The determinant representation of the gauge transformation for the discrete KP hierarchy

The determinan t represen tation of the gau ge transformation for the di screte KP hierarc h y Liu Shao w ei † , Cheng Yi † , He Jingsong ‡ ∗ , † Departmen t of Mat hematics, Unive rsity of Sc ience and T ec hnology of China, Hefei, 230 026 Anhui,P .R. China ‡ Departmen t of Mathematics, Ningbo Uni v ersity , Ningbo, 315211 Zhejiang, P .R. China Abstract A successiv e gauge transformation operator T n + k for the discrete KP(dKP) hierarch y is defined, whic h is in v olv ed with tw o types of gauge transformations o p erators. The determinan t representa tion of the T n + k is established,and then it is u sed to get a n ew tau function τ ( n + k ) △ of the dKP hierarc h y from an initial τ △ . In this pro cess, we in tro du ce a generalized discrete W ronskian determinan t and some useful pr op erties of discrete difference op erator. Keyw ords gauge transf orm ation, dKP h ierarch y , tau fun ction 2000 MR Sub ject Classification 35Q51, 37K10 ∗ Corresponding author, Email:hejingsong@n bu.edu.cn,jshe@ustc.edu.cn 1 § 1. In tro duction The d iscrete K P(dKP) h ierarc hy[1, 2, 3, 4] is an interesting ob ject in the researc h of the discrete in tegrable sys tems and the discretizatio n of the in tegrable s y s tems[5, 6, 7 , 8]. Naturally , th ere exist s ome similar pr op erties b etw een dKP and KP hierarc h y[9], suc h as tau fun ction[2, 9, 10], Hamiltonian structures[3, 4] and gauge transformation[1, 11, 12], etc. In p articular, Gauge transformation is on e kind of effectiv e wa y to construct the solution of the integrable sys- tems, b oth contin uous and discrete. Since Chau et al.[1 1 ] h a ve introdu ced tw o t yp es of gauge transformation op erators for the KP hierarc h y , the similar op er ator for the constrained K P hierarc hy[13, 14, 15, 16], q-KP hierarch y [17, 18, 19] and discrete KP hierarc hy [1] ha v e also b een giv en. Oevel [1] h as giv en explicitly three t yp es of gauge transforation op erators of th e dKP hierarc h y , wh ic h are called DT, adjoint DT and binary DT. Ho we v er, DT and adjoin t DT are t w o elemen tary gauge transformation op erators, b ecause a bin ary DT consists of a DT and an adjoin t DT. Here DT and adjoint DT of the dKP hierarc hy are regarded as discrete analogues of the T D and T I [11] of the KP hierarch y , so w e sh all denote it by T d and T i resp ectiv ely in follo wing sections. F urthermore, in theorem 3 of [1] Oevel has also considered n-fold iteration of T d and binary DT for the dKP hierarc h y [1]. 1 On the other hand, the determinant repre- sen tation [12] of the gauge transformation op erators p r o vides a simple m etho d to construct the transformed τ f unction [16, 19, 20, 21] for sev eral sp ecial cases of the KP hierarch y and q-KP hierarc hy . In this pap er we shall extend these r esu lts to discrete KP hierarc hy . W e com bine the t wo elemen tary types of gauge transf ormation op erators ( T d and T i ) for the dKP hierarc hy and let them act n times and k times resp ectiv ely . So w e get the combined gauge op erator T n + k and its the d eterminan t repr esen tation, then use it to constru ct n ew tau fun ction τ ( n + k ) △ of the dKP hierarc hy from an initial tau f u nction. The organization of the pap er is as follo ws. In section 2, w e giv e a brief description of the discrete KP hierarc hy and pr o ve some usefu l prop erties of the discrete op erators. In section 3, based on the Oev el’s t w o t yp es element ary gauge transforation op erator[1], T d and T i , we gi v e the determinant represen tation of the gauge trans f ormation op erator T n + k . In section 4, we shall construct the τ ( n + k ) △ function of th e dKP after th e successiv e gauge transform ations b y using pr evious results. Section 5 is dev oted to the conclusions and discu s sions. § 2. The discrete KP and op erators T o b e self-con tained, w e giv e a brief introdu ction to dKP hierarch y b ased on detailed researc h in [2]. Here we d enote b y Γ and △ resp ectiv ely , the shif t and the difference op erators acting on the asso ciativ e ring F of fun ctions. Where F = { f ( n ) = f ( n, t 1 , t 2 , · · · , t j , · · · ); n ∈ Z , t i ∈ R } , and Γ f ( n ) = f ( n + 1) , 1 The ex p ression of dKP an d n-fold iteration of gauge transformation in [1] is differen t from here, but th e essence is th e same. 2 △ f ( n ) = f ( n + 1 ) − f ( n ) = (Γ − I ) f ( n ) . Where I is the identit y op erator. Define the f ollo wing op eration, f or an y j ∈ Z △ j ◦ f = ∞ X i =0  j i  ( △ i f )( n + j − i ) △ j − i ,  j i  = j ( j − 1) · · · ( j − i + 1) i ! . (2.1) So we obtain an associativ e ring F ( △ ) of formal p seudo difference op erators, with the op eration “ + ” and “ ◦ ” F ( △ ) =    R = d X j = −∞ f j ( n ) △ j , f j ( n ) ∈ R, n ∈ Z    . W e also d enote by R + = P d j = 0 f j ( n ) △ j , the p ositiv e difference part of R and b y R − = P − 1 j = −∞ f j ( n ) △ j , the V olterra part of R . Also d efine th e adjoin t op erator to the △ op er ator by △ ∗ , △ ∗ f ( n ) = (Γ − 1 − I ) f ( n ) = f ( n − 1) − f ( n ) . Where Γ − 1 f ( n ) = f ( n − 1), and the corresp ond ing “ ◦ ” op eration is △ ∗ ◦ f = ∞ X i =0  j i  ( △ ∗ i f )( n + i − j ) △ ∗ j − i . Then w e obtain the adjoin t ring F ( △ ∗ ) to the F ( △ ), and the formal adjoin t to R ∈ F ( △ ) is R ∗ ∈ F ( △ ∗ ), defined by R ∗ = P d j = −∞ △ ∗ j ◦ f j ( n ). The ∗ oper ation satisfies ( F ◦ G ) ∗ = G ∗ ◦ F ∗ for t w o op erators and f ( n ) ∗ = f ( n ) for a function. The discrete KP -hierarc hy [2] is a family of evol ution equation in in finitely man y v ariables t = ( t 1 , t 2 , · · · ) ∂ L ∂ t i = [( L i ) + , L ] . (2.2) Where L is a general fi r st-order pseud o difference op erator L = △ + ∞ X j = 0 f j ( n ) △ − j . (2.3) Similar to KP , L also can b e generate by d ressing op erator W W ( n ; t ) = 1 + ∞ X j = 1 w j ( n ; t ) △ − j , and L = W ◦ △ ◦ W − 1 . (2.4) There are w av e function w ( n ; t, z ) an d adjoin t wa v e f unction w ∗ ( n ; t, z ), w ( n ; t, z ) = W ( n ; t )(1 + z ) n exp ( ∞ X i =1 t i z i ) 3 = (1 + w 1 ( n ; t ) z + w 2 ( n ; t ) z 2 + · · · )(1 + z ) n exp ( ∞ X i =1 t i z i ) (2.5) and w ∗ ( n ; t, z ) = ( W − 1 ( n − 1; t )) ∗ (1 + z ) − n exp ( ∞ X i =1 − t i z i ) = (1 + w ∗ 1 ( n ; t ) z + w ∗ 2 ( n ; t ) z 2 + · · · )(1 + z ) − n exp ( ∞ X i =1 − t i z i ) . (2.6) Also a tau function τ △ = τ ( n ; t ) for dKP exists [2], wh ic h s atisfies that w ( n ; t, z ) = τ ( n ; t − [ z ]) τ ( n ; t ) (1 + z ) n exp ( ∞ X i =1 t i z i ) (2.7) and w ∗ ( n ; t, z ) = τ ( n ; t + [ z ]) τ ( n ; t ) (1 + z ) − n exp ( ∞ X i =1 − t i z i ) , (2.8) where [ z ] = ( 1 z , 1 2 z 2 , 1 3 z 3 , · · · ). By comparing (2.5) with (2.7 ) one find s that w i are expressed by τ ( n ; t ),and further obtai ns dressing op er ator W ( n ; t ) as follo wing form W ( n ; t ) = 1 − ( 1 τ ( n ; t ) ∂ t 1 τ ( n ; t )) △ − 1 + ( 1 2 τ ( n ; t ) ( ∂ 2 t 1 − ∂ t 2 ) τ ( n ; t )) △ − 2 + · · · . (2.9) And taking it b ac k in to (2.4), then comparin g (2.4) and (2.3), all of dynamical v ariables { f i ( n ) } can b e exp anded by τ ( n ; t ). Th e first few of { f i } can b e written out explicitly b elo w, f 0 ( n ) = w 1 ( n ) − w 1 ( n + 1) = △ ∂ t 1 ln τ ( n ; t ) , (2.10) f − 1 ( n ) = w 1 ( n + 1) w 1 ( n ) − w 2 ( n + 1) − w 2 1 ( n ) + w 2 ( n ) − ( △ w 1 )( n ) = ( △ w 1 ( n ))( w 1 ( n ) − 1) − △ w 2 ( n ) =  △ ∂ t 1 ln τ ( n ; t )  ∂ t 1 ln τ ( n ; t ) + 1  − △  1 2 τ ( n ; t ) ( ∂ 2 t 1 − ∂ t 2 ) τ ( n ; t )  (2.11) . . . On the other hand , from the Sato equation of the dKP hierarch y , we can get an imp ortant form ula on r esidue of L n ,i.e. ∂ 2 t 1 t i ln τ ( n ; t ) = r esL i , i ≥ 1 , (2.12) whic h leads to another u seful forms of f i through τ ( n ; t ).F or examples, f − 1 ( n ) = ∂ 2 t 1 ln τ ( n ; t ) , (2.13) 4 ∂ 2 t 1 ,t 2 ln τ ( n ; t ) = △ f − 1 ( n ) + f − 2 ( n + 1) + f 0 ( n ) f − 1 ( n ) + f − 1 ( n ) f 0 ( n − 1) + f − 2 ( n ) , (2.14) . . . Note that,in general, (2.12) implies recurs ion relation of f − i ( n ) if i ≥ 2, one can not get explicit form of f − i form it except f − 1 . This is a crucial difference of dK P hierarc hy f rom KP hierarc hy . No w w e pr o ve some useful pr op erties for the op erators which are used later. Lemma 2.1 F or f ∈ F and △ , Γ as ab o v e, the follo wing identitie s hold. (1) △ ◦ Γ = Γ ◦ △ (2.15) (2) △ ∗ = −△ ◦ Γ − 1 (2.16) (3) ( △ − 1 ) ∗ = ( △ ∗ ) − 1 = − Γ ◦ △ − 1 (2.17) (4) f ◦ △ − 1 = X i ≥ 0 △ − i − 1 ◦ △ i (Γ f ) (2.18) (5) △ − 1 ◦ f ◦ △ − 1 = ( △ − 1 f ) ◦ △ − 1 − △ − 1 ◦ Γ( △ − 1 f ) (2.19) Pro of: F or ∀ g ( n ) ∈ F , (1) ( △ ◦ Γ) g ( n ) = g ( n + 2) − g ( n + 1) = Γ( g ( n + 1) − g ( n )) = (Γ ◦ △ ) g ( n ) (2) △ ∗ g ( n ) = − ( g ( n ) − g ( n − 1)) = ( −△ ◦ Γ − 1 ) g ( n ) (3) I = ( △ ∗ ) − 1 ◦ △ ∗ = ( − Γ ◦ △ − 1 ) ◦ ( −△ ◦ Γ − 1 ) (4) X i ≥ 0 △ − i − 1 ◦ △ i (Γ f ) = X i ≥ 0 ∞ X i =0  − i − 1 j  ( △ i + j f )( n − i − j ) ◦ △ − i − j − 1 ( by (2.1)) = ∞ X l =0 l X j = 0  j − l − 1 j  ( △ l f )( n − l ) ◦ △ − l − 1 ( let l = i + j ) = f ◦ △ − 1 . ( S ince when l ≥ 1 l X 0  j − l − 1 j  = l X 0  l j  ( − 1) j = 0) (5)W e d efi ne 2 △ − 1 = (Γ − I ) − 1 = Γ − 1 I − Γ − 1 = ∞ X i =1 (Γ − 1 ) i . (2.20) 2 This definition is coincide with (2.1) 5 And th us ( ∞ X i =1 f ( n − i ))( ∞ X j = 1 g ( n − j )) = ∞ X i =1 Γ − i ( f ( n ) ∞ X m =1 g ( n − m )) + ∞ X j = 1 Γ − j ( g ( n ) ∞ X m =1 f ( n − m + 1)) . (2.21) By usin g (2.20), equation (2.21 ) implies (( △ − 1 f ) ◦ △ − 1 ) g ( n ) = ( △ − 1 ◦ f ◦ △ − 1 ) g ( n ) + ( △ − 1 ◦ Γ( △ − 1 f )) g ( n ) , whic h finish es the pro of b ecause g ( n ) is an arbitrary function.  § 3. The determinan t represen tation In this secti on, we sh all constru ct the determin ant rep resen tation of the gauge transformations. T o this end, w e start with t wo elemen tary t yp es of gauge tr ansformation op erators of the dKP hierarc hy[1]: T d ( φ ) = Γ( φ ) ◦ △ ◦ φ − 1 = △ − △ φ φ , (3.1) T i ( ψ ) = Γ − 1 ( ψ − 1 ) ◦ △ − 1 ◦ ψ = ( △ + △ (Γ − 1 ( ψ )) ψ ) − 1 . (3.2) Where φ and ψ are called eigenfunctions and adjoin t eigenfunctions of L for gauge transforma- tion, which satisfy the f ollo wing d ynamic equation of dKP sys tem ∂ t n φ = ( L n ) + φ, ∂ t n ψ = − ( L n + ) ∗ ψ . Here the defi n ition of adjoint eigenfunction are differen t from (2.8). Set { φ 1 , φ 2 , · · · , φ n } and { ψ 1 , ψ 2 , · · · , ψ n } b e tw o sets of n different eigenfunctions and n d if- feren t adjoin t eigenfunctions of the dKP hierarch y asso ciated with L . Then under the transfor- mation (3.1), the corresp ond in g transformations of { L, φ, ψ } are L → L (1) = T d ( φ 1 ) ◦ L ◦ T d ( φ 1 ) − 1 , φ → φ (1) = T d ( φ 1 ) φ = Γ φ − (Γ φ 1 ) φ − 1 1 φ, (3.3) ψ → ψ (1) = ( T d ( φ 1 ) ∗ ) − 1 ψ = Γ( △ − 1 ( φ 1 ψ )) Γ φ 1 . (3.4) Similarly u nder the transformation (3.2), the corresp onding transformations of { L, φ, ψ } are L → L (1) = T i ( ψ 1 ) ◦ L ◦ T i ( ψ 1 ) − 1 , 6 φ → φ (1) = T i ( ψ 1 ) φ = △ − 1 ( φψ 1 ) Γ − 1 ( ψ 1 ) , (3.5) ψ → ψ (1) = ( T i ( ψ 1 ) ∗ ) − 1 ψ = − ψ ψ − 1 1 Γ − 1 ( ψ 1 ) + Γ − 1 ( ψ ) . (3.6) Where φ (1) , ψ (1) are corresp ondin g eigenfunction and adjoint eigenfun ction to L (1) . Note T d and T i will annihilate their generation f u nction,i.e., T d ( φ 1 ) φ 1 = 0 , ( T − 1 i ( ψ 1 )) ∗ ψ 1 = 0 , (3.7) whic h is a crucial fact to find the r epresen tation of th e T n + k b elo w . No w we consider the successiv e gauge transf orm ation including the t wo t yp e transformations. In [1 ], only n-fold T d or n-fold binary DT was considered. Ho w ev er we shall study a general gauge trans formation op erator T n + k whic h consists of n-fold T d and k-fold T i in the follo wing. W e d efi ne the op erator as follo wing, T n + k = T i ( ψ ( n + k − 1) k ) ◦ · · · ◦ T i ( ψ ( n ) 1 ) ◦ T d ( φ ( n − 1) n ) ◦ · · · ◦ T d ( φ (1) 2 ) ◦ T d ( φ 1 ) , (3.8) in which φ ( i − 1) i = T d ( φ ( i − 2) i − 1 ) ◦ T d ( φ ( i − 3) i − 2 ) · · · T d ( φ (1) 2 ) ◦ T d ( φ 1 ) · ψ i , ψ ( n ) j = ( T d ( φ ( n − 1) n ) − 1 ) ∗ · · · ( T d ( φ (1) 2 ) − 1 ) ∗ ◦ ( T d ( φ 1 ) − 1 ) ∗ · ψ j , ψ ( n + l ) j = ( T i ( ψ ( n + l − 1) l ) − 1 ) ∗ · · · ( T i ( ψ ( n +1) 2 ) − 1 ) ∗ ◦ ( T i ( ψ ( n ) 1 ) − 1 ) ∗ · ψ ( n ) j . That means under transformation (3.7), we hav e L T d ( φ 1 ) − − − − → L (1) T d ( φ (1) 2 ) − − − − − → L (2) → · · · → L ( n − 1) T d ( φ ( n − 1) n ) − − − − − − − → L ( n ) T i ( ψ ( n ) 1 ) − − − − − → L ( n +1) T i ( ψ ( n +1) 2 ) − − − − − − → L ( n +2) → · · · → L ( n + k − 1) T i ( ψ ( n + k − 1) k ) − − − − − − − − → L ( n + k ) . (3.9) and { φ ( n ) j , ψ ( n ) j } are corresp onding eigenfunctions and adjoint eige nfun ctions of L ( n ) . No w we sho w that the op erator T n + k has a determinan t representa tion, w h ic h giv es th e exp licit form of T n + k and is useful for direct computation. Firs t we in tro duce a discrete W ronskian determinan t. Defin e W △ n ( φ 1 , · · · , φ n ) =          φ 1 φ 2 · · · φ n △ φ 1 △ φ 2 · · · △ φ n . . . . . . · · · . . . △ n − k − 1 φ 1 △ n − k − 1 φ 2 · · · △ n − k − 1 φ n          , (3.10) and more general 7 I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) =                 △ − 1 ψ k φ 1 △ − 1 ψ k φ 2 · · · △ − 1 ψ k φ n . . . . . . · · · . . . △ − 1 ψ 1 φ 1 △ − 1 ψ 1 φ 2 · · · △ − 1 ψ 1 φ n φ 1 φ 2 · · · φ n △ φ 1 △ φ 2 · · · △ φ n . . . . . . · · · . . . △ n − k − 1 φ 1 △ n − k − 1 φ 2 · · · △ n − k − 1 φ n                 , (3.11) whic h shall b e used in the follo w ing theorem. Theorem 3.1 If n > k , T n + k and T − 1 n + k has the follo w ing determinan t repr esen tation: T n + k = 1 I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) ·                 △ − 1 ψ k φ 1 · · · △ − 1 ψ k φ n △ − 1 ◦ ψ k . . . · · · . . . . . . △ − 1 ψ 1 φ 1 · · · △ − 1 ψ 1 φ n △ − 1 ◦ ψ 1 φ 1 · · · φ n 1 △ φ 1 · · · △ φ n △ . . . · · · . . . . . . △ n − k φ 1 · · · △ n − k φ n △ n − k                 , (3.12) T − 1 n + k =          φ 1 ◦ △ − 1 Γ( △ − 1 ψ k φ 1 ) · · · Γ( △ − 1 ψ 1 φ 1 ) Γ( φ 1 ) · · · Γ( △ n − k − 2 φ 1 ) φ 2 ◦ △ − 1 Γ( △ − 1 ψ k φ 2 ) · · · Γ( △ − 1 ψ 1 φ 2 ) Γ( φ 2 ) · · · Γ( △ n − k − 2 φ 2 ) . . . . . . · · · . . . . . . · · · . . . φ n ◦ △ − 1 Γ( △ − 1 ψ k φ n ) · · · Γ( △ − 1 ψ 1 φ n ) Γ ( φ n ) · · · Γ( △ n − k − 2 φ n )          · ( − 1) n − 1 Γ( I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) . (3.13) here the determinan t of T n + k expand b y the last column and the functions are on the left h and with the action ” ◦ ”. As f or T − 1 n + k the determinant expand by the fi rst column and the functions are on the righ t h an d w ith the action ” ◦ ” to o. Pro of: F or n > k , we know the highest order item of T n + k is △ n − k . So w e can assum e T n + k has the follo wing form T n + k = n − k X i =0 a i ◦ △ i + − 1 X i = − k a i ◦ △ − 1 ◦ ψ | i | , a n − k = 1 , (3.14) T − 1 n + k = n X j = 1 φ j ◦ △ − 1 ◦ b j . (3.15) 8 It is easy to find T n + k · φ i = 0 , i = 1 , 2 , · · · , n, (3.16) ( T − 1 n + k ) ∗ · ψ j = 0 , j = 1 , 2 , · · · , k , (3.17) from (3.7), wh ic h giv e the follo wing algebraic sys tem on a i and b j , a − k △ − 1 ( ψ k φ i ) + · · · + a − 1 △ − 1 ( ψ 1 φ i ) + a 0 φ i + · · · + a n − k − 1 △ n − k − 1 φ i = −△ n − k φ i (3.18) i = 1 , 2 , · · · , n , b 1 Γ( △ − 1 ( ψ j φ 1 )) + b 2 Γ( △ − 1 ( ψ j φ 2 )) + · · · + b n Γ( △ − 1 ψ j φ n )) = 0 (3.19) j = 1 , 2 , · · · , k. Then with the help of (3.14), b y Gramer rule the solution of (3.18) is giv en by a − i = − 1 I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) ·        △ − 1 ψ k φ 1 · · · △ − 1 ψ i +1 φ 1 △ n − k φ 1 △ − 1 ψ i − 1 φ 1 · · · △ − 1 ψ 1 φ 1 φ 1 · · · △ n − k − 1 φ 1 . . . · · · . . . . . . . . . · · · . . . . . . · · · . . . △ − 1 ψ k φ n · · · △ − 1 ψ i +1 φ n △ n − k φ n △ − 1 ψ i − 1 φ n · · · △ − 1 ψ 1 φ n φ n · · · △ n − k − 1 φ n        , (3.20) a i = − 1 I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) ×        △ − 1 ψ k φ 1 · · · △ − 1 ψ 1 φ 1 φ 1 · · · △ i − 1 φ 1 △ n − k φ 1 △ i +1 φ 1 · · · △ n − k − 1 φ 1 . . . · · · . . . . . . · · · . . . . . . . . . · · · . . . △ − 1 ψ k φ n · · · △ − 1 ψ 1 φ n φ n · · · △ i − 1 φ n △ n − k φ n △ i +1 φ n · · · △ n − k − 1 φ n        . (3.21) So w e obtain the representat ion of T n + k as (3.12). Next w e pr o ve the determinan t representat ion of T − 1 n + k . Bec ause T n + k ◦ T − 1 n + k = 1, F or the rationalit y of the definition of T − 1 n + k in (3.15) w e ha v e to verify ( T n + k ◦ T − 1 n + k ) − = I − = 0 . T aking (3.14) and (3.15) into the left h and side ,then ( T n + k ◦ T − 1 n + k ) − = n − k X i =0 n X j = 1 a i ( △ i φ j ) ◦ △ − 1 ◦ b j + − 1 X i = − k n X j = 1 a i ◦ △ − 1 ◦ ψ | i | φ j ◦ △ − 1 ◦ b j . (3.22) F urthermore, w e can rewrite the second term in (3.22) fr om (2.19) ( T n + k ◦ T − 1 n + k ) − = n − k X i =0 n X j = 0 a i ( △ i φ j ) ◦ △ − 1 ◦ b j 9 + n X j = 0 a − k ◦ ( △ − 1 ψ k φ j ) ◦ △ − 1 ◦ b j − n X j = 0 a − k ◦ △ − 1 ◦ Γ( △ − 1 ψ k φ j ) b j + · · · + n X j = 0 a − 1 ◦ ( △ − 1 ψ 1 φ j ) ◦ △ − 1 ◦ b j − n X j = 0 a − 1 ◦ △ − 1 ◦ Γ( △ − 1 ψ 1 φ j ) b j . (3.23) Rearranging the right hand sid e, and using w ith (3.18) and (3.19), then (3.23) b ecomes ( T n + k ◦ T − 1 n + k ) − = n X j = 1 T n + k ( φ j ) ◦ △ − 1 ◦ b j (3.24) − k X i =1 a i ◦ △ − 1 ◦ ( T − 1 n + k ) ∗ ( ψ i ) (3.25) = 0 , (3.26) due to (3.16 ) and (3.17). On th e other hand , b y lemma 2.1, T − 1 n + k has the follo wing form T − 1 n + k = n − k − 2 X i =0 n X j = 1 △ − 1 − i ◦ Γ( △ i φ j ) b j + △ − n + k ◦ n X j = 0 Γ( △ n − k − 1 φ j ) b j + ∞ X i = n − k n X j = 0 △ − 1 − i ◦ Γ( △ i φ j ) b j , (3.2 7) and the T − 1 n + k ma y b e exp ressed as T − 1 n + k = △ − n + k + ( l ow er oder item ) , (3.28) according to the T n + k ◦ T − 1 n + k = I and the form of T n + k . Th en we get the follo wing algebraic equations on b i ,                                  Γ( △ − 1 ( ψ k φ 1 )) b 1 + Γ( △ − 1 ( ψ k φ 2 )) b 2 + · · · + Γ( △ − 1 ψ k φ n )) b n = 0 . . . Γ( △ − 1 ( ψ 1 φ 1 )) b 1 + Γ ( △ − 1 ( ψ 1 φ 2 )) b 2 + · · · + Γ( △ − 1 ψ 1 φ n )) b n = 0 Γ( φ 1 ) b 1 + Γ( φ 2 ) b 2 + · · · + Γ( φ n ) b n = 0 Γ( △ φ 1 ) b 1 + Γ( △ φ 2 ) b 2 + · · · + Γ( △ φ n ) b n = 0 . . . Γ( △ n − k − 2 φ 1 ) b 1 + Γ( △ n − k − 2 φ 2 ) b 2 + · · · + Γ( △ n − k − 2 φ n ) b n = 0 Γ( △ n − k − 1 φ 1 ) b 1 + Γ( △ n − k − 1 φ 2 ) b 2 + · · · + Γ( △ n − k − 1 φ n ) b n = 1 . (3.29) Note the first k equations are giv en by (3.19) and the last n − k equations are giv en by comparing the corresp onding terms in (3.27) with (3.28). Solving (3.29) giv es b i = 1 Γ( I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n )) · 10               Γ( △ − 1 ψ k φ 1 ) · · · Γ( △ − 1 ψ k φ i − 1 ) 0 Γ ( △ − 1 ψ k φ i +1 ) · · · Γ( △ − 1 ψ k φ n ) . . . · · · . . . . . . . . . · · · . . . Γ( △ − 1 ψ 1 φ 1 ) · · · Γ( △ − 1 ψ 1 φ i − 1 ) 0 Γ ( △ − 1 ψ 1 φ i +1 ) · · · Γ( △ − 1 ψ 1 φ n ) Γ( φ 1 ) · · · Γ( φ i − 1 ) 0 Γ ( φ i +1 ) · · · Γ( φ n ) . . . · · · . . . . . . . . . · · · . . . Γ( △ n − k − 1 φ 1 ) · · · Γ( △ n − k − 1 φ i − 1 ) 1 Γ( △ n − k − 1 φ i +1 ) · · · Γ( △ n − k − 1 φ n )               , (3.30) and tak e it bac k to (3.15 ), we obtain (3.13). So w e fin ish the pro of.  W e n otice that when use ∂ instead of △ , and set n = 0 , f 0 = 0, all results will approac h to the case of K P automatical ly[12 ].F or the case of n = k , we ha ve the f ollo wing Theorem 3.2 When n = k , T n + n = 1 I W △ n + n ( ψ n , · · · , ψ 1 ; φ 1 , · · · , φ n ) ·          △ − 1 ψ n φ 1 · · · △ − 1 ψ n φ n △ − 1 ◦ ψ n . . . · · · . . . . . . △ − 1 ψ 1 φ 1 · · · △ − 1 ψ 1 φ n △ − 1 ◦ ψ 1 φ 1 · · · φ n 1          , (3.31) T − 1 n + n =          − 1 ψ n · · · ψ 1 φ 1 ◦ △ − 1 Γ( △ − 1 ψ n φ 1 ) · · · Γ( △ − 1 ψ 1 φ 1 ) . . . . . . · · · . . . φ n ◦ △ − 1 Γ( △ − 1 ψ n φ n ) · · · Γ( △ − 1 ψ 1 φ n )          · ( − 1) Γ( I W △ n + n ( ψ n , · · · , ψ 1 ; φ 1 , · · · , φ n ) . (3.32 ) Pro of: Assu me T n + n = 1 + − 1 X i = − n a i ◦ △ − 1 ◦ ψ | i | , T − 1 n + n = 1 + n X j = 1 φ j ◦ △ − 1 ◦ b j . The remaining p ro of is as Theorem 3.1, so we omit it. No w we obtain the determinan t represen tation of ( T − 1 n + k ) ∗ b y a d irect co mputation, w h ic h is useful in the follo wing cont ext. Corollary 3.3 When n > k , ( T − 1 n + k ) ∗ = − 1 Γ( I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n )) ·        Γ( △ − 1 ψ k φ 1 ) · · · Γ( △ − 1 ψ 1 φ 1 ) Γ( φ 1 ) · · · Γ( △ n − k − 2 φ 1 ) Γ ◦ △ − 1 ◦ φ 1 . . . · · · . . . . . . · · · . . . . . . Γ( △ − 1 ψ k φ n ) · · · Γ( △ − 1 ψ 1 φ n ) Γ ( φ n ) · · · Γ( △ n − k − 2 φ n ) Γ ◦ △ − 1 ◦ φ n        . 11 § 4 T he transformed dKP h ierarc h y W e are no w in a p osition to give the transformed KP by u sing results in last section. In other hand w e fin d the τ ( n + k ) △ function und er the transf ormation of T n + k . The existence of τ △ function for dKP is pro v ed in [2]. Lemma 4.1 Under the T d ( φ 1 ), we ha ve τ △ → τ (1) △ = φ 1 τ △ , (4.1) and un der T i ( ψ 1 ), τ △ → τ (1) △ = ψ 1 ( n − 1) τ △ . (4.2) Pro of : T aking an in itial Lax op erator L , we get a transform ed one L (1) = △ + f (1) 0 + f (1) − 1 △ − 1 + · · · . By directly computation L (1) = T d ( φ 1 ) ◦ L ◦ T d ( φ 1 ) − 1 = △ + △ ( △ φ 1 φ 1 ) + f 0 ( n + 1) + . . . So f (1) 0 = △ ( △ φ 1 φ 1 ) + f 0 ( n + 1) . This implies f (1) 0 = △ ∂ t 1 ln φ 1 + f 0 , (4.3) b y using the equation ( φ 1 ) t 1 = L + φ 1 = ( △ + f 0 ) φ 1 . Moreo v er, tak e f 0 = △ ( ∂ t 1 ln τ △ ) bac k into the ab ov e equation, then f (1) 0 = △ ∂ t 1 ln φ 1 + △ ∂ t 1 ln τ △ = △ ∂ t 1 ln φ 1 τ △ . and this sh o ws τ (1) △ = φ 1 τ △ , b ecause f (1) 0 = △ ∂ t 1 ln τ (1) △ . T he pro of of (4.2) is analogous to the ab o ve. So it is omitted.  By a direct computation w ith the help of ab o v e theorem, we ha v e the f ollo wing theorem. Theorem 4.2 When k = 0 u nder the tr an s formation of T n +0 = T n , there are φ ( n ) = T n · φ = W △ n +1 ( φ 1 , , · · · , φ n , φ ) W △ n ( φ 1 , · · · , φ n ) , (4.4) ψ ( n ) = ( T − 1 n ) ∗ · ψ = ( − 1) n Γ( I W △ n +1 ( ψ , φ 1 , · · · , φ n )) Γ( W △ n ( φ 1 , · · · , φ n )) , (4.5) 12 and τ ( n ) △ = W △ n ( φ 1 , · · · , φ n ) · τ △ . (4.6) Pro of: The determinant representat ion of T n is giv en by T n + k | k =0 , then substituting it into φ ( n ) = T n φ and ψ ( n ) = ( T − 1 n ) ∗ ψ , we can get r esu lts of this theorem. F urthermore, τ ( n ) △ is giv en b y applying the gauge trans f ormation rep eatedly , i.e., τ ( n ) △ = φ ( n − 1) n τ ( n − 1) △ = φ ( n − 1) n φ ( n − 2) n − 1 τ ( n − 2) △ = φ ( n − 1) n φ ( n − 2) n − 1 · · · φ (1) 2 φ 1 τ △ = W △ n ( φ 1 , · · · , φ n ) W △ n − 1 ( φ 1 , · · · , φ n − 1 ) · W △ n − 1 ( φ 1 , · · · , φ n − 1 ) W △ n − 2 ( φ 1 , · · · , φ n − 2 ) · · · W △ 2 ( φ 1 , φ 2 ) W △ 1 ( φ 1 ) · φ 1 · τ △ = W △ n ( φ 1 , · · · , φ n ) τ △ . (4.7)  Theorem 4.3 When n > k , und er the T n + k , φ ( n + k ) = T n + k · φ = I W △ ( n +1)+ k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n , φ ) I W △ n + k ( ψ k , · · · , ψ 1 ; φ n , · · · , φ 1 ) , (4.8) ψ ( n + k ) = ( T − 1 n + k ) ∗ ψ = ( − 1) n Γ( I W △ n +( k +1) ( ψ , ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n )) Γ( I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n )) (4.9) ( ψ 6 = ψ i , i = 1 , · · · , k ) , and τ n + k △ = ( − 1) k n I W △ k ,n ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) · τ △ . (4.10) Pro of: T he φ ( n + k ) and ψ ( n + k ) are obtained b y a direct application of determinan t of T n + k and ( T − 1 n + k ) ∗ . Moreo ver, τ ( n + k ) △ can b e derived b y follo w ing w ay , τ ( n + k ) △ = Γ − 1 ( ψ ( n + k − 1) k ) τ ( n + k − 1) △ = Γ − 1 ( ψ ( n + k − 1) k )Γ − 1 ( ψ ( n + k − 2) k − 1 ) τ ( n + k − 2) △ = ψ ( n + k − 1) k ( n − 1) ψ ( n + k − 2) k − 1 ( n − 1) · · · ψ ( n +1) 2 ( n − 1) ψ ( n ) 1 ( n − 1) τ ( n ) △ = ( − 1) n I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) I W △ n +( k − 1) ( ψ k − 1 , · · · , ψ 1 ; φ 1 , · · · , φ n ) · ( − 1) n I W △ n +( k − 1) ( ψ k − 1 , · · · , ψ 1 ; φ 1 , · · · , φ n ) I W △ n +( k − 2) ( ψ k − 2 , · · · , ψ 1 ; φ 1 , · · · , φ n ) · · · · · · · ( − 1) n I W △ n +1 ( ψ 1 ; φ 1 , · · · , φ n ) W △ n ( φ 1 , · · · , φ n ) W △ n ( φ 1 , · · · , φ n ) τ △ = ( − 1) nk I W △ n + k ( ψ k , · · · , ψ 1 ; φ 1 , · · · , φ n ) τ △ . (4.11 )  13 § 5 Conc lusions and discu ssions W e ha v e established determinant representa n t of the gauge transform ation op erator T n + k of the dKP hierarc hy , and further giv en th e transformed τ ( n + k ) △ from kno w n τ △ . The adv an tage of this represen tation is compact an d systematic by comparing with an iterativ e w a y of gauge transformation. Also it is con venien t for find solution of d K P hierarc hy . T o illustrate our approac h , we j ust ha v e giv en some results on a sp ecial chain of the gauge transformations. Of course, it is p ossible to consider other co mplicated c hain if it is needed. W e n otice that u nder the map: △ → ∂ , n ≡ 0, f 0 = 0, the wh ole d KP hierarc hy approac h es to the KP hierarc hy , and our results in previous sections can go b ack to the case of KP hierarc hy to o. Because of the differences b etw een op erators △ and ∂ in d KP and K P hierarc h y r esp ectiv ely , and the existence of { f 0 } , w e kno w it is not a easy task to get simple closed form of the dKP equation by f − 1 , and then to get explicit s olution of d KP equation by gauge transform ation as w e h a ve done for the KP equation. Ho wev er, we can get an explicit solution of f − 1 through τ ( n + k ) △ starting from a kn o wn τ △ = 1, which can b e regarded as discr ete analog ue of u 1 in KP hierarc hy with a Lax op erator L = ∂ + u 1 ∂ − 1 + u 2 ∂ − 2 + · · · , and then to sho w the difference b et w een f − 1 and u 1 . This is a wa y for u s to understand the discrete effect of the KP hierarch y . On the other hand, it is p ossible to find the add itional symm etry and its algebraic structures of the dKP hierarch y du e to the existence of the tau f unction τ △ = τ ( n ; t ) [2]. W e shall try to d o it in a near futur e. Ac knowledgemen t This work is supp orted partly b y the NSF C grant of China under No.106 71187 . He is also supp orted by the Pr ogram for NCET under Grant No.NCET-08- 0 515. W e thank Pr o fessor Li Yishen(USTC, China) for long -term encour agements and suppor ts. References [1] W.Oev el, Darb oux transformations for integ rable lattice systems, Nonlinear Physics: Th e- ory an d Exp eriment,E.Al finito, L. Martina and F.Pempinelli(eds)( W orld Scien tific, Sin- gap ore,1996 ), 233-240. [2] L.Haine, P .Iliev, Comm utativ e Rings of Difference O p erators and an Adelic Flag Manifold, In ternational Mathematics Researc h Notices,No.6,( 2000),281-323. [3] L.A.Dic k ey , Mo difi ed KP and discrete KP , L ett.Math.Phys.4 8(1999 ),277-289. [4] B.A. Kup er s himidt, Discrete Lax equations and differenece calculus, Asterisque No.123 ,1985, 1-212. [5] Adeler V.E., Bob enko A.I,Suris Y u.B, Classification of inte grable equation on quad-graphs. 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