66) to the Pl ¨ uc ker relation, 0 = | 0 , · · · , N − 3 , N − 2 , N − 2 n − 1 | ×     0 , · · · , N − 3 , N − 2 m − 1 , d N − 2 m − 1     − | 0 , · · · , N − 3 , N − 2 , N − 2 m − 1 | ×     0 , · · · , N − 3 , N − 2 n − 1 , d N − 2 m − 1     +     0 , · · · , N − 3 , N − 2 , d N − 2 m − 1     × | 0 , · · · , N − 3 , N − 2 n − 1 , N − 2 m − 1 | . (67) 3.3 Reduction fr om the KP hierarch y throu gh the pote ntial f orm In this section, we consider the following di ff erence equation u m + 1 n + 1 − u m n = 1 a 2 m − 1 b 2 n ! 1 u m + 1 n − u m n + 1 , (68) which is closely related to eq.(1) as 1 a m − 1 b n ! 1 v m n = u m n + 1 − u m + 1 n . (69) The autonomo us version of eq.(68) is known as the potential discre te KdV equ ation[16]. W e call eq.(68) the no n- autonom ous potential discrete KdV equatio n. Casorati determinant solution to eq.(68) is gi ven by u m n = ρ m n τ m n − m − 1 X i = m 0 1 a i − n − 1 X j = n 0 1 b j , (70) where ρ m n =              ϕ ( s ) 1 ( m , n ) · · · ϕ ( s + N − 2) 1 ( m , n ) ϕ ( s + N ) 1 ( m , n ) ϕ ( s ) 2 ( m , n ) · · · ϕ ( s + N − 2) 2 ( m , n ) ϕ ( s + N ) 2 ( m , n ) . . . . . . · · · . . . ϕ ( s ) N ( m , n ) · · · ϕ ( s + N − 2) N ( m , n ) ϕ ( s + N ) N ( m , n )              , (71) and ϕ ( s ) r ( m , n ) ( r = 1 , . . . , N ), τ m n are defined by eqs.(39) and (38), respectively . Equation (68) is deriv ed from the following bilinear equation s for ρ m n and τ m n ρ m + 1 n τ m n + 1 − ρ m n + 1 τ m + 1 n = 1 a m − 1 b n !  τ m + 1 n τ m n + 1 − τ m + 1 n + 1 τ m n  , (72) ρ m + 1 n + 1 τ m n − ρ m n τ m + 1 n + 1 = 1 a m + 1 b n !  τ m + 1 n + 1 τ m n − τ m + 1 n τ m n + 1  , (73) throug h the dependent variable tran sformation (70). In particu lar , eq.( 69) a lso fo llows from eq .(72). Therefor e, we may regard eqs.(72) and (73) as yet another bilinearization of the non-autonom ous discrete KdV equa tion (1). W e can sh ow th at ρ m n and τ m n satisfy eqs. (72) and (73) as follo ws. Ap plying the di ff er ence formu las (5 4), (5 5), (58) and − ( a m − 1 ρ m − 1 n + τ m − 1 n ) = | 0 , 1 , · · · , N − 3 , N − 1 , N − 2 m − 1 | , (74 ) − ( b n − 1 ρ m n − 1 + τ m n − 1 ) = | 0 , 1 , · · · , N − 3 , N − 1 , N − 2 n − 1 | , (75) to the Pl ¨ uc ker relation 0 = | 0 , · · · , N − 3 , N − 2 , N − 1 | × | 0 , · · · , N − 3 , N − 2 n − 1 , N − 2 m − 1 | − | 0 , · · · , N − 3 , N − 2 , N − 2 n − 1 | × | 0 , · · · , N − 3 , N − 1 , N − 2 m − 1 | + | 0 , · · · , N − 3 , N − 2 , N − 2 m − 1 | × | 0 , · · · , N − 3 , N − 1 , N − 2 n − 1 | , (76) 8 we have e q. (72). Sim ilarly , we obtain eq.(7 3) by applying t he form ulas (55), (56), (59), (75) and A ( m ) − 1 ( a m ρ m + 1 n − τ m + 1 n ) = | 0 , · · · , N − 3 , N − 1 , g N − 2 m + 1 | , (77) to the Pl ¨ uc ker relation 0 = | 0 , · · · , N − 3 , N − 2 , N − 1 | ×     0 , · · · , N − 3 , N − 2 n − 1 , g N − 2 m + 1     − | 0 , · · · , N − 3 , N − 2 , N − 2 n − 1 | ×     0 , · · · , N − 3 , N − 1 , g N − 2 m + 1     +     0 , · · · , N − 3 , N − 2 , g N − 2 m + 1     × | 0 , · · · , N − 3 , N − 1 , N − 2 n − 1 | . (78) W e finally remark that if we introd uce the continuous independent v ariables t 1 , t 3 , · · · thr ough ϕ ( s ) r ( m , n ) as ϕ ( s ) r ( m , n ) = α r p s r m − 1 Y i = m 0 (1 + a i p r ) n − 1 Y j = n 0 (1 + b j p r ) e p r t 1 + p 3 r t 3 + ··· + β r ( − p r ) s m − 1 Y i = m 0 (1 − a i p r ) n − 1 Y j = n 0 (1 − b j p r ) e − p r t 1 − p 3 r t 3 + ··· , (79) then τ m n becomes the τ fun ction of the KdV hierarchy . In this case, ρ m n and u m n can be expressed as ρ m n = ∂τ m n ∂ t 1 , u m n = ∂ ∂ t 1 log τ m n , (80) respectively , and u m n satisfies the potential KdV equation ∂ u m n ∂ t 3 − 3 2 ∂ u m n ∂ t 1 ! 2 − 1 4 ∂ 3 u m n ∂ t 3 1 = 0 . (8 1) This is co nsistent with th e fact that (auto nomou s v ersion of) e q.(68) is d erived as the B ¨ ack lund tr ansforma tion of the potential KdV equation[1 6 ]. 4 Concludin g remarks In this article, we have considered the bilinearization of the non-au tonom ous discrete KdV equation and constructed Casorati determ inant solution . W e have presen ted th ree d i ff erent bilinea rizations, each of wh ich h as di ff e rent origin . Although we have co nstructed o nly Casorati deter minant solution, namely , s oliton type solu tion, it migh t not be di ffi cult to discuss oth er ty pes o f solution s, such as rational solutions or period ic solutions, based on th e bilinear equations that have been o btained in this article. Also, we expect that oth er non -auton omous discrete in tegrable systems on two-dimensional lattice can be in vestigated in similar mann er . As was me ntioned in Section 3 .2, the τ function s in the second b ilinearization resemb le tho se in the theory o f R I and R I I biortho gonal functions, b ut the explicit r elation is not clear yet. It migh t be a n intr iguing pro blem to stu dy underly ing structure of the second bilineariza tion. Finally , recently T akahash i and Hirota have succeed ed in constructin g the soliton solutions of the ultradiscrete KdV eq uation in p ermanen t form[17]. It migh t be an interesting problem to investigate the perma nent type solution s for the non-a utonom ous case. Ackno wledgments The author s would like to express their sincere gratitude to Pro fessor N. Matsuura for stimulating discussions and fruitful informa tions which mo ti vated this work . They are also g rateful to Pro fessor A. Nakayash iki for valuable discussions and useful comments. 9 A Proof of di ff erence f ormulas In the append ix, we give the proo f of the di ff erence formulas of τ fu nctions which ha ve been used in the deriv ation of bilinear equ ations fro m th e Pl ¨ ucker relations fo r comp leteness. For late r convenience, we first pr epare the following two equations for ϕ ( s ) r and ψ ( s ) r which are derived from eqs.(48)-(50): ψ ( s ) r ( m , n ) + a m − 1 ψ ( s + 1) r ( m , n ) = ϕ ( s ) r ( m + 1 , n ) , ( 82) ϕ ( s ) r ( m + 1 , n ) − a m ϕ ( s + 1) r ( m + 1 , n ) = A r ( m ) ϕ ( s ) r ( m , n ) . (83) Equations (54) and (55) W e have τ m − 1 n = | 0 m − 1 , 1 m − 1 , · · · , N − 2 m − 1 , N − 1 m − 1 | . (84) Adding the ( i + 1)-th co lumn multiplied by a m − 1 to the i -th column for i = 1 , 2 , . . . , N − 1 and using eq.(4 8), we ha ve τ m − 1 n = | 0 , 1 , · · · , N − 2 , N − 1 m − 1 | . (85) Multiplying a m − 1 to the N -th co lumn and using eq.(48) we obtain a m − 1 τ m − 1 n = | 0 , 1 , · · · , N − 2 , a m − 1 × ( N − 1) m − 1 | = | 0 , 1 , · · · , N − 2 , ( N − 2 ) m − ( N − 2) m − 1 | = − | 0 , 1 , · · · , N − 2 , N − 2 m − 1 | , which is eq.(54). Equ ation (55) can be sho wn in a similar manner by shifting n and using eq.(51). Equation (56) W e have τ m + 1 n = | 0 m + 1 , 1 m + 1 , · · · , N − 2 m + 1 , N − 1 m + 1 | . (86 ) Adding the ( i + 1)-th co lumn multiplied by − a m to the i -th column for i = 1 , 2 , . . . , N − 1 and using eq.(8 3), we have τ m + 1 n =    0 , 1 , · · · , N − 2 , N − 1 m + 1    , k =              A 1 ( m ) ϕ ( s ) 1 ( m , n ) . . . A N ( m ) ϕ ( s ) N ( m , n )              . Multiplying a m to the N -th co lumn and using eq.(83) we have a m τ m + 1 n =    0 , 1 , · · · , N − 2 , a m × ( N − 1) m + 1    =    0 , 1 , · · · , N − 2 , N − 2 m + 1    = A ( m ) ×     0 , 1 , · · · , N − 2 , g N − 2 m + 1     , which is eq.(56). Equation (57) W e have σ m n =     b 0 , b 1 , · · · , d N − 1     , (87) which is rewritten by using eq. (82) from the first column to the N -th colum n as σ m n =     0 m + 1 , 1 m + 1 , · · · , N − 2 m + 1 , d N − 1 m     . (88) Now , from eqs.(50) and (82) we ha ve A r ( m ) ϕ ( s ) r ( m − 1 , n ) = − ( a m + a m − 1 ) ψ ( s + 1) r ( m , n ) + ϕ ( s ) r ( m + 1 , n ) . (89) Applying eq. (89) to the N -th colum n of the right hand side of eq.(88) we obtain ( a m + a m − 1 ) σ m n = −    0 m + 1 , · · · , N − 2 m + 1 , N − 2 m − 1    = −    0 m , · · · , N − 3 m , N − 2 m + 1 , N − 2 m − 1    = − A ( m ) ×     0 m , · · · , N − 3 m , g N − 2 m + 1 , N − 2 m − 1     , which is eq.(57). 10 Equation (58) Shifting n in eq.( 54 ) we have − a m − 1 τ m − 1 n − 1 =     0 n − 1 , 1 n − 1 , · · · , N − 3 n − 1 , N − 2 n − 1 , N − 2 m − 1 n − 1     . (90 ) Eliminating ϕ ( s + 1) r ( m − 1 , n − 1) from eqs.(48) m − 1 n − 1 and (51) m − 1 n − 1 we get ( a m − 1 − b n − 1 ) ϕ ( s ) r ( m − 1 , n − 1) = a m − 1 ϕ ( s ) r ( m − 1 , n ) − b n − 1 ϕ ( s ) r ( m , n − 1) . (91) Using eq.(91) to the N -th column of the right hand side of eq.(90), we have − a m − 1 ( a m − 1 − b n − 1 ) τ m − 1 n − 1 = a m − 1 | 0 n − 1 , 1 n − 1 , · · · , N − 3 n − 1 , N − 2 n − 1 , N − 2 m − 1 | . Adding ( i + 1)- th column multiplied by a m − 1 to i -th column for i = 1 , 2 , . . . , N − 1 and using eq .(48), we obtain − ( a m − 1 − b n − 1 ) τ m − 1 n − 1 = | 0 , 1 , · · · , N − 3 , N − 2 n − 1 , N − 2 m − 1 | , which is eq.(58). Equation (59) Shifting n in eq.( 56 ) we have A ( m ) − 1 a m τ m + 1 n − 1 =     0 n − 1 , 1 n − 1 , · · · , N − 3 n − 1 , N − 2 n − 1 , g N − 2 m + 1 n − 1     =     0 , 1 , · · · , N − 3 , N − 2 n − 1 , g N − 2 m + 1 n − 1     , or a m τ m + 1 n − 1 =     0 , 1 , · · · , N − 3 , N − 2 n − 1 , N − 2 m + 1 n − 1     . (92) Eliminating ϕ ( s + 1) r ( m + 1 , n − 1) from eqs.(83) n − 1 and (51) m + 1 n − 1 we get b n − 1 A r ( m ) ϕ ( s ) r ( m , n − 1) + a m ϕ ( s ) r ( m + 1 , n ) = ( a m + b n − 1 ) ϕ ( s ) r ( m + 1 , n − 1) . (93) Applying eq.(93) to the N -th colum n of the rig ht hand side of eq.(92), we ha ve a m ( a m + b n − 1 ) τ m + 1 n − 1 = a m ×    0 , · · · , N − 3 , N − 2 n − 1 , N − 2 m + 1    , which yields eq.(59) ( a m + b n − 1 ) τ m + 1 n − 1 = A ( m ) ×     0 , · · · , N − 3 , N − 2 n − 1 , g N − 2 m + 1     . W e omit the pro of of other di ff eren ce formulas, since they can be prov ed in a similar manner . Refer ences [1] N. Matsuur a: Book o f Ab stracts, DMHF2 007: COE Co nference on th e Development of Dyn amic Mathem atics with High Functionality (Kyushu University , 2007) p.93. [2] R. Hirota: J. Phys. Soc. Jpn. 43 (19 77) 142 4. [3] Y . Ohta, R. Hirota, S. Tsujimoto and T . Imai: J. Phys. Soc. Jpn. 62 (1993 ) 1 872. [4] K. Kajiwara and J. Satsuma: J. Ph ys. Soc. Jpn. 60 (1991 ) 5 06. [5] K. Kajiwara, Y . Ohta and J. Satsuma: Phys. Lett. A180 (19 93) 2 49. [6] A. Nagai, T . T okihiro , J. Satsuma, R. W illox and K. Kajiwara: Phys. Lett. A234 ( 1997 ) 3 01. [7] R. 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