Extensions of the finite nonperiodic Toda lattices with indefinite metrics
aa r X i v : . [ n li n . S I] M a y Extensions of the finite nonperiodic Toda lattices withindefinite metrics
Jian Li, Chuanzhong Li ∗ School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China
Abstract
In this paper, we firstly construct a weakly coupled Toda lattices with indefinite met-rics which consist of 2 N different coupled Hamiltonian systems. Afterwards, we considerthe iso-spectral manifolds of extended tridiagonal Hessenberg matrix with indefinite met-rics what is an extension of a strict tridiagonal matrix with indefinite metrics. For theinitial value problem of the extended symmetric Toda hierarchy with indefinite metrics,we introduce the inverse scattering procedure in terms of eigenvalues by using the Ko-dama’s method. In this article, according to the orthogonalization procedure of Szeg¨o,the relationship between the τ -function and the given Lax matrix is also discussed. Wecan verify the results derived from the orthogonalization procedure with a simple exam-ple. After that, we construct a strongly coupled Toda lattices with indefinite metrics andderive its tau structures. At last, we generalize the weakly coupled Toda lattices withindefinite metrics to the Z n -Toda lattices with indefinite metrics. Mathematics Subject Classifications (2010) : 37K05, 37K10, 35Q53.
Keywords:
Hamiltonian systems, indefinite metrics, coupled Toda lattices, inverse scat-tering method, τ -function. Contents Z n -Toda lattices with indefinite metrics 15 ∗ Corresponding author:[email protected]. Introduction
Toda system is one of the most important integrable systems in mathematical physics.Many mathematicians have made significant contributions to the Toda equations and itsgeneralization [1], such as M. Toda, Y. Kodama, etc. In recent years, some senior math-ematicians have used different methods to study the Toda equations, such as symmetryand bilinear method, etc. This paper aims to extend the Toda equations via an extendedalgebra group [2], and the solutions can be pasted together to constitute a compact man-ifold. On the basis of [3, 4], the Toda lattices are produced by semisimple Lie algebra. Inthe process of solving the extended Toda equations, we use the inverse scattering method,it also promotes the development of mathematical physics, integrable systems and Liealgebras [5, 6]. According to the Hamiltonian systems of 2 N particles which described bythe finite nonperiodic Toda lattice hierarchy [7], then we introduce a pair of Hamiltoniansystems ( H, b H ) given by ( H = P Ni =1 y i + P Ni =1 exp( x i − x i +1 ) , b H = P Ni =1 y i b y i + P Ni =1 ( b x i − b x i +1 ) exp( x i − x i +1 ) . (1.1)The topology of an iso-spectral set of tridiagonal Hessenberg matrices was considered [7],and it has distinct real eigenvalues in the following form, L H = α ··· β α ··· ... ... ... ··· ··· α N −
10 0 0 0 ··· ··· β N − α N , (1.2) b L H = b α ··· b β b α ··· ... ... ... ··· ··· b α N −
10 0 0 0 ··· ··· b β N − b α N , (1.3)where the variables b α k and b β k in L H and b L H are expressed as s k b a k = − b y k and b b k = b x k − b x k +1 exp( x k − x k +1 ) with s i = ±
1. while different signs of s i may create different systems.The initial value problem of Toda equations were studied by applying the inverse scatteringmethod in [8], we generalize the above method and get general results on the basis of themethod mentioned in the references. With the help of the defined inner product in thispaper, the elements of L and b L can be expressed in a simple way.This paper is arranged as follows. In Section 2, we construct the weakly coupled Todaequations via the transformation [9], by calculating the equations (2.10), the elements of L and b L can be expressed through an inner product and the initial value of the weaklycoupled Toda lattices. Finally, we briefly describe the relationships between the elementsof L H , b L H and τ -functions, and analyse specific relationships between b τ i and b D i . Insection 3, we give a proof that the wave functions of the weakly coupled Toda lattices canbe solved by the inverse scattering method with the Gram-Schmidt’s orthogonalization.In Section 4, we illustrate these results with a specific example, and some properties ofthe elements in the example are discussed. In section 5, we introduce the strongly coupledToda lattices with indefinite metrics, and give some different conclusions compare withthe weakly coupled Toda lattices. In section 6, first we give a definition of the Z n -Todaequations by the algebraic transformation, and the solutions of the Z n -Toda equationsare obtained according to the initial value. Weakly coupled Toda lattices with indefinitemetrics
In this section, we define a weakly coupled Toda lattices with indefinite metrics. Forthe Hamiltonian (1.1), a transformation of variables will be introduced similarly as theone from Flaschka [9], which is about the classical Toda lattices with indefinite metrics: ( s k a k = − y k ,s k b a k = − b y k , k = 1 , . . . , N ; (2.1) ( b k = exp( x k − x k +1 ) , b b k = b x k − b x k +1 exp( x k − x k +1 ) , k = 1 , . . . , N − . (2.2)Then the extended Toda equations are written in this form with b = b b = b N = b b N = 0, ( da k dt = ( s k +1 b k − s k − b k − ) , d b a k dt = s k +1 b k b b k − s k − b k − b b k − ; (2.3) ( db k dt = b k ( s k +1 a k +1 − s k a k ) , d b b k dt = [( s k b a k − s k +1 b a k +1 ) b k + ( s k a k − s k +1 a k +1 ) b b k ] . (2.4)The equations (2.3) and (2.4) can also be expressed as the following Lax equations: ( ddt L = [ B, L ] , ddt b L = [ b B, L ] + [ B, b L ] , (2.5)where L and b L are a N × N tridiagonal matrix with real entries, L = s a s b ··· s b s a ... ... ... ··· s N − a N − s N b N − ··· s N − b N − s N a N , (2.6) b L = s c a s b b ··· s b b s c a ··· ... ... ··· s N − b a N − s N b b N − ··· s N − b b N − s N b a N , (2.7) B and b B are the projection of L , b L given by ( B := [( L ) > − ( L ) < ] , b B := [( b L ) > − ( b L ) < ] . (2.8)Note that, LS − and b LS − are symmetric tridiagonal matrix and S is a diagonal matrix S = diag( s , s , ..., s N ). In the this section, the extended Hamilton equations (1.1) can be xpressed by Lax equations (2.5) and the matrices (1.2), (1.3). In fact, the variables in(1.2) and (1.3) are given by α k = s k a k , b α k = s k b a k ,β k = s k s k +1 b k , b β k = 2 s k s k +1 b k b b k , (2.9)and there is no doubt that they are equivalent. In order to solve the problem of the Laxequations (2.5), we can use the inverse scattering method to construct a specific formulafrom [8]. There are four linear equations that are contained in (2.5), L Φ = ΦΛ , b L Φ + L b Φ = b ΦΛ , ddt Φ = B Φ , ddt b Φ = b B Φ + B b Φ , (2.10)where Φ is the eigenmatrix of L , and b Φ is the eigenmatrix of b L , Λ = diag( λ , ..., λ N − , λ N )is a diagonal matrix, and Φ, b Φ also satisfy the following relationship: Φ T S Φ = S, Φ T S b Φ + b Φ T S Φ = 0 , Φ S − Φ T = S − , Φ S − b Φ T + b Φ S − Φ T = 0 . (2.11)Particularly, if S = I (the identity matrix), then (2.10) shows that L can be diagonalizedby using an orthogonal matrix O ( N ); if S = diag(1 , ..., , − , ..., − O ( p, q ) with p + q = N . From theorthogonality of (2.11), we obtain the eigenmatrix Φ( ˆΦ) of L ( ˆ L ). Although eigenvaluesof L are real, the elements in Φ T S Φ differs from s i . The eigenmatrixs Φ and b Φ consist ofthe eigenvectors of L and b L , and considering the following system of linear equations ( Lφ = λφ, b Lφ + L b φ = λ b φ, (2.12)which the φ and b φ consist of Φ, b Φ are given in the following form,Φ = φ ( λ ) φ ( λ ) · · · φ ( λ N )... ... ... φ N ( λ ) φ N ( λ ) · · · φ N ( λ N ) , (2.13) b Φ = b φ ( λ ) b φ ( λ ) · · · b φ ( λ N )... ... ... b φ N ( λ ) b φ N ( λ ) · · · b φ N ( λ N ) . (2.14)From the first two equations of (2.11), we get something that looks like an “orthogonality”relationship as follows, (P Nk =1 s − k φ i ( λ k ) φ j ( λ k ) = δ ij s − i , P Nk =1 s − k [ b φ i ( λ k ) φ j ( λ k ) + φ i ( λ k ) b φ j ( λ k )] = 0 . (2.15) lso, we can get the similarly relationship from the another two equations of (2.11), (P Nk =1 s k φ k ( λ i ) φ k ( λ j ) = δ ij s i , P Nk =1 s k [ b φ k ( λ i ) φ k ( λ j ) + φ k ( λ i ) b φ k ( λ j )] = 0 . (2.16)According to (2.15), we extend the inner product with four functions of λ from [2], ( < f, g > := P Nk =1 s − k f ( λ k ) g ( λ k ) ,< f, b g > + < b f , g > := P Nk =1 s − k [ f ( λ k ) b g ( λ k ) + b f ( λ k ) g ( λ k )] , (2.17)where λ are arbitrary. The elements of L and b L can be expressed: ( a ij := ( L ) ij = s j < λφ i φ j >, b a ij := ( b L ) ij = s j < λ b φ i φ j > + s j < λφ i b φ j >. (2.18)Thus, the elements of L and b L can be expressed by the above inner product with φ i and b φ i . In fact, a lot of work has been finished in this area. Not only the Φ can be obtainedby the orthonormalization procedure of G. Szeg¨o [10], but there is another way to get theorthonormalization procedure, which is introduced by Kodama and Mclauglin [11]. Thespecific forms of φ ( t ) and b φ ( t ) are given below, φ i ( λ, t ) = e λt p D i ( t ) D i − ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c s c · · · s i − c ,i − φ ( λ )... ... . . . ... ... s c i s c i · · · s i − c i,i − φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.19)and b φ i ( λ, t ) = e λt [ D i ( t ) D i − ( t )] i − X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s k b c ,k · · · s i − c ,i − ... ... . . . ... ... s c · · · s k b c ,k · · · s i − c i,i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · b φ ( λ )... ... ... s c i · · · b φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − [ b D i ( t ) D i − ( t ) + D i ( t ) b D i − ( t )] e λt D i ( t ) D i − ( t )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · φ ( λ )... . . . ... s c i · · · φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.20)where c ij ( t ) = < φ i φ j e λt > , b c ij ( t ) = < b φ i φ j e λt > + < φ i b φ j e λt > . The D k ( t ) and b D k ( t )are expressed by the determinant of the k × k matrix with entries s i c ij ( t ) and s i b c ij ( t ), D k ( t ) = (cid:12)(cid:12) ( s i c ij ) ≤ i,j ≤ k (cid:12)(cid:12) , (2.21) b D k ( t ) = i X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s k b c ,k · · · s i c ,i ... ... ... ... s c i · · · s k b c i,k · · · s i c i,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.22)Note that D k (0) = 1 and b D k (0) = 0 for any k . With the formulas (2.19) and (2.20), weget the solutions for the problem of (2.10), we can derive the following proposition withthe above formulas. roposition 1. L ( t ) and b L ( t ) blow up to infinity at t while b D i ( t ) = D i ( t ) = 0 withsome t and i . For the case of matrices L and b L , the b D i ( t ) and D i ( t ) can be writtenwith the τ -functions, and the solutions α i , b α i , β i and b β i can be expressed in the followingforms: ( α i = s i a i = ddt log τ i τ i − , b α i = s i b a i = ddt b τ i τ i − − τ i b τ i − τ i τ i − , (2.23) β i = s i s i +1 b i = τ i +1 τ i − τ i , b β i = 2 s i s i +1 b i b b i = b τ i +1 τ i − + τ i +1 b τ i − τ i − τ i +1 τ i − b τ i τ i . (2.24)The derivative of the weakly coupled Toda equations (2.3) and (2.4) are expressed in thebilinear form, ( τ i τ ′′ i − ( τ ′ i ) = τ i +1 τ i − , b τ i τ ′′ i + τ i b τ ′′ i − b τ ′ i τ ′ i = b τ i +1 τ i − + b τ i − τ i +1 . (2.25)The τ i and b τ i can be written as determinants refer to [12, 13]: τ i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ τ ′ · · · τ ( i − τ ′ τ ′′ · · · τ ( i )1 ... ... . . . ... τ ( i − τ ( i )1 · · · τ (2 i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.26) b τ i = i − X k =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ · · · b τ ( k )1 · · · τ ( i − τ ′ · · · b τ ( k +1)1 · · · τ ( i )1 ... ... ... ... ... τ ( i − · · · b τ ( k + i − · · · τ (2 i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.27)where τ , b τ are given by τ = c = < φ φ e λt > = s − D , b τ = b c = < φ b φ e λt > = s − b D . From (2.21) and (2.22), we know that there is a certain relationship between τ i , b τ i and D i , b D i , τ i = s i [ Q i − k =1 ( β k ) i − k ] D i , b τ i = s i Q i − k =1 [( i − k ) b β k ( β k ) i − k − D i + ( β k ) i − k b D i ] , (2.28)where β k = β k (0) and b β k = b β k (0). From (2.28), we know that τ i , b τ i turn out to theblow-ups with both α i , b α i and β i , b β i at t for D ( t ) = b D ( t ) = 0. If there are some k thatmake β k = b β k = 0, we can not decide β k , b β k in the form of (2.25), the reason is that theoriginal system will be split into many small subsystems. Next, we are going to use the inverse scattering method to give the expressions of φ i ( λ, t )and b φ i ( λ, t ), which is mainly used in the article of Kodama [2], and the time variable ofΦ( t ) can be obtained by the orthonormalization procedure of Szeg¨o [10]. According tothe reference [2], we know that B = L − diag( L ) − L ) < , b B = b L − diag( b L ) − b L ) < . ue to the first two equations of (2.10): L Φ = ΦΛ and b L Φ + L b Φ = b ΦΛ, we get B =ΦΛΦ − − diag( L ) − L ) < , and b B = ΦΛ b Φ − + b ΦΛΦ − − diag( b L ) − b L ) < . Then thelatter two equations of (2.10) can be written as: ( ddt Φ = ΦΛ − [diag( L ) + 2( L ) < ]Φ , ddt b Φ = b ΦΛ − [diag( b L ) + 2( b L ) < ]Φ + [diag( L ) + 2( L ) < ] b Φ . (3.1)The elements of the Φ t and b Φ t are expressed by the right side of the equations (3.1), andthe vectors φ i ( λ k , t ), b φ i ( λ k , t ) ( k = 1 , ..., N ) of the first line in Φ t and b Φ t are given by ( ddt φ ( λ k ) = [ λ k − s < λφ ( λ ) > ] φ ( λ k ) , ddt b φ ( λ k ) = [ λ k − s < λφ ( λ ) > ] b φ ( λ k ) − s < λ b φ ( λ ) φ ( λ ) > φ ( λ k ) . (3.2)Then (3.2) can be readily solved in the form φ ( λ k , t ) = ψ ( λ k ,t ) √ s <ψ ( λ,t ) > , b φ ( λ k , t ) = b ψ ( λ k ,t ) √ s <ψ ( λ,t ) > − s ψ λk,t ) <ψ λ,t ) b ψ λ,t ) > [ s <ψ λ,t ) > ] 32 , (3.3)with ψ ( λ k , t ) = φ ( λ k ) e λt and b ψ ( λ k , t ) = b φ ( λ k ) e λt . The elements of the second line inΦ t , b Φ t are expressed, ddt φ ( λ k ) = [ λ k − s < λφ ( λ ) > ] φ ( λ k ) − s < λφ ( λ ) φ ( λ ) > φ ( λ k ) , ddt b φ ( λ k ) = ( λ k − s [ < λφ ( λ ) > b φ ( λ k ) − s < λ b φ ( λ ) φ ( λ ) > ] φ ( λ k ) − s < λφ ( λ ) φ ( λ ) > φ ( λ k ) − s < λφ ( λ ) b φ ( λ ) + λ b φ ( λ ) φ ( λ ) > φ ( λ k ) . (3.4)Through the integration, the φ ( λ k ), b φ ( λ k ) can be written as φ ( λ k , t ) = ψ ( λ k ,t ) √ s <ψ ( λ,t ) > , b φ ( λ k , t ) = b ψ ( λ k ,t ) √ s <ψ ( λ,t ) > − s ψ λk,t ) <ψ λ,t ) b ψ > ( λ,t )[ s <ψ λ,t ) > ] 32 , (3.5)where ψ ( λ k , t ) = φ ( λ k ) e λt − s < φ ( λ ) φ ( λ, t ) e λt >φ ( λ k , t ) , b ψ ( λ k , t ) = b φ ( λ k ) e λt − s [ < φ ( λ, t ) φ ( λ, t ) e λt > b φ ( λ k , t )+ s < φ ( λ ) b φ ( λ, t ) e λt + b φ ( λ ) φ ( λ, t ) e λt > ] φ ( λ k , t ) . (3.6)Generally, the i th lines in (3.1) satisfy ddt φ i ( λ k ) = [ λ k − s i < λφ i ( λ ) > ] φ i ( λ k , t ) − P i − j =1 [ s j < λφ i ( λ ) φ j ( λ ) > ] φ j ( λ k , t ) , ddt b φ i ( λ k ) = [ λ k − s i < λφ i ( λ ) > b φ i ( λ k , t ) − s i < λ b φ i ( λ ) φ i ( λ ) > φ i ( λ k , t ) − P i − j =1 [ < λφ i ( λ ) φ j ( λ ) > b φ j ( λ k , t )+ < λ b φ i ( λ ) φ j ( λ ) + λφ i ( λ ) b φ j ( λ ) >φ j ( λ k , t )] . (3.7)Then, it is the same with the procedures above, φ i ( λ k ) = ψ i ( λ k ,t ) √ s i <ψ i ( λ,t ) > , b φ i ( λ k ) = b ψ i ( λ k ,t ) √ s i <ψ i ( λ,t ) ψ j ( λ,t ) > − siψi ( λk,t ) < b ψi ( λ,t ) ψi ( λ,t ) > [ si<ψ i ( λ,t ) > ] 32 , (3.8) ψ i ( λ k , t ) = φ i ( λ k ) e λt − P i − j =1 s j < φ i ( λ ) φ j ( λ, t ) e λt >φ j ( λ k , t ) , b ψ i ( λ k , t ) = b φ i ( λ k ) e λt − P i − j =1 s j [ < φ i ( λ ) φ j ( λ, t ) e λt > b ψ i ( λ k , t )+ < b φ i ( λ ) φ j ( λ, t ) + φ i ( λ ) b φ j ( λ, t ) >e λt ] φ j ( λ k , t ) . (3.9)Now, we are going to give a further proof, there are two linear equations given by ( Φ = T Ψ , b Φ = b T Ψ + T b Ψ , (3.10)then we give Ψ = ψ ( λ ) · · · ψ ( λ N ) ψ ( λ ) · · · ψ ( λ N )... ψ N ( λ ) · · · ψ N ( λ N ) , (3.11) b Ψ = b ψ ( λ ) · · · b ψ ( λ N ) b ψ ( λ ) · · · b ψ ( λ N )... b ψ N ( λ ) · · · b ψ N ( λ N ) , (3.12)and T = diag[ s i (cid:10) ψ i (cid:11) ] − , i = 1 , , · · · , N, b T = diag (cid:0) − s i h b ψ i ψ i i [ s h ψ i i ] (cid:1) , i = 1 , , · · · , N. (3.13)According to (3.10), the expressions of the equations (2.9) can be expressed as ( T − LT )Ψ = ΨΛ , ( T − LT ) b Ψ + ( b T − LT + T − b LT + T − L b T )Ψ = b ΨΛ , ddt Ψ = ( T − BT )Ψ − ( ddt log T )Ψ , ddt b Ψ = ( T − BT ) b Ψ + ( b T − BT + T − b BT + T − B b T )Ψ − ( ddt (log b T Ψ + log T b Ψ) . (3.14)From (3.1), we find T − BT = − T − LT ) < + T − LT − ( T − diag( L ) T ) , b T − BT + T − b BT + T − B b T = − b T − LT + T − b LT + T − L b T ) < + b T − LT + T − L b T − ( b T − diag( L ) T + T − diag( b L ) T + T − diag( L ) b T ) . (3.15)Further, from (3.14), we have dψdt = − T − LT ) < ψ + λψ − (diag( L ) + ddt log T ) ψ, d b ψdt = − b T − LT ) + ( T L b T )] < ψ − T − LT ) < b ψ + λ b ψ − (diag( L ) + ddt log T ) b ψ − ( ddt b TT ) ψ, (3.16)where the structure of b T is similar to T . According to (3.16), the solutions with ψ , b ψ expressed as dψ i dt = − P i − j =1 <λψ i ψ j ><ψ j >ψ j + λψ i , d b ψ i dt = − P i − j =1 < λψ i ψ j > − < b ψ j ψ j > [ <ψ j > ] ψ j + <λψ i b ψ j > + <λ b ψ i ψ j ><ψ j > ψ j + <λψ i ψ j ><ψ j > b ψ j + λ b ψ j , (3.17) nd ( ddt log < ψ i > = s j < λφ i > = a ij , ddt log2 < ψ i b ψ i > = s j [ < λ b φ i φ i > + < λ b φ j φ i > ] = b a ij . (3.18)Obviously, (3.18) can be evolved from (3.17). And through simple calculation, ( ψ ( λ, t ) = Q ( t ) φ ( λ ) e λt , b ψ ( λ, t ) = [ b Q ( t ) φ ( λ ) + Q ( t ) b φ ( λ )] e λt , (3.19)where Q ( t ), b Q ( t ) are lower triangular matrices and φ ( λ ) = φ ( λ, b φ ( λ ) = b φ ( λ, ψ ( λ, t ), b ψ ( λ, t ) which satisfy ψ ( λ,
0) = φ ( λ ), b ψ ( λ,
0) = b φ ( λ ), and s i < ψ i ψ j > = s i < φ i φ j > = δ ij ( t = 0), s i < b φ i φ j + φ i b φ j > = 0( t =0). For the “orthogonality” relations of (2.12) and (2.13), it show that the < ψ i ψ j > = 0, < b ψ i ψ j + ψ i b ψ j > = 0 for i = j , so the “orthogonality” relations can be written like this: ( < ψ i φ j e λt > = P Nk =1 s − k < ψ i ( λ k , t ) φ j ( λ k ) e λt > = 0 ,< b ψ i φ j e λt > + < ψ i b φ j e λt > = P Nk =1 (cid:16) s − k < b ψ i φ j e λt > + s − k < ψ i b φ j e λt > (cid:17) = 0 . (3.20)The solutions between ψ i ( λ, t ) and b ψ i ( λ, t ) of (3.18) are given from [2], ψ i ( λ, t ) = e λt D i − ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c s c · · · s i − c ,i − φ ( λ )... ... . . . ... ... s c i s c i · · · s i − c i,i − φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.21)where c ij = D φ i φ j e λt E , and the elements of D k ( t ) are s j c ij ( t ), i = 1 , , ..., N , D k ( t ) = | ( s i c ij ( t )) ≤ i,j ≤ k | . (3.22) Lemma 1.
According to (3.17) , the b ψ i can be expressed b ψ i ( λ, t ) = e λt D i − ( t ) i − X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s k b c ,k · · · φ ( λ ) ... ... . . . ... ... s c i · · · s k b c i,k · · · φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + e λt D i − ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · b φ ( λ ) ... ... ... s c i · · · b φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − b D i − ( t ) e λt D i − ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s i − c ,i − φ ( λ ) ... . . . ... ... s c i · · · s i − c i,i − φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.23) where b c ij ( t ) = < b φ i φ j e λt > + < φ i b φ j e λt > , and k represents the number of columns inthe determinant above, b D k ( t ) = i X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s k b c ,k · · · s i c ,i ... ... ... ... s c i · · · s k b c i,k · · · s i c i,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.24) Proof.
From (3.20) and (3.19), we have ( s j P ik =1 Q ik c kj ( t ) = 0 ,s j P ik =1 [ b Q ik c kj ( t ) + Q ik b c kj ( t )] = 0 , ≤ j ≤ i − . (3.25) olving (3.25) for Q ik and b Q ik , we have Q ik = − D ki − ( t ) D i − ( t ) , b Q ik = − b D ki − ( t ) D i − ( t ) + b D i − ( t ) D ki − ( t ) D i − ( t ) . (3.26)In fact, b D ki − ( t )( D ki − ( t )) is the substitution of the k th and the i th row of b D i − ( t )( D i − ( t )).From (3.19), we have b ψ i = e λt i X k =1 ( Q ik b φ k + b Q ik φ k ) (3.27)= − e λt i − X k =1 b D i − ( t ) D ki − ( t ) − D i − ( t ) b D ki − ( t ) D i − ( t ) φ k + e λt D ki − ( t ) D i − ( t ) b φ k ! + e λt D i − ( t ) D i − ( t ) b φ i (3.28)= e λt D i − i − X k =1 ( − i + k b φ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s c i · · · s c ,i − ... · · · ... · · · ... s i − c i − , · · · s i − c i − ,i · · · s i − c i − ,i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + b φ i D i − ( t ) (3.29) − e λt i − X k =1 b D i − ( t ) D ki − ( t ) − D i − ( t ) b D ki − ( t ) D i − ( t ) φ k , (3.30)which is just (3.23). From (3.21) and (3.23), < ψ i > , < b ψ i ψ i > can be expressedwith D i and b D i < ψ i > = D i s i D i − ,< b ψ i ψ i > = D i − b D i − b D i − D i s i D i − . (3.31)Then we can obtain the formulas that we have mentioned above, φ i ( λ, t ) = e λt p D i ( t ) D i − ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c s c · · · s i − c ,i − φ ( λ )... ... . . . ... ... s c i s c i · · · s i − c i,i − φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.32) b φ i ( λ, t ) = e λt [ D i ( t ) D i − ( t )] i − X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c ,i − · · · s k b c ,k · · · s i − c ,i − ... ... . . . . . . ... s c ,i − · · · s k b c i,k · · · s i − c i,i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · b φ ( λ )... ... ... s c i · · · b φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − [ b D i ( t ) D i − ( t ) + D i ( t ) b D i − ( t )] e λt D i ( t ) D i − ( t )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · φ ( λ )... . . . ... s c i · · · φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.33)By using the formulas (3.33) and (3.32), we get the solutions for (2.10). The methodabove is similar to szeg¨o [10], and it is also equivalent to the procedure of Gram-Schmidt[7]. .1 Example Next, we give a simple example to verify our results, and talk about the properties ofthe solutions. Let L , b L are a 2 × S = diag(1 , − L , b L are givenby L = (cid:18) a − b b − a (cid:19) , (3.34) b L = b a − b b b b − b a ! . (3.35)According to (2.3) and (2.4), we have da dt = − b − b b , d b a dt = − b b b , db dt = − b ( a + a ) − b b ( b a + b a ) , d b b dt = − [( a + a ) b b + ( b a + b a ) b ] , da dt = − b , d b a dt = − b b b − b b , (3.36)note that, b = b b = 0. From (3.36), if the initial value of ( a + a ) is positive, one canfind ( a + a ) −→ ∞ , then b , b b increase bigger and faster, and the corresponding resultis that b a −→ ∞ . On the contrary, if the initial value of a + a −→ −∞ , then b and b b −→ a i , b a i , b i and b b i for (i=1, 2), and keepone parameter m in L , then we discuss its eigenvalues and eigenvectors of L and theirproperties at the same time. While a , b a and b b are equal to 0, take b and b a are equalto 1, and take a as a parameter, then we have L = (cid:18) − − m (cid:19) , (3.37) b L = (cid:18) − (cid:19) . (3.38)And what is more, we can get the eigenvalues and eigenvectors of the particular matrix,which the specific forms are given as follow, λ = ( √ m − − m ) ,λ = ( −√ m − − m ) , b λ = 0 , b λ = − , (3.39)Φ = (cid:18) λ λ − − (cid:19) , (3.40) b Φ = (cid:18) (cid:19) , (3.41)where λ i and b λ i ( i = 1 ,
2) are characteristic value. Then we discuss the value of m below,different values of m produce different results: (1) when m ≥
2, then 0 ≥ λ ≥ λ ; (2) hen | m | <
2, then λ and λ are complex; (3) when m ≤ −
2, then λ > λ > λ and λ are real, that is exactly the cases we need. According to (3.39), (3.40) and (3.41), wecan get some specific results:Φ ( t ) = 1 e ( λ +2 λ ) t (cid:18) λ e λ t λ e λ t − e λ t − e λ t (cid:19) , (3.42) b Φ ( t ) = λ +1 e λ t + λ + e λ t (2 e λ t ) ( λ +1 e λ t + λ + e λ t (2 e λ t ) ) e t ( λ +1) e ( λ − ( λ + λ ) e λ t [( λ + λ +2) e (2 λ λ t ] ( λ +1) e ( λ − ( λ + λ ) e λ t [( λ + λ +2) e (2 λ λ t ] e t = b Φ , b Φ , b Φ , b Φ , ! , (3.43)where b Φ i,j ( t )(1 ≤ i, j ≤
2) are obtained from (3.43). The solutions of the extended Todaequations are obtained from (2.19) and (2.20), L ( t ) = (cid:18) λ e λ + λ ) t + λ e λ t − e λ + λ ) t − e λ t − e λ + λ ) t − e λ t λ e λ + λ ) t + λ e λ t (cid:19) , (3.44) b L ( t ) = 1 e ( λ +2 λ ) t (cid:18)b a , ( t ) b a , ( t ) b a , ( t ) b a , ( t ) (cid:19) , (3.45)where b a , ( t ) = [ λ e (3 λ +2 λ ) t + λ e λ ( t ) ] b Φ , ( t ) − [ e (3 λ +2 λ ) t + λ e λ ( t ) ] b Φ , ( t )+[ λ e (3 λ +2 λ ) t + λ e λ ( t ) − e λ t ] b Φ , ( t ) − [ e (3 λ +2 λ ) t + λ e λ ( t ) ] b Φ , ( t ) , (3.46) b a , ( t ) = [ λ e (3 λ +2 λ ) t + λ e (4 λ + λ )( t ) ] b Φ , ( t ) − [ λ e (3 λ +2 λ ) t + e (4 λ + λ ) t ] b Φ , ( t )+[ λ e (3 λ +2 λ ) t + λ e (4 λ + λ ) t ] b Φ , ( t ) − [ λ e (3 λ + λ ) t + e (4 λ + λ ) t − λ e λ t ] b Φ , ( t ) , (3.47) b a , ( t ) = [ λ e (3 λ +2 λ ) t + λ e λ ( t ) ] b Φ , ( t ) − [ λ e (3 λ +2 λ ) t + e λ ( t ) ] b Φ , ( t )+[ λ e (3 λ +2 λ ) t + λ e λ ( t ) − e λ t ] b Φ , ( t ) − [ e (3 λ +2 λ ) t + e λ ( t ) ] b Φ , ( t ) , (3.48) b a , ( t ) = [ λ e (3 λ +2 λ ) t + λ e (4 λ + λ )( t ) ] b Φ , ( t ) − [ e (3 λ +2 λ ) t + λ e (4 λ + λ ) t ] b Φ , ( t )+[ λ e (3 λ +2 λ ) t + λ e (4 λ + λ ) t − e λ t ] b Φ , ( t ) − [ e (3 λ +2 λ ) t + e (4 λ + λ ) t ] b Φ , ( t ) . (3.49)In addition, according to (3.43), there is a situation that is m = ±
2, one of the things topay attention is that L ( t ) ± diag(1 , In this section, we introduce a new strongly coupled Toda lattices with indefinitemetrics. For the Hamiltonian (1.1), we give a extended transformation of variables, ( s k a k = − y k ,s k e a k = − e y k , k = 1 , . . . , N, (4.1) ( b k = exp( x k − x k +1 ) cosh( e x k − e x k +1 ) , e b k = exp( x k − x k +1 ) sinh( e x k − e x k +1 ) , k = 1 , . . . , N − . (4.2) n addition, the strongly coupled Toda lattices with indefinite metrics can be expressedas: ( da k dt = [ s k +1 ( b k + e b k ) − s k − ( b k − + e b k − )] , d e a k dt = 2 s k +1 b k e b k + 2 s k − b k − e b k − , (4.3) ( db k dt = [ s k +1 ( b k a k +1 + e b k e a k +1 ) − s k ( b k a k + e b k e a k )] , d e b k dt = [ s k +1 ( e b k a k +1 + b k e a k +1 ) − s k ( e b k a k + b k e a k )] , (4.4)where b = e b = b N = e b N = 0. According to the strongly coupled Toda lattices withindefinite metrics above, we can use Lax pair to express it as following, ( ddt L = [ B, L ] + [ e B, e L ] , ddt e L = [ e B, L ] + [ B, e L ] , (4.5)where L , e L have the following form: L = s a s b ··· s a s b ··· s b s a ··· s b s a ··· ... ... ... ··· ··· s N a N ··· ··· s N b N − s N a N , (4.6) e L = s e a s e b ··· s e a s e b ··· s e b s e a ··· s e b s e a ··· ... ... ... ··· ··· s N − e b N − s N e a N ··· ··· s N e a N , (4.7)and B , e B are given by ( B := [( L ) > − ( L ) < ] , e B := [( e L ) > − ( e L ) < ] . (4.8)For Lax equations (4.5), we get some linear equations, L Φ + e L e Φ = ΦΛ , e L Φ + L e Φ = e ΦΛ , ddt Φ = B Φ + e B e Φ , ddt e Φ = e B Φ + B e Φ , (4.9)where Φ is the eigenmatrix of L , e Φ is the eigenmatrix of e L , and Λ is a diagonal matrix.Thus, e Φ and Φ satisfy some relationships: Φ S − Φ T + e Φ S − e Φ T = S − , e Φ S − Φ T + Φ S − e Φ T = 0 , Φ T S Φ + e Φ T S e Φ = S, e Φ T S Φ + Φ T S e Φ = 0 , (4.10) here ( Φ = [ φ i ( λ j )] ≤ i,j ≤ N , e Φ = [ e φ i ( λ j )] ≤ i,j ≤ N . (4.11)Form the equations of (4.10), one can get the relationships, P Nk =1 s − k [ φ i ( λ k ) φ j ( λ k ) + e φ i ( λ k ) e φ j ( λ k )] = δ ij s − i , P Nk =1 s − k [ e φ i ( λ k ) φ j ( λ k ) + φ i ( λ k ) e φ j ( λ k )] = 0 , P Nk =1 s k [ φ k ( λ i ) φ k ( λ j ) + e φ k ( λ i ) e φ k ( λ j )] = δ ij s i , P Nk =1 s k [ e φ k ( λ i ) φ k ( λ j ) + φ k ( λ i ) e φ k ( λ j )] = 0 . (4.12)So, the extended matrices of L and e L are expressed by ( a ij := ( L ) ij = s j < λφ i φ j + λ e φ i e φ j >, e a ij := ( e L ) ij = s j < λ e φ i φ j + λφ i e φ j >. (4.13)From the inverse scattering method, two new explicit forms of Φ( t ), e Φ( t ) are given asfollowing φ i ( λ, t ) = M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s i − c ,i − φ ... ... ... ... s c i · · · s i − c i,i − φ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + M [ i ] X q =0 X i − P j =1 k j =2 q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k ]11 · · · s i − c [ k i − ]1 ,i − φ ... ... ... ... s c [ k ] i · · · s i − c [ k i − ] i,i − φ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)c M [ i ] X q =0 X i − P j =1 k j =2 q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k ]11 · · · s i − c [ k i − ]1 ,i − φ ... ... ... ... s c [ k ] i · · · s i − c [ k i − ] i,i − φ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.14)where M = H e λt H − H , c M = − H e λt H − H , and e φ i ( λ, t ) = c M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c · · · s i − c ,i − φ ... ... ... ... s c i · · · s i − c i,i − φ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + c M [ i ] X q =0 X i − P j =1 k j =2 q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k ]11 · · · s i − c [ k i − ]1 ,i − φ ... ... ... ... s c [ k ] i · · · s i − c [ k i − ] i,i − φ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + M [ i ] X q =0 X i − P j =1 k j =2 q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k ]11 · · · s i − c [ k i − ]1 ,i − φ ... ... ... ... s c [ k ] i · · · s i − c [ k i − ] i,i − φ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.15)where H = √ b q a +( a − b ) , H = q a +( a − b ) √ , and a = D i ( t ) D i − ( t ) + e D i ( t ) e D i − ( t ), b = e D i ( t ) D i − ( t ) + D i ( t ) e D i − ( t ), and e D k ( t ) = [ i ] X q =0 X i P j =1 k j =2 q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k ]11 · · · s p c [ k p ]1 ,p · · · s i c [ k i ]1 ,i ... ... ... ... s c [ k ] i · · · s p c [ k p ] i,p · · · s i c [ k i ] i,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.16) k ( t ) = [ i ] X q =0 X i P j =1 k j =2 q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k ]11 · · · s p c [ k p ]1 ,p · · · s i c [ k i ]1 ,i ... ... ... ... s c [ k ] i · · · s p c [ k p ] i,p · · · s i c [ k i ] i,i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.17)and c [ k p ] ij = ( c ij , k p = 0 e c ij , k p = 1 . Thus we obtain the solutions (4.14) and (4.15) for the problem(4.9). In fact, for the matrices L , e L , the determinants D i ( t ), e D i ( t ) can be written with τ -functions, then α i , e α i , β i and e β i are expressed as α i = s i a i = ddt log ( τ i − e τ i )( τ i − − e τ i − ) τ i − − e τ i − , e α i = s i e a i = ddt log ( τ i + e τ i )( τ i − − e τ i − )( τ i − e τ i )( τ i − + e τ i − ) ,β i = s i s i +1 ( b i + e b i ) = ( τ i +1 τ i − + e τ i +1 e τ i − )( τ i + e τ i ) − e τ i +1 τ i − + τ i +1 e τ i − ) τ i e τ i ( τ i − e τ i ) , e β i = 2 s i s i +1 b i e b i = ( e τ i +1 τ i − + τ i +1 e τ i − )( τ i + e τ i ) − τ i +1 τ i − + e τ i +1 e τ i − ) τ i e τ i ( τ i − e τ i ) . (4.18)The strongly coupled Toda equations also can be expressed by ( τ i τ ′′ i + e τ i e τ ′′ i − ( τ ′ i ) + ( e τ ′ i ) = τ i +1 τ i − + e τ i +1 e τ i − , e τ i τ ′′ i + τ i e τ ′′ i − τ ′ i e τ ′ i = e τ i +1 τ i − + τ i +1 e τ i − . (4.19)From [12–14], the τ -functions are written in the form with a simple structure, τ i = [ i ] X q =0 X i P j =1 k j =2 q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ , [ k ] τ ′ , [ k ] · · · τ ( i − , [ k i − ] τ ′ , [ k ] τ ′′ , [ k ] · · · τ ( i )1 , [ k i − ] ... ... . . . ... τ ( i − , [ k ] τ ( i )1 , [ k ] · · · τ (2 i − , [ k i − ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.20) e τ i = [ i ] X q =0 X i P j =1 k j =2 q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ , [ k ] τ ′ , [ k ] · · · τ ( i − , [ k i − ] τ ′ , [ k ] τ ′′ , [ k ] · · · τ ( i )1 , [ k i − ] ... ... . . . ... τ ( i − , [ k ] τ ( i )1 , [ k ] · · · τ (2 i − , [ k i − ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.21)Similarly, τ ( i )1 , [ k p ] = ( τ ( i )1 , k p = 0 e τ ( i )1 , k p = 1 , where τ is given by τ = c := < ( φ ) + ( e φ ) > e λt , e τ = e c := < φ e φ e λt > , and τ ( i )1 = d i τ dt i , e τ ( i )1 = d i e τ dt i . Therefore, the relationshipsbetween β i , e β i , D i and e D i are given by τ i = s i [ Q i − k =1 ( β k + e β k ) i − k ( D i + e D i ) + ( β k − e β k ) i − k ( D i − e D i )] , e τ i = s i [ Q i − k =1 ( β k + e β k ) i − k ( D i + e D i ) + ( β k − e β k ) i − k ( e D i − D i )] . (4.22) Z n -Toda lattices with indefinitemetrics In the next part, we will give a new finite nonperiodic Z n -Toda lattices with indefinitemetrics as following. efinition 1. According to (2.3) , we define finite nonperiodic Z n -Toda lattice equationswith indefinite metrics as: da k,l dt = s k +1 P p + q = l +1 b k,p b k,q − s k − P p + q = l +1 b k − ,p b k − ,q , db k,l dt = ( s k +1 P p + q = l +1 b k,p a k +1 ,q − s k P p + q = l +1 b k,p b k,q ) . (5.1)When l = 1, a k,l and b k,l are equivalent to a k and b k [2]. In fact, before defining finitenonperiodic Z n -Toda lattice equations, we introduce a more general transformation ofvariables, the specific transformation is given by s k a k,l = − y k,l , ( k = 1 , , · · · , N ) b k,l = P i k + ··· + i j k j = k b x k i ··· b x kjij k ! ··· k p ! exp( x k,l − x k +1 ,l ) , ( k = 1 , , · · · , N −
1) (5.2)where b x k,l = ( x k,l − x k +1 ,l ). Meanwhile, the Z n -Hamilton quantity is as H k = 12 N X k =1 X p + q = k y k,p y k,q + X i k + ··· + i j k j = k x k i · · · x k j i j k ! · · · k p ! exp( x k, − x k +1 , ) , (5.3)where x i j = x i,j − x i +1 ,j . According to the definition of the finite nonperiodic Z n -Todalattice equations with indefinite metrics (5.1), we can obtain its Lax equations, ddt L k = X p + q = k +1 [ B p , L q ] , (5.4)where L k = s P a ,k Γ k − s P b ,k Γ k − ··· s P b ,k Γ k − s P a ,k Γ k − s P b ,k Γ k − ··· ... ... ... ··· s N − P a N − ,k Γ k − s N P b N − ,k Γ k − ··· s N − P b N − ,k Γ k − s N P a N,k Γ k − , (5.5)and B k = [( L k ) > − ( L k ) < ]. According to the extended general variables, α k,l , β k,l , a k,l and b k,l are given by ( α k,l = s k a k,l ,β k,l = P p + q = k +1 s k s k +1 b p,l b q,l . (5.6)For (5.4), linear equations produced from the inverse scattering method can be expressedas: P p + q = k +1 L p Φ q = ΛΦ k , ddt Φ k = P p + q = k +1 B p Φ q , (5.7)where Φ k is the eigenmatrix of L k , and Φ k ≡ [ φ [ k ] ( λ ) , · · · , φ [ k ] ( λ N )] = [ φ [ k ] i ( λ j )] ≤ i,j ≤ N .Further more, Φ k satisfies Φ T S Φ = S, P p + q = k Φ Tp S Φ q = 0 , k = 2 , · · · , N, Φ S − Φ T = S − , P p + q = k Φ p S − Φ Tq = 0 , k = 2 , · · · , N. (5.8) rom (5.8), one can get the “orthogonality” relations: P Nk =1 s − k φ [1] i ( λ k ) φ [1] j ( λ k ) = δ ij s − i , P Nk =1 s − k N P p + q =3 φ [ p ] i ( λ k ) φ [ q ] j ( λ k ) ! = 0 , P Nk =1 s k φ [1] k ( λ i ) φ [1] k ( λ j ) = δ ij s i , P Nk =1 s k N P p + q =3 φ [ p ] k ( λ i ) φ [ q ] k ( λ j ) ! = 0 . (5.9)So, the L k can be expressed by L k = X p + q = k +1 Φ p ΛΦ Tq . (5.10)According to the proof of section 3, the specific form of φ [ k ] i ( λ, t ) can be given as φ [ k ] i ( λ, t ) = D ( t ) N − X j =1 X [ p + ··· + p i − − ( i − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k p ]11 · · · s i − c [ k pi − ]1 ,i − φ ( λ )... ... . . . s c [ k p ] i · · · s i − c [ k pi − ] i,i − φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.11)+ D ( t ) N − X j =1 X [ p + ··· + p i − − ( i − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k p ]11 · · · s i − c [ k pi − ]1 ,i − φ ( λ )... ... . . . s c [ k p ] i · · · s i − c [ k pi − ] i,i − φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.12)+ · · · + D n − ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s c [ k ]11 · · · s i − c [ k i − ]1 ,i − φ ( λ )... ... . . . s c [ k ] i · · · s i − c [ k i − ] i,i − φ i ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5.13)where D i ( t ) is the n -th order Frobenius form of e λt √ D i ( t ) D i − ( t ) , and the specific forms aregiven by D i ( t ) = P k + k + ··· + k p = i − − i v k v k ··· v ki − i − v i e λt . For v i above, we used a variablereplacement: u i = P p + q = i +1 D i,p ( t ) D i − ,q ( t ), so we can get the relationship between v k and u k by iterative methods. Acknowledgements:
Chuanzhong Li is supported by the National Natural ScienceFoundation of China under Grant No. 11571192 and K. C. Wong Magna Fund in NingboUniversity.
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