An application of the Casoratian technique to the 2D Toda lattice equation

An application of the C asoratian technique to the 2D T o da lattice equation W en-Xiu Ma a,b ∗ a Department of Mathematics and Statistics, University of South Florida, T ampa, FL 3 3620 -5700 , USA b State K ey Lab o ratory of Scientific and Eng ineering Computing, Chinese Academy o f Sciences , P .O. Box 2719, Be ijing 10 0080 , PR C hina Abstract A general Casora tian formulation is pro po sed for the 2D T o da lattice equatio n, whic h inv olves coupled eig enfunction sys tems. V ar ious Caso ratian type solutions ar e generated, thro ug h solving the r e sulting linear conditions and us ing a B ¨ acklund transformation. MSC: 3 7K10, 35Q 58, 35Q51 Key w ords. The 2D T o da lattice equation, Casoratian form ulation, soliton, complexiton ∗ Email: mawx@cas.usf.edu 1 1 In tro duction It i s w ell-kno wn that W rons kian form ulations sho w a common c haracteristic feature of con tin uous soliton equations, an d p ro vide a p ow erfu l tool to constru ct exact solutions to con tinuous soliton equations [1]-[7]. The r esu lting tec hniqu e has app lied to man y con tin uous soliton equations such as the KdV, MKdV, NLS, d eriv ativ e NLS, Bo ussin esq, KP , sine-Go rd on and sinh-Gordon equat ions. With W ron s kian form ulations, soliton solutions and rational solutions are u sually expressed as some kind of logarithmic d er iv ativ es of W ronskian type determinan ts with resp ect to sp ace v ariables, and the in v olv ed d eterminan ts are generated b y eigenfunctions satisfying linear systems of d ifferen tial equations. A great h elp is that W rons k ian form ulations tran s form nonlinear problems in to linear problems, an d th us c ont inuous soliton equatio ns can b e treated b y means of linear theories. There is a discrete v ersion of W ronskian f orm ulations, called Casoratian formulati ons, for d is- crete soliton equ ations su c h as th e V olttera, nonlinear electrical net work, and T o da lattice equations (see, for example, [8]-[11]). With Casoratia n formulations, solit on solutions and rat ional solutions are often expressed as some kind of r ational functions of C asoratian t yp e determinants, and the in v olv ed determinants are m ade of eigenfunctions satisfying linear s y s tems of differen tial-difference equations. Therefore, the Casoratian tec h nique offe rs a direct approac h for constructing exac t solutions t o discrete soliton equ ations. Besides solit on solutions and rat ional solutions, the W rons kian and C asoratian te c hniqu es can b e used to construct positon solutio ns [12]-[1 5 ], i.e., solutions in vo lving one kind of t ranscend ental functions: trigonometric f unctions. More generally , a no v el kind of solutions called comp elxiton solutions has b een in tro du ced and generated using suc h tec h niques for con tin uous and d iscr ete soliton equat ions [3, 8] and soliton equat ions with sources [16]. Those so lutions con tain tw o kinds of transcendental w a v es: e xp onen tial wa ves and trigonometric w a v es, with differen t sp eeds, and they c orresp ond to complex eigen v alues of asso ciated c h aracteristic linear problems and generate solitons and p ositons as limit cases of the complex eige nv alues [5, 17]. One of in triguing discrete soliton equations is the 2D T o da lattic e equation [1 8] ∂ 2 Q n ∂ s∂ x = V n +1 − 2 V n + V n − 1 , Q n = ln (1 + V n ) , (1) where x, s ∈ R and n ∈ Z . Thr ou gh the d ep end ent v ariable transformatio n V n = ∂ 2 ∂ s∂ x ln τ n , (2) the e quation (1) ma y be in tegrated with resp ect to x and s to obtain 1 + ∂ 2 ∂ s∂ x ln τ n = τ n +1 τ n − 1 τ 2 n , (3) where th e constan ts of i nteg ration are set to zero. T h is equation is equiv alen t to ∂ 2 τ n ∂ s∂ x τ n − ∂ τ n ∂ s ∂ τ n ∂ x = τ n +1 τ n − 1 − τ 2 n , (4) 2 whic h ca n b e wr itten as D x D s τ n • τ n = 2( τ n +1 τ n − 1 − τ 2 n ) , (5) in terms of Hirota’s op er ator [1 8]: ( D z f • g ) = ( ∂ z − ∂ z ′ ) f ( z ) g ( z ′ ) | z ′ = z . (6) If we set y n = ln τ n τ n +1 , (7) then we obtain another form for the 2D T od a lattice equatio n: ∂ 2 y n ∂ s∂ x = e y n − 1 − y n − e y n − y n +1 . (8) Tw o forms (1) and (8) of the 2D T o da la ttice equatio n are linked through ∂ 2 y n ∂ s∂ x = V n − V n +1 . In this pap er, we would like to establish a general Casoratian form u lation for the 2D T o da la ttice equation (4) and analyze its exact sol utions based on the resulting Casoratian form ulation and a B¨ ac klu n d transformation. The pap er is orga nized as follo ws . In Section 2, a general Casoratian form u lation is presen ted for the bilinear 2D T o da lattice equation (4) . In Sectio n 3, some sp ecific cases of lin ear conditions are discussed and a B¨ ac klund trans formation is fu rnished to constru ct exact solutions, and v arious examples of Casoratian type solutions are presen ted. Concluding r emarks are given finally in Section 4 . 2 A general Casorati an form ulation The N - soliton solution to the bilinear 2D T o d a lattice equation (4) is expressed as a Casorati determinan t [19] τ n = Cas( φ 1 , φ 2 , · · · , φ N ) =              φ 1 ( n ) φ 1 ( n + 1) · · · φ 1 ( n + N − 1) φ 2 ( n ) φ 2 ( n + 1) · · · φ 2 ( n + N − 1) . . . . . . . . . φ N ( n ) φ N ( n + 1) · · · φ N ( n + N − 1)              , (9) where eac h φ i ( n ) = φ i ( n, x, s ) s atisfies the linear differentia l-difference equations ∂ φ i ( n ) ∂ x = φ i ( n + 1) , ∂ φ i ( n ) ∂ s = − φ i ( n − 1) , 1 ≤ i ≤ N . (10) 3 The ab o v e Casorat i determinan t has b een used in the theory of the 1D lattice equations [8]-[11]. W e will adopt the notati on [8] k ..l = k , k + 1 , · · · , l (11) where k < l , and d enote the ge neralized Casorati determinant by | i 1 , · · · , i N | = det([ i 1 , · · · , i N ]) , (12) where i j ∈ Z , 1 ≤ j ≤ N , and the matrix [ i 1 , · · · , i N ] is defined by [ i 1 , · · · , i N ] =          φ 1 ( n + i 1 ) φ 1 ( n + i 2 ) · · · φ 1 ( n + i N ) φ 2 ( n + i 1 ) φ 2 ( n + i 2 ) · · · φ 2 ( n + i N ) . . . . . . . . . φ N ( n + i 1 ) φ N ( n + i 2 ) · · · φ N ( n + i N )          . (13) Ob viously , the stand ard Casorati determinan t is giv en b y Cas( φ 1 , φ 2 , · · · , φ N ) = | 0 ..N − 1 | . Theorem 1. L et ε = ± 1 a nd δ = ± 1 , i.e., ( ε, δ ) = (1 , 1) , (1 , − 1) , ( − 1 , 1) or ( − 1 , − 1) . If a set o f functions φ i ( n ) = φ i ( n, x, s ) , 1 ≤ i ≤ N , satisfies the fol lowing c ouple d line ar differ ential-differ enc e e quations: ∂ φ i ( n ) ∂ x = εφ i ( n + δ ) + N X j =1 λ ij ( x ) φ j ( n ) , 1 ≤ i ≤ N , (14) ∂ φ i ( n ) ∂ s = − εφ i ( n − δ ) + N X j =1 µ ij ( s ) φ j ( n ) , 1 ≤ i ≤ N , (15) wher e λ ij ( x ) and µ ij ( s ) , 1 ≤ i, j ≤ N , ar e arbitr ary r e al functions, then τ n = | 0 ..N − 1 | d efine d by (9) solves the biline ar 2D T o da lattic e e q uation (4) . Pr o of: Under an exc h ange of the v ariables x and s , the cases of linear conditions (14) and (15) with differen t v alues δ = ± 1 are t ransf orm ed int o ea c h other, b ut the bilinear 2D T o da lattice equation (4) is in v ariant. Therefore, w e only n eed to c h ec k the c ase under δ = 1. In what follo ws, we set δ = 1. Let u s u s e ( E f )( n ) = f ( n + 1) and define ( L x φ i )( n ) = N X i =1 λ ij φ j ( n ) , ( L s φ i )( n ) = N X i =1 µ ij φ j ( n ) , 1 ≤ i ≤ N . (16) 4 Then, using (14), w e can compute t hat ∂ τ n ∂ x = N X i =1                  φ 1 ( n ) φ 1 ( n + 1) · · · φ 1 ( n + N − 1) . . . . . . . . . ∂ x φ i ( n ) ∂ x φ i ( n + 1) · · · ∂ x φ i ( n + N − 1) . . . . . . . . . φ N ( n ) φ N ( n + 1) · · · φ N ( n + N − 1)                  = ε N X i =1                  φ 1 ( n ) φ 1 ( n + 1) · · · φ 1 ( n + N − 1) . . . . . . . . . ( E φ i )( n ) ( E φ i )( n + 1) · · · ( E φ i )( n + N − 1) . . . . . . . . . φ N ( n ) φ N ( n + 1) · · · φ N ( n + N − 1)                  + N X i =1                  φ 1 ( n ) φ 1 ( n + 1) · · · φ 1 ( n + N − 1) . . . . . . . . . ( L x φ i )( n ) ( L x φ i )( n + 1) · · · ( L x φ i )( n + N − 1) . . . . . . . . . φ N ( n ) φ N ( n + 1) · · · φ N ( n + N − 1)                  = ε N X j =1              φ 1 ( n ) φ 1 ( n + 1) · · · ( E φ 1 )( n + j − 1) · · · φ 1 ( n + N − 1) φ 2 ( n ) φ 2 ( n + 1) · · · ( E φ 2 )( n + j − 1) · · · φ 2 ( n + N − 1) . . . . . . . . . . . . φ N ( n ) φ N ( n + 1) · · · ( E φ N )( n + j − 1) · · · φ N ( n + N − 1)              + N X i =1                  φ 1 ( n ) φ 1 ( n + 1) · · · φ 1 ( n + N − 1) . . . . . . . . . λ ii φ i ( n ) λ ii φ i ( n + 1) · · · λ ii φ i ( n + N − 1) . . . . . . . . . φ N ( n ) φ N ( n + 1) · · · φ N ( n + N − 1)                  = ε | 0 ..N − 2 , N | +  N X i =1 λ ii  τ n . Using a lmost the same argument, we can ob tain ∂ τ n ∂ s = − ε | − 1 , 1 ..N − 1 | +  N X i =1 µ ii  τ n . 5 F urther, w e can similarly compute that ∂ 2 τ n ∂ s∂ x = − | − 1 , 1 ..N − 2 , N | − τ n + ε  N X i =1 µ ii  | 0 ..N − 2 , N | +  N X i =1 λ ii  − ε | − 1 , 1 ..N − 1 | +  N X i =1 µ ii  τ n  = − | − 1 , 1 ..N − 2 , N | − τ n + ε  N X i =1 µ ii  | 0 ..N − 2 , N | − ε  N X i =1 λ ii  | − 1 , 1 ..N − 1 | +  N X i =1 λ ii  N X i =1 µ ii  τ n . Plugging these results into the bilinear equation (4 ) giv es ∂ 2 τ n ∂ s∂ x τ n − ∂ τ n ∂ s ∂ τ n ∂ x − τ n +1 τ n − 1 + τ 2 n = − | − 1 , 1 ..N − 2 , N || 0 ..N − 1 | + | 0 ..N − 2 , N || − 1 , 1 ..N − 1 | − | 1 ..N || − 1 ..N − 2 | . This sum is the Laplace expansion by N × N minors of the foll o wing 2 N × 2 N d eterminan t: − 1 2       [ − 1 , 0 , 1 ..N − 2 ] [ ∅ , N − 1 , N ] [ − 1 , 0 , ∅ ] [ 1 ..N − 2 , N − 1 , N ]       , where ∅ in d icates the N × ( N − 2) zero matrix, and [ ∅ , N − 1 , N ] = [ ∅ , Φ( n + N − 1) , Φ( n + N )] and [ − 1 , 0 , ∅ ] = [Φ( n − 1) , Φ ( n ) , ∅ ] with Φ( m ) = ( φ 1 ( m ) , · · · , φ N ( m )) T . Obvio usly , th is determinan t is zero. Therefore, the s olution is verified. ✷ The linear conditions (14) and (15) in the case of ( ε, δ ) = (1 , 1) is a generalizatio n of the conditions (10). Theorem 1 tells us that if a set of functions φ i ( n ) , 1 ≤ i ≤ N , satisfies all linear conditions in (14) and (15), then w e can get a Casoratian solution τ n = | 0 ..N − 1 | to the bilinear 2D T o da lattice equation (4). If we exc hange x and s in τ n , w e can get another Casoratian solution, based on Theorem 1. Let us observe ho w the Casoratian form ulation generate s solutions a little bit more carefully . F rom the compatibilit y cond itions φ i,xs = φ i,sx , 1 ≤ i ≤ N , of the conditions (14) and (15), we ha v e the equalities N X j,k =1 ( λ ij µ j k − µ ij λ j k ) φ k = 0 , 1 ≤ i ≤ N , (17) and th us w e see that the Caso rati determinan t Cas( φ 1 , φ 2 , · · · , φ N ) b ecomes zero at a p oin t ( x, s ) where the coefficient matrices A = A ( x ) = ( λ ij ( x )) N × N and B = B ( s ) = ( µ ij ( s )) N × N do not comm ute. Therefore, if A and B are constan t and don’t co mmute, then τ n = | 0 ..N − 1 | is zero. This sho ws that the redu ced case of (14) and (15) under A ( x ) B ( s ) − B ( s ) A ( x ) = 0 (18) 6 is imp ortant in generating non-trivial C asoratian solutions to the bilinear 2D T o da lattice equation (4). 3 Casoratian t yp e solutions W e w ould lik e to construct exac t solutions of the b ilinear 2D T od a latti ce equation (4) b y using the resulting Casoratian formulatio n and introdu cing a B¨ ac klund transformation. Theorem 2. If A ( x ) = ( λ ij ( x )) N × N and B ( s ) = ( µ ij ( s )) N × N ar e c ontinuous and sa tisfy (18) and A ( x ) Z x 0 A ( x ′ ) dx ′ = Z x 0 A ( x ′ ) dx ′ A ( x ) , (19) B ( s ) Z s 0 B ( s ′ ) ds ′ = Z s 0 B ( s ′ ) ds ′ B ( s ) , (20) then the line ar differ ential-differ enc e e quations (14) and (15) have the fol lowing solution Φ = Φ( n ) = exp( Z x 0 A ( x ′ ) dx ′ + Z s 0 B ( s ′ ) ds ′ )( p n 1 e ε ( p δ 1 x − p − δ 1 s )+ q 1 , · · · , p n N e ε ( p δ N x − p − δ N s )+ q N ) T , (21) wher e Φ = ( φ 1 , · · · , φ N ) T and p i 6 = 0 , q i , 1 ≤ i ≤ N , ar e arbitr ary r e al c onstants. Pr o of: The co ndition (18) implies that exp( Z x 0 A ( x ′ ) dx ′ + Z s 0 B ( s ′ ) ds ′ ) = exp ( Z x 0 A ( x ′ ) dx ′ ) exp( Z s 0 B ( s ′ ) ds ′ ) = exp ( Z s 0 B ( s ′ ) ds ′ ) exp( Z x 0 A ( x ′ ) dx ′ ) , A ( x ) exp( Z s 0 B ( s ′ ) ds ′ ) = exp( Z s 0 B ( s ′ ) ds ′ ) A ( x ) , B ( s ) exp( Z x 0 A ( x ′ ) dx ′ ) = exp( Z x 0 A ( x ′ ) dx ′ ) B ( s ) . The other t wo conditions (19) and (20) guaran tee that ∂ x exp( Z x 0 A ( x ′ ) dx ′ ) = A ( x ) exp ( Z x 0 A ( x ′ ) dx ′ ) , (22) ∂ x exp( Z s 0 B ( s ′ ) ds ′ ) = B ( s ) exp( Z s 0 B ( s ′ ) ds ′ ) , (23) resp ectiv ely . F urther, a d ir ect computation sho w s that ∂ Φ( n ) ∂ x = ε Φ( n + δ ) + A ( x )Φ( n ) , ∂ Φ( n ) ∂ s = − ε Φ( n − δ ) + B ( s )Φ( n ) . This v erifies t he solution in (21). ✷ 7 Noting that (14) and (15) are lin ear, any linear com bination of Φ defin ed by (21) with different sets of p i and q i , 1 ≤ i ≤ N , is again a solution to (14) and (15) . One example is the set of fu nctions φ i = M X j =1 p n ij e ( εp δ ij + λ i ) x − ( εp − δ ij − µ i ) s + q ij , 1 ≤ i ≤ N , (24) where the p ij ’s are arbitrary non-zero real constants and the q ij ’s, λ i ’s and µ i ’s are arbitrary real constan ts. Actually , Φ = ( φ 1 , · · · , φ N ) T satisfies the lin ear conditions (14) and (15) with A = diag( λ 1 , · · · , λ N ) and B = diag( µ 1 , · · · , µ N ). Thus, we ha ve a Casoratian solution τ n = | 0 ..N − 1 | = Cas( φ 1 , · · · , φ N ). The N -solit on solutions co rresp ond to M = 2 [2 0]. Th e situation with a general int eger M yields new Casoratian solutions in v olving man y free parameters. If for eac h l ≤ N , we fur ther tak e λ i = µ i , 1 ≤ i ≤ l , then Φ = ( φ 1 , ∂ λ 1 φ 1 , · · · , 1 k ! ∂ k 1 λ 1 φ 1 ; · · · ; φ l , ∂ λ l φ l , · · · , 1 k ! ∂ k l λ l φ l ) , (25) where k 1 + · · · + k l = N , satisfies the linear co nditions (14) and (15) with A = B = diag( C 1 , · · · , C l ) , C i =          λ i 0 1 λ i . . . . . . 0 1 λ i          , 1 ≤ i ≤ l. Th us, this giv es us the fo llo wing Casoratian solution τ n = C as( φ 1 , ∂ λ 1 φ 1 , · · · , 1 k ! ∂ k 1 λ 1 φ 1 ; · · · ; φ l , ∂ λ l φ l , · · · , 1 k ! ∂ k l λ l φ l ) . (26) Theorem 3. If τ n = τ n ( x, s ) solves the biline ar 2D T o da lattic e e quation (4 ) , and σ n = σ n ( x, s ) satisfies ∂ 2 σ n ∂ s∂ x σ n = ∂ σ n ∂ x ∂ σ n ∂ s , σ n +1 σ n − 1 = σ 2 n , (27) then the function ˜ τ n define d by ˜ τ n = ˜ τ n ( x, s ) = σ n ( αx, α − 1 s ) τ n ( αx, α − 1 s ) , (28) wher e α is a non-zer o r e al c onstant, pr esents another solution to the biline ar 2D T o da lattic e e quation (4) . Pr o of: Under th e first condition in (27), a direct compu tation t ells that ( ∂ 2 ˜ τ n ∂ s∂ x ˜ τ n − ∂ ˜ τ n ∂ x ∂ ˜ τ n ∂ s )( x, s ) = [ σ 2 n ( ∂ 2 τ n ∂ s∂ x τ n − ∂ τ n ∂ x ∂ τ n ∂ s )]( αx, α − 1 s ) . Th us, the second c ondition in (27) ensures that ( ∂ 2 ˜ τ n ∂ s∂ x ˜ τ n − ∂ ˜ τ n ∂ x ∂ ˜ τ n ∂ s − ˜ τ n +1 ˜ τ n − 1 + ˜ τ 2 n )( x, s ) = [ σ 2 n ( ∂ 2 τ n ∂ s∂ x τ n − ∂ τ n ∂ x ∂ τ n ∂ s − τ n +1 τ n − 1 + τ 2 n )]( αx, α − 1 s ) = 0 . 8 The theorem is prov ed. ✷ This theorem pro vides us with an auto-B¨ ac klund transform ation of the bilinear 2D T o da lattice equation (4). Generally , it also g enerates new solutions to the nonlinear 2D T o da lattice equations (1) and (8) from a giv en solution to the b ilinear 2D T od a lattice equation (4), through the trans- formations giv en in the introd uction. Ho we ve r, the case of α = 1 do esn ’t lead to new solutions to the nonlinear 2D T o da lattice equation (1). A p articular selection of σ n in Th eorem 3 engenders the follo wing corollary . Corollary 1. L et τ n = τ n ( x, s ) b e a solution to th e biline ar 2 D T o da la ttic e e quation (4) and α b e a non-zer o r e al c onstant. If a n ( x ) and b n ( s ) satisfy a n +1 ( x ) a n − 1 ( x ) b n +1 ( s ) b n − 1 ( s ) = ( a n ( x )) 2 ( b n ( s )) 2 , (29) then ˜ τ n with σ n ( x, s ) = a n ( x ) b n ( s ) : ˜ τ n = ˜ τ n ( x, s ) = a n ( αx ) b n ( α − 1 s ) τ n ( αx, α − 1 s ) (30) solves the biline ar 2D T o da la ttic e e quation (4) . In p articular, if a ( x ) , b ( s ) , f ( x ) and g ( s ) ar e r e al functions but f ( x ) and g ( s ) ar e p ositive or ne gative, then ˜ τ n with a n ( x ) = a ( x )( f ( x )) n and b n ( s ) = b ( s )( g ( s )) n : ˜ τ n = ˜ τ n ( x, s ) = a ( αx ) b ( α − 1 s )( f ( αx )) n ( g ( α − 1 s )) n τ n ( αx, α − 1 s ) (31) solves the biline ar 2D T o da la ttic e e quation (4) . In the ab o v e corollary , the assump tion that f ( x ) and g ( s ) are p ositiv e or negativ e is just to guaran tee that ˜ τ n is well defin ed o v er the domain of x, s ∈ R and n ∈ Z . A com bination of Theorems 1 , 2 an d 3 offers us an approac h for constructing Casoratia n t yp e solutions t o the bilinear 2 D T od a lattice equation (4). If we tak e a ( x ) = b ( s ) = 1, f ( x ) = x β and g ( s ) = s α , the resulting solution ˜ τ n with α = 1 g iv es the so lution presen ted in [20]. Let u s take ¯ φ i = p n i e λ i x + µ i s + q i = e − εp δ i x + εp − δ i s φ i , 1 ≤ i ≤ N , (32) where p i 6 = 0, q i , λ i and β i , 1 ≤ i ≤ N , are arbitrary real constan ts. The set of functions { φ i } N i =1 satisfies (14) and (15) with A = diag( λ 1 , · · · , λ N ) and B = diag( µ 1 , · · · , µ N ) as s ho w ed b efore, and ob viously , w e ha v e ¯ τ n = C as( ¯ φ 1 , · · · , ¯ φ N ) = exp  N X i =1 ( λ i x + µ i s + q i )  N Y i =1 p n i Y i>j ( p i − p j ) = exp ( − ε N X i =1 p δ i x ) exp( ε N X i =1 p − δ i s )Cas( φ 1 , · · · , φ N ) . (3 3) 9 The last equalit y in (33) also tells us a formula for τ n = C as( φ 1 , · · · , φ N ), where the φ i ’s are defined b y (24) with M = 1. It follo ws from the ab o v e corollary with α = 1 that ¯ τ n is a C asoratian solution to the bilinear 2D T o da lat tice equation (4). Ag ain from the ab o v e corollary , we h a v e a c lass of Casoratian type solutions to the bilinear 2D T o da la ttice equation (4) : ˜ τ n = a ( αx ) b ( α − 1 s )( f ( αx )) n ( g ( α − 1 s )) n exp  N X i =1 ( λ i αx + µ i α − 1 s + q i )  N Y i =1 p n i Y i>j ( p i − p j ) . (34) Ob viously , these solutions ˜ τ n are all just sp ecial cases of (31 ) with τ n = 1. T hey generate non- constan t s olutions to the non lin ear 2D T o da lattice equati on (8), b ut only th e zero solutio n to the nonlinear 2D T o d a la ttice equation (1). If we now tak e the functions φ i , 1 ≤ i ≤ N , in (24 ) with M = 2, i.e. , φ i = p n i 1 e ( εp δ i 1 + λ i ) x − ( εp − δ i 1 − µ i ) s + q i 1 + p n i 2 e ( εp δ i 2 + λ i ) x − ( εp − δ i 2 − µ i ) s + q i 2 , 1 ≤ i ≤ N , (35) then b y the abov e corollary , w e ha v e a class of Casoratian t yp e solutions to the bilinear 2D T o da lattice e quation (4): ˜ τ n = ˜ τ n ( x, s ) = a ( αx ) b ( α − 1 s )( f ( αx )) n ( g ( α − 1 s )) n Cas( φ 1 , · · · , φ N )( αx, α − 1 s ) . (36) A general case of M in (24) can pro duce more general Casoratian t yp e solutions to the bilinear 2D T o da la ttice equatio n (4). Su c h solutions ˜ τ n can also generate n ew solutions to the nonlinear 2D T o da la ttice equation (8), and if α 6 = 1, new solutions to the n onlinear 2 D T o d a lattice equation (1). 4 Concluding remarks A general Caso ratian form ulation of the bilinear 2D T o da la ttice equation (4) has b een present ed b y means of the bilinear form of (4). The resulting theory pro vides us with an e ffectiv e appr oac h for constructing exact solutions to the bilinear 2D T o da lattice equation (4). Sp ecial classes of functions satisfying (14) and (15), e.g., the fun ctions d efi ned b y (24 ) and (25), w ere used to gen- erate Casoratian solutions, and further using the B¨ ac klund transformation in Theorem 3, v arious examples of Casoratia n t yp e solutions were presente d. W e remark that the solutions ˜ τ n present ed in Corollary 1 ma y not b e exactly Casoratian, ev en if τ n is Casoratian. F or example, ˜ τ n is n on-Casoratian w h en f ( x ) and g ( s ) are not constan t fu n ctions. On the other hand, taking different t yp es of fun ctions for a ( x ) , b ( s ) , f ( x ) and g ( s ) can yield p ositon and complexiton t yp e solutio ns. There are also tw o other questions that we are int erested in. Th e fi rst question i s ho w to solve the system o f different ial-difference equations in (14) a nd (15) generally , in particular, in t he case where the conditions (19) and (20) are not satisfied, or more generally , the equations (22) and 10 (23) don’t h old. This will bring us v ery d ifferen t Casoratian solutions to the b ilinear 2D T o da lattice equation (4). Th e second qu estion is w h at kind of Casoratian form ulations can exist for Pfaffianization of discrete soliton equations [21, 22], for example, for Pfaffianization of the 2D T o da lattice equation [21]. Any answers to th ese t wo questions will enhance our un derstanding of b oth div ersit y of Casoratian t yp e solutions and universit y of Casoratian form u lations. Ac knowledgements: The work was in part supported b y State Key Lab orator y of Scientific and Eng i- neering Computing, Chinese Acade my o f Sciences, Beijing, P R China and the University of South Florida , T ampa, Florida , USA. 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