Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations

Transcription

1 Closed-fom expessions fo integals of taveling wave eductions of integable lattice equations Dinh T Tan, Pete H van de Kamp, G R W Quispel Depatment of Mathematics, La Tobe Univesity, Victoia 386, Austalia td2tan@studentslatobeeduau Abstact We give a method to calculate closed-fom expessions in tems of multisums of poducts fo integals of odinay diffeence equations which ae obtained as taveling wave eductions of integable patial diffeence equations Impotant ingedients ae the staicase method, a non-commutative Vieta fomula, and cetain splittings of the Lax matices The method is applied to all equations of the Adle- Bobenko-Suis classification, with the exception of Q 4 PACS numbes: 23Ik Submitted to: J Phys A: Math Gen

2 Closed-fom expessions fo integals 2 1 Intoduction Two main classes of discete integable systems that may be distinguished ae odinay diffeence equations O Es and patial diffeence equations P Es By imposing peiodic initial conditions, a P E educes to an O E The staicase method which was intoduced in [7, 8] povides us with a tool to constuct integals of O Es, o mappings, deived as eductions of integable P Es These integals ae obtained by expanding the tace of a monodomy matix in powes of the spectal paamete Recently in [6], elegant and succinct fomulas fo integals of sine-godon and modified Koteweg-de Vies mkdv maps have been given fo the fist time The integals ae expessed in tems of multi-sums of poducts, Theta, which ae defined in section 2 These multi-sums of poducts wee discoveed by inspection and poved by induction Popeties of these multi-sums of poducts wee used to pove the invaiance of the integals independently of the staicase method and make it possible to pove functional independence and involutivity of the integals diectly In a ecent pape by Adle, Bobenko and Suis ABS, multi-linea equations on quad-gaphs ae classified with espect to consistency aound the cube [1] It is also known that fo P Es on quad-gaphs which satisfy the consistency popety, we can obtain a Lax pai algoithmically [2, 3] Hence, the staicase method can be applied to obtain integals of taveling wave eductions of the equations in the ABS list Some questions aise hee: is it possible to give simila explicit expessions fo these integals as was done fo mkdv and sine-godon? If so, is thee any method to obtain closed-fom expessions fo integals diectly, athe than using inspection and induction? This pape answes both these questions in the affimative Based on a non-commutative Vieta fomula, and cetain splittings of the Lax-matices, we explain how multi-sums of poducts emege and we give a method to actually deive closed-fom expessions fo the integals 2 Outline of this pape In this pape, we estict ouselves to so-called 1, z 2 taveling wave eductions z 2 N + which we now explain With l, m Z Z, we conside a 2-dimensional P E with field vaiable v, fv l,m, v l+1,m, v l,m+1, v l+1,m+1 ; α, 1 and paametes α α 1, α 2,, α k Now we intoduce a eduction v l,m v n, whee n l + mz 2 Then, v l,m satisfies the peiodicity v l,m v l+z2,m 1, and the P E educes to an O E fv n, v n+1, v n+z2, v n+z2 +1; α 2 We suppose that ou equation 1 aises as the compatibility condition of two linea equations, that is, it has a Lax pai A Lax pai L l,m, M l,m fo a P E 1 is a pai of

3 Closed-fom expessions fo integals 3 matices that satisfy, cf [9], L l,m M 1 l,m M 1 l+1,m L l,m+1 3 Similaly, a O E has a Lax pai L n, M n if they ae non-singula matices that satisfy M n L n L n+1 M n 4 Using the 1, z 2 taveling wave eduction, the Lax pai fo a P E educes to matices L n, M n which satisfy the following equation L n M 1 n M 1 n+1l n+z2 5 The monodomy matix L n fo the 1, z 2 eduction is given by, cf [8], L n M 1 n z 2 1 L i+n 6 i whee the invesely odeed poduct is b L i : L b L b 1 L a+1 L a 7 i1 Taking M n L n, we obtain a Lax pai L n, M n fo the educed O E 2 fom the Lax pai of the coesponding P E 1 Recently, a desciption of z 1, z 2 -eduction, with geneal z 1, z 2 Z N is given in [11] This genealizes the eductions in [8] whee the case z 1, z 2 N 2 with z 1, z 2 co-pime was consideed In [11], it was poved that fo any z 1, z 2 -eduction, thee is a M with whom the monodomy matix L foms a Lax pai fo the eduction and explicit fomulas fo L, M in tems of the educed Lax matices L, M wee povided It follows fom equation 4 that the tace of L n is invaiant unde the map obtained fom a O E Since the educed Lax matices geneally depend on a spectal paamete, integals fo the O E 2 ae obtained by expanding the tace of the monodomy matix in powes of the spectal paamete Theefoe, to give explicit expessions fo these integals, we need to expand a poduct of L matices We split the L matices in the fom L i i λx i + Y i, whee λ is a function of the spectal paamete Next, we conside the fomal expansion of the matix poduct in a non-commutative Vieta fomula: whee Z a,b Z a,b b λx i + Y i a i 1 <i 2 < <i b a i 1 <i 2 < <i b b a+1 b a+1 λ b a+1 Z a,b 8a λ Za,b, 8b X b X b 1 X i+1y i X i 1 X i1 +1Y i1 X i1 1 X a, Y b Y b 1 Y i+1x i Y i 1 Y i1 +1X i1 Y i1 1 Y a 9a 9b

4 Closed-fom expessions fo integals 4 This is a genealization of the Vieta expansion with commutative vaiables: b λ + f i b a+1 whee the multi-sums of poducts s a,b s a,b {f i } : a i 1 <i 2 < <i b j1 λ b a+1 s a,b, 1 ae the elementay symmetic functions, f ij 11 In this pape we will conside two diffeent foms of X i and Y i with special popeties such that the elements of the matices Z a,b a,b o Z can be expessed in tems of multisums of poducts Theta, espectively Phi Fo the educed mkdv equation [6] α 1 v n v n+z2 v n+1 v n+z α 2 v n v n+1 α 3 v n+z2 v n+z2 +1, 12 and the educed sine-godon equation [6] β 1 v n v n+z2 +1 v n+1 v n+z2 + β 2 v n v n+1 v n+z2 v n+z2 +1 β 3, 13 the L matix can be witten as L i λx i + Y i, whee 1 v i /v i+1 X i : J, Y i 1 v i+1 /v i, 14 and λ is the spectal paamete Substituting 14in the equations 9a and 9b, we deive multi-sums of poducts, Theta, with diffeent aguments,,ɛ { v i 1i } v i j 1i j 1 j+ɛ, 15 v i+1 v ij +1,ɛ {v i v i+1 } a i 1 <i 2 < <i b j1 a i 1 <i 2 < <i b j1 v ij v ij +1 1j+ɛ, 16 see Lemma 1 and Lemma 4 The late one is the definition of Theta given in [6] The geneal definition of Theta,,ɛ {f i } : f ij 1j+ɛ, 17 a i 1 <i 2 < <i b j1 whee f is a function on [a, b] Z, is obtained by eplacing Y i with f 1i i D i 18 f 1i+1 i whee Substituting X i J and Y i D i into the Vieta expansion 9a we obtain Z a,b J b a+1, 19,a 1{f i },a{f i } 2

5 Closed-fom expessions fo integals 5 The equation 19 can be genealized to 2n 2n matices J and D, povided they satisfy the following popeties J 2 I, D i JD i J, and D i is diagonal To illustate the second way of splitting L, we wite the L matices of the mkdv and sine-godon equations in a diffeent way with X i H and Y i s i A i i, whee H and A i j ae defined as H : 1, A i j a i a i b j 1 b j 21 Then, with special popeties of H and A we obtain multi-sums of poducts called Psi which ae intoduced below In paticula, if we take s i f i, a i 1/f i and b i, then the fomula 9b can be evaluated as Z a,b Φ a,b 1 f b Φ a,b 2 Φ a+1,b 1 1 f b Φ a+1,b 2 1 whee multi-sums of poducts, Φ, ae defined as: Φ a,b {f i } : a i 1,i 1 +1<i 2,i 2 +1<<i 1,i 1 +1<i b j1, 22 with f a function on [a, b] Z Moe geneally, when H and A satisfy f ij, 23 H 2, A n ma k l α k,m A n l, HA n mh H, and A n mha k l A n l, 24 substituting X i H and Y i s i A i i in equation 9b yields whee Ψ is Z a,b Ψ a,b Ψ a+1,b 2 1 HAa b 1 : s a 1 Φ a,b { + Ψ a+2,b 1 1 A b a+1h + Ψ a+2,b 2 2 H + Ψ a+1,b 1 A b a, 25 1 b+1 } s i α i 1,i 26 s i+1 α i 1,i α i,i+1 The elementay symmetic functions satisfy the following ecusive fomulas fo all c such that c b a + 1, s a,b s a,b m 2 s a+c,b i im 1 m 2 s b c+1,b i im 1 s a,a+c 1 i, 27a s a,b c i, 27b whee m 1 max, + a + c b 1, m 2 min, c These ecusive fomulas ae obtained by using the Vieta expansion 1 and witing b z i b +c z i a+c 1 z i, 28

6 Closed-fom expessions fo integals 6 whee z i λ + f i in this case Taking c, these ecusive fomulas povide us with an efficient way of poducing and stoing the elementay symmetic functions We will obtain simila ecusive fomulas fo Z a,b a,b and Z by using Vieta expansion and the matix analogue of 28 Recusive fomulas fo Theta and Phi ae deived fom those fo Z a,b a,b and Z, espectively The ecusivity of these multi-sums of poducts gives us a convenient way of computing these multi-sums of poducts The est of this pape is oganized as follows In section 3, we explain how Theta aises fom equation 9a using popeties of J 14 and D i 18 Then, the ecusive fomulas fo Theta ae obtained We give two closed-fom expessions fo integals of the mkdv equation in tems of Theta, one of which coincides with the fom discoveed in [6], using the two Vieta fomulas 9a, 9b In section 4, we show how the multi-sums of poducts, Phi, emege as enties of matix coefficients The ecusive fomulas fo Phi ae given and we pove equation 25 As a fist application, we expess integals of mkdv equation in tem of Psi Next, we pesent a geneal fomula fo integals fo all equations whose educed Lax matices can be witten in the fom i λh + s i A i i 29 We explicitly wite the educed Lax matices fo the equation in the ABS clasification in the fom 29, with the exception of Q 4 So we obtain close-fom expessions fo integals of these equations fom the geneal fomula The final section compaes two sets of integals expessed in tems of Theta and Psi, espectively This section also discusses possible futue wok 3 Multi-sums of poducts, Θ In this section, we fist pesent popeties of the multi-sums of poducts, Theta Then, an application of Theta is given We deive two closed-fom expessions fo integals of mkdv 31 Popeties of the multi-sums of poducts, Θ The integals in tems of the multi-sums of poducts 17 with f i v i v i+1, fist intoduced in [6], wee discoveed by inspection and poved by induction We found that the multi-sums of poducts can actually be deived fom the Vieta like fomula 9a, 9b and special popeties of the educed Lax matices Also we show that the following popeties, which wee used in the poofs in [6, identity 3,4], n,ɛ Θn,ɛ a+1,b n,ɛ 1 n,ɛ + f a 11+ɛ Θ a+1,b n 1,ɛ±1, 3 + f 1n+ɛ b 1 n 1,ɛ, 31 ae special cases of the moe geneal identities 33a and 33b given below These identities 33a, 33b ae simila to the ecusive fomulas fo the symmetic functions 27a and 27b and can be used to efficiently compute the multi-sums of poducts,

7 Closed-fom expessions fo integals 7 Theta They also play an impotant ole in calculating gadients fo poving the functional independence and the involutivity of integals [13, 14] Lemma 1 Let J and D i be as in 14 and18 espectively The multi-sums of poducts,ɛ defined by 17 ae the enties in the matix coefficients in the expansion b λj + D i whee is defined in 2 b a+1 λj b a+1 32 Poof We use Vieta expansion 8a, 9a to expand the left hand side of equation 32 b λj + D i b a+1 λ b a+1 Z a,b Using popeties of J and D i such as D i J k J k D 1k i and J 2 I, we have Z a,b JJ JD i J JD i 1 J JD i1 J J Now since we have a i 1 <i 2 <i b a i 1 <i 2 <i b a i 1 <i 2 <i b a i 1 <i 2 <i b a i 1 <i 2 <i b D 1k i a i 1 <i 2 <i b J b i D i J i i 1 1 D i 1 J i 2 i 1 1 D i1 J i 1 a J b i D i J i i 1 1 D i 1 J i 3 i 2 1 D i2 J i 2 a 1 D 1i 1 a i 1 J b i D i J i i 1 1 D i 1 J i 3 a 2 D 1i 2 a 1 i 2 J b a+1 D 1i 1 a i f 1i+k i f 1i+k+1 i f 1 a+1 i f 1 1 a+1 i 1 f 1 1 a+1 i 1 f 1 a+2,a 1,a, i 1i 1 2 a D i 1 f 1 1 a+2 i 1 1i1 a D i 1 1i1 a D i 1 f 1 1 a+2 i 1 Using Lemma 1 and the matix analogue of 28 we obtain the following ecusive fomulas fo Theta

8 Closed-fom expessions fo integals 8 Poposition 2 Fo any c b a + 1 we have m 2,ɛ Θ a+c,b,ɛ Θ a,a+c 1 i,ɛ im 1 m 2 i,ɛ+i, c i,ɛ Θb c+1,b i,+ɛ+i, im 1 whee m 1 max, + a + c b 1 and m 2 min, c 33a 33b Poof It is easy to see that 33b follows fom 33a by substituting c b a c + 1 in 33a, changing vaiable i j and using the fact that j minx, y j max x, y By using the Vieta expansion 8a and the poduct stuctue 28, the ecusive fomulas 27a and 27b still hold if we eplace s by Z Theefoe, using Lemma 1 we have J b a+1 m 2 J b a+c+1 i Θ a+c,b i im 1 If b a + 1 is even and using equation 2, we have,a 1 Θ a+c,b i,a+c 1,a Θ a+c,b i,a+c Theefoe, we get + m 1 i m 2, c i even m 1 i m 2, c i odd,a 1,a m 2 Θ a,a+c 1 i,a 1 im 1 m 2 Θ a,a+c 1 i,a im 1 Since {a 1, a} {, 1} mod 2, we obtain,ɛ m 2 Θ a,a+c 1 i,ɛ im 1 The poof fo odd b a + 1 is simila Θ a+c,b i,a+c Θ a+c,b i,a+c 1 Θ a+c,b i,a+i 1, Θ a+c,b i,a+i Θ a+c,b i,ɛ+i J c i Θ a,a+c 1 i 34 Θ a,a+c 1 i,a 1 Θ a,a+c 1 i,a Θ a,a+c 1 i,a Θ a,a+c 1 i,a 1 Note that fom the definition of Theta 17, o fom equation 32, we have the following popeties,ɛ if < o > b a + 1,,ɛ 1, Θ a,a 1,ɛ f 11+ɛ a If we conside these popeties as initial values of Theta, then we can efficiently calculate Theta fom the ecusive fomulae 33a by taking c b a

9 Closed-fom expessions fo integals 9 32 Applications of Θ to the mkdv equation The 1,z 2 -eductions of the Lax matices fo mkdv ae given by vn v n+z2 v L n n+1 k α3 and Mn 1 v n α 1 k v k v n+1 /v n α 1 k α n 35 2 v n+z2 Indeed, we have L n Mn 1 of 12, and N n M 1 n+1l n+z2 F mkdv N n, whee F mkdv is the left hand side k v n+1 v n+z2 k v nv n+z2 +1 Using Lemma 1 with f i v i /v i+1 1i, a, b z 2 1, and taking the tace of the monodomy matix 6, we obtain the following esult Poposition 3 The closed-fom expessions fo integals of the mkdv equation ae given as follows whee z 2 is odd I α 1 Θ,z 2 1, + Θ,z 2 1 v,1 + α 2 Θ,z 2 1 v z2 1, + α 3 Θ,z 2 1 1,1, 36 v z2 v Howeve, in [6], integals of the mkdv equation ae expessed in tems of Theta with diffeent aguments, namely f i v i v i+1 To deive the late close-fom expessions fo integals of the mkdv equation, the following Lemma was used [6] Lemma 4 We have b 1 L i whee f i v i v i+1, and Z a,b 1 when is even, and when is odd Z a,b 1 b a i Z a,b 1 λ, 37 va v b 1, v b v a 1,1 1 v av b 1,1 v a v b 1, Wheeas fo Lemma 1 we used the Vieta expansion 9a, this Lemma is poved using the Vieta expansion 9b, see Appendix A Multiplying both sides of equation 37 by M 1 and taking the tace we obtain the following esult, which is Theoem 3 in [6]

10 Closed-fom expessions fo integals 1 Poposition 5 The tace of the monodomy matix of the mkdv equation is whee with f i v i v i+1 TL z 2 +1/2 i k 2 I, I α 1 v v z2 Θ,z , + 1 v v z2 Θ,z ,1 + α 2 Θ,z 2 1 2,1 + α 3 Θ,z 2 1 2,, 38 It follows that the coefficients I fom a set of integals fo the mkdv equation Thei invaiance can also be poven diectly By using the popeties 3 and 31 of multi-sums of poducts it was shown in [6] that SI I F mkdv Λ, 39 whee S is the shift opeato Sv i v i+1, and Λ 1 v v 1 v z2 v z2 +1 Θ 1,z ,1 Θ 1,z , 4 is called an integating facto It is impotant to note that the set of integals expessed in Theta with f i v i /v i+1 1i is the same as the one expessed in Theta with f i v i v i+1 due to the following identities,a+ɛ{ v i v i+1 1i } if b a + 1 is even, and if b a + 1 is odd va v b+1 1ɛ+1 b a+1,ɛ+1 {v iv i+1 },,a+ɛ{ v i v i+1 1i } v a v b+1 1ɛ+1 b a+1,ɛ+1 {v iv i+1 } 4 Multi-sums of poducts, Φ Witing the educed Lax matices as linea combinations of ank one 2 2 matices, gives ise to closed-fom expessions in tems of multi-sums of poducts Phi, defined in equation 23 In this section, we fist give ecusive fomulas fo Phi Then, by using the non-commutative Vieta expansion, we give the fomula 25 fo poducts of L i i λh + s i A i i matices in tems of Psi 26 whee A i j is defined in 21 This fomula is fist applied to the mkdv equation At the end we give the analogous esults fo nealy all equations in the Adle-Bobenko-Suis list [1]

11 Closed-fom expessions fo integals Recusive fomulas fo the multi-sums of poducts, Φ The Lemma below explains how Phi is deived fom the Vieta expansion 8b and 9b Lemma 6 Let H and F i be defined by 1 H, F i 1 f i and let f be a function on [a, b + 1] Z The multi-sums of poducts, Φ, defined by 23 appea in the enties of the matix coefficients in the expansion whee b+1 λh + F i W a,b+1 b a+2 Φ a,b f b+1 Φ a,b 1 λ W a,b+1, Φ a+1,b 1 f b+1 Φ a+1,b 1 1 Poof Using the popeties H 2, F i F j F i, F i HF j f j F i, HF i H f i H and the non-commutative Vieta fomula 9b, we have W a,b+1 HF b F i 2 +1HF i 2 1 F i1 +1HF i1 1 F a+1 H a+2 i 1,i 1 +1<i 2,i 2 +1<<i 2 b 1 a+1 i 1,i 1 +1<i 2,i 2 +1<<i 1 b 1 a+2 i 1,i 1 +1<i 2,i 2 +1<<i 1 b a+1 i 1,i 1 +1<i 2,i 2 +1<<i b f b Φ a+1,b 2 2 f b Φ a,b Φ a,b 1 f b+1 Φ a,b 1 + f b Φ a,b 2 1 HF b F i 1 +1HF i 1 1 F i1 +1HF i1 1 F a F b+1 F i 1 +1HF i 1 1 F i1 +1HF i1 1 F a+1 H F b+1 F i+1hf i 1 F i1 +1HF i1 1 F a f b Φ a+1,b Φ a+1,b 1 1 f b+1 Φ a+1,b Φ a+1,b 1 1 Fom the definition of Phi, we have the following popeties Φ a,b Φ a,b Theefoe, we obtain Φ a,b 1 f a Φ a+2,b 1 W a,b+1 1 f b+1 + Φ a,b 1 1 f b+1 + f b Φ a,b 2 1, 41 + Φ a+1,b 42 Φ a,b f b+1 Φ a,b 1 Φ a+1,b 1 f b+1 Φ a+1,b 1 1 Using this Lemma, we deive ecusive fomulas fo Phi

12 Closed-fom expessions fo integals 12 Poposition 7 With a 1 c b + 1 we have Φ a,b Φ a,c 1 i Φ c+1,b i + Φ a,c 2 i 1 Φc,c 1 Φ c+2,b i 43a i i Φ a,c 1 i Φ c+1,c+1 1 Φ c+3,b i 1 + Φ a,c i Φc+2,b i 43b We note that popeties 41 and 42 ae special cases of these ecusive fomulas 43a and 43b whee c b and c a 1, espectively Poof We use ecusive fomula 33a with W eplacing s We have W a,b+1 W c+1,b+1 i W a,c i i i Φ c+1,b i f b+1 Φ c+1,b 1 i Φ c+2,b i 1 f b+1 Φ c+2,b 1 i 1 Equating the enty in the fist column and fist ow, we get Φ a,b Φ a,c 1 i Φ c+1,b i + Φ a,c 2 i 1 f cφ c+2,b i, i Φ a,c 1 i f c Φ a,c 2 i Φ a+1,c 1 i 1 f c Φ a+1,c 2 i 1 which is the fist ecusive fomula 43a The second ecusive fomula 43b is obtained fom the fist one by using the following Φ c+1,b i Φ c+2,b i Φ a,c 2 i 1 Φc,c 1 Φ a,c i Φa,c 1 i + Φ c+1,c+1 1 Φ c+3,b i 1, Note that fom the definition of Phi 23, we have the following popeties Φ a,b when < o > b a + 1/2, Φ a,b 1, Φ a,a 1 f a Once again, these popeties can be consideed as initial values to calculate Phi using the ecusive fomulas fo Phi povided 42 Poduct of L matices and multi-sum of poducts, Ψ Let A i j and H be matices that ae defined in 21 Now using the Vietaexpansion 8b,9b and popeties of H and A, we obtain the following lemma Lemma 8 Let L i i λh + s i A i i We have b L i b a+1 X a,b b λ i,

13 Closed-fom expessions fo integals 13 with X a,b whee c i s i a i 1 + b i and Ψ a,b Ψ a+1,b 2 1 HAa b 1 + Ψ a+2,b 1 1 : s a 1 Φ a,b { s b+1 i+1 } c i c i+1 The geneal case 25 is obtained in the same way A b a+1h + Ψ a+2,b 2 2 H + Ψ a+1,b 1 A b a, 44 c i 45 Poof Fo bevity of notation, we wite B i : s i A i i Using the popeties of H and A n m we have k B k B k 1 B l s k A k k s l A l l s l A k l c so when l < i < k we have Theefoe, we obtain B k B k 1 B i+1 HB i 1 B l s i+1 A k i+1 s i+1 c i c i+1 s l A k l k i+2 l+1 k l+1 c Hs l A i 1 c s l f i A k l l i 1 l+1 k l+1 c c B k B i+1hb i 1 B i1 +1HB i1 1 B l s l A k l k f ij c j1 l+1 when l < i 1, i < i 2, < i i < k, and it equals when i j i j fo some j Now applying these fomulas and the fomula 9b we have X a,b HB b 1 B i 1 +1HB i 1 1 B i1 +1HB i1 1 B a sa a+1 i 1,i 1 +1<i 2,<i 1 b 2 a+2 i 1,i 1 +1<i 2,<i 1 b 1 a+2 i 1,i 1 +1<i 2,<i 2 b 2 a+1 i 1,i 1 +1<i 2,<i b 1 Φ a+1,b 2 1 c b Ψ a+1,b 2 1 HAa b 1 HA b 1 B b B i 1 +1HB i 1 1 B i1 +1HB i1 1 B a+1 H HB b 1 B i 2 +1HB i 2 1 B i1 +1HB i1 1 B a+1 H B b B i+1hb i 1 B i1 +1HB i1 1 B a a + s a+1 Φ a+2,b 1 1 A b c a+1h + s a+1 Φ a+2,b 2 2 H + s a Φ a+1,b 1 a+1 c a+1 c b + Ψ a+2,b 1 1 A b a+1h + Ψ a+2,b 2 2 H + Ψ a+1,b 1 A b a A b a b +1 Note that if > b a + 1/2 we have X a,b We also note that Lemma 6 is a special case of this Lemma with a i 1/f i and b i and s i f i c i

14 Closed-fom expessions fo integals Application of Ψ to the mkdv equation In this section, we pesent closed-fom expessions fo integals of mkdv in tems of the multi-sums of poducts Psi The educed Lax pai fo the mkdv equation given by 35 gives ise to the multisums of poducts, Theta Now we use a gauge tansfomation to obtain a new educed Lax pai which gives the multi-sums of poducts, Psi Recall that fo a P E 1 with a Lax pai L l,m, M l,m, a gauge matix G l,m gives us a new Lax pai fo the equation, that is L l,m G l+1,m L l,m G 1 l,m, Ml,m G l,m+1 M l,m G 1 l,m These matices educe to matices L, M of the coesponding O E Using the gauge matix 1/v l,m G l,m, 1/k we have a new educed Lax pai fo the mkdv equation vn+1 k 2 v L n v n v n 1 α2 n k v and M n n+z2 α 2 1 v n+z2 v n+1 1 α 1 v n α 3 We note that the tace of the monodomy matix is invaiant unde gauge tansfomations Now we wite the educed Lax pai fo the mkdv equation as follows whee and 46 L i i s i A i i + λh, M i 1 i s i Ă i i + λh, 47 Ă i j ă i ă i bj 1 bj λ 1 k 2, i 1 v i, s i v i v i+1, a i 1 v i, b i 1 v i+1, λ λ + α 2α 3 α 2 1 α 1, i 1 v i+z2, s i α 1 v i v i+z2, ă i α 2 α 1 v i+z2, bi α 3 α 1 v i Using Lemma 8 we expand the tace of the monodomy matix in powes of λ Afte 1 multiplying with M and taking the tace, we get whee Ĩ α 1 Ψ 1,z T L z 2 +1 i λ i Ĩ i, + α 2 v + α 1v v z2 Ψ 1,z v 1 α 2v + α 3v z2 v z2 1 v 1 + α 1v v z2 v 1 v z2 1 + α 3 v z2 + α 1v v z2, + α 1 Ψ 1,z 2 2 v z2 1 z 2 i Ψ 2,z α 1 v v z2 Ψ 2,z v 1 i, 48

15 Closed-fom expessions fo integals 15 with Ψ a,b v a 1 v a Φ a,b { b+1 v i 1 v i+2 v i 1 + v i+1 v i + v i+2 } v i v i 1 + v i+1 v i 1 49 Note that if > z 2 + 1/2, then Ĩ It is clea by the staicase method that Ĩ is an integal of the coesponding mkdv map Howeve, using popeties of Psi one can also show that Ĩ is invaiant unde the mkdv map By doing so, we obtain an integating facto [6] which we do not get fom the staicase Poposition 9 Fo z 2 + 1/2, Ĩ is an integal of the mkdv map with integating facto Λ whee Ψ is given by 49 1 v v 1 v z2 v z2 +1 Ψ 1,z v 1 v z2 Ψ 2,z v 1 Ψ 2,z 2 1 The poof of this Poposition is given in the Appendix B v z2 Ψ 1,z Geneal closed-fom expessions fo integals of all equations in the ABS list, except Q 4 z 2 i1 v 1 i, In this section, we give simila esults as we did fo the mkdv equation fo almost all equations in the ABS list We pesent a geneal fomula fo closed-fom expessions fo integals of all equations whose educed Lax pais can be witten in the fom 47 Then, we wite the educed Lax matices of ABS equations with the exception of Q 4 in this fom, so that we can apply the geneal expessions to obtain integals 441 Geneal closed-fom expessions fo integals Assume that the educed Lax matices ae witten in the fom 47, with λ gλ + h We now expand the tace of the monodomy matix L in tems of powes of λ Theoem 1 Let L be given by 6, whee L, M 1 ae matices which can be witten in the fom 47 Then, we have TL z 2 +1 i λ I, whee I is expessed in tems of Ψ 45 I gψ 1,z hψ 1,z s ă + b Ψ 1,z s ă + b a z2 1 + b Ψ 1,z s b + a z2 1Ψ 2,z 2 2 z 2 1 Poof We use Lemma 8 to expand the monodomy matix We have L s Ă + gλ + hh z 2 z 2 1 λ X,z 2 1 i i 1 + s Ψ 2,z i 5 z 2 +1 i z 2 1 λ W i i, 51

16 Closed-fom expessions fo integals 16 whee W s Ă + hhx,z 2 1 Using popeties of H and A, we have + ghx,z THX,z 2 1 Ψ 1,z 2 2, THX,z Ψ 1,z 2 2 1, TĂ X,z 2 1 ă + b Ψ 1,z b + a z2 1Ψ 2,z Ψ 2,z ă + b a z2 1 + b Ψ 1,z 2 2, which we use to evaluate the tace of 51 Now fom this theoem, if the I do not depend on λ, then fom the staicase method we have I is an invaiant Hence, we have the following coollay Coollay 11 Suppose that a i 1 +b i, ă +b, a z2 1+ b, i, s i, i, s i, g and h do not depend on the spectal paamete k Then I given by 5 is an integal Hee we give a diect poof that I is an integal of the equation deived fom the Lax equation L M 1 M1 1 L z2 Poof Since L M 1 M 1 1 L z2, we have s A + λh s Ă + gλ + hh 1 z2 s 1 Ă gλ + hhs z2 A z 2 z 2 + λh If a i, ă i, b i, b i do not depend on k the cases H 1 and H 3, equating coefficients of λ in both sides we obtain E i, i 1, 2,, 6, with E 1 : s 1 z2 gs z2, E 2 : gs 1 z2 s 1, E 3 : s b + gs a 1 z2 s 1 ă 1 + gs z2 b z2, E 4 : s s b + ă d 1 s 1 s z2 a z2 + b 1, E 5 : s s b + ă b + h 1 z2 s 1 s z2 b z2 a z2 + b 1, E 6 : s a s b + ă 1 z2 s z2 s 1 a z2 + b 1 ă 1 + h Fo the case whee a i, b i, ă i, b i depend on k, we wee able to check that the identities E i hold fo the cases Q 1 and Q 3 Fo the est of the equations, calculations get complicated as we have to deal with squae oots Using the fouth and fifth identities, we have h s b + ă b z2 b Similaly, using the fouth and sixth identities, we have h s 1 a ă 1 a z2 + b 1 We have SI gψ 2,z hψ 2,z s 1 ă 1 + b 1 Ψ 2,z s 1 b 1 + a z2 Ψ 3,z s 1 Ψ 3,z s 1 ă 1 + b 1 a z2 + b 1 Ψ 2,z 2 1 z 2 1 i i1

17 Closed-fom expessions fo integals 17 Now we wite SI I A + B z 2 1 i1 i whee A : 1 z2 gψ 2,z s 1 ă 1 + b 1 Ψ 2,z gψ 1,z s b + a z2 1Ψ 2,z B : 1 z2 h + s 1 ă 1 + b 1 a z2 + b 1 Ψ 2,z s 1 Ψ 3,z h + s ă + b a z2 1 + b Ψ 1,z s Ψ 2,z s 1 b 1 + a z2 Ψ 3,z s ă + b Ψ 1,z Fom the popeties of Phi 41 and 42, we obtain the following popeties of Psi Ψ n,m Ψ n,m s m+1 a m + b m+1 Ψ n,m 1 s n 1 a n 1 + b n Ψ n+1,m + s m+1 Ψ n,m 2 1, 52 + s n 1 Ψ n+2,m 1 53 whee n m and Using these popeties 52 and 53, we get A 1 z2 gs z2 a z2 1 + b z2 + s 1 ă 1 + b 1 gs a + b 1 + s b + a z2 1 Ψ 2,z z2 gs z2 s Ψ 2,z z2 s 1 gs Ψ 3,z B a z2 1Ψ 2,z z2 gs z2 s + b 1 Ψ 2,z Ψ 3,z z2 s 1 gs + 1 z2 gs z2 a z2 + s 1 ă 1 gs a + s b Ψ 2,z a z2 1Ψ 2,z We also have + Ψ 2,z Ψ 2,z E 1 b 1 Ψ 2,z z2 h + s 1 a z2 + b 1 a ă Ψ 2,z 2 1 E 4 s s z2 Ψ 1,z 2 1 That poves ou statement + Ψ 3,z E 2 Ψ 2,z E 3 h + s ă + b b b z2 Ψ 1,z 2 2 Fom this poof, one can deive an integating facto by dividing E 1, E 2, E 3, E 4 and h + s 1 a z2 + b 1 a ă, h + s ă + b b b z2 by the coesponding equation 442 Application to equations in the ABS list In Table 1, we give a table fo witing the educed Lax matices fo equations H 1, H 2, H 3 and Q 1, Q 2, Q 3, see [1], in the fom 47 All the educed Lax pais given satisfy the conditions in Coollay 11, so that we obtain closed-fom expessions fo the integals fom the fomula 5

18 Closed-fom expessions fo integals 18 E H1 H2 H3 Q1 Q2 Q 3 Q3 λ g ai bi i si h ăi bi i si p k 1 vi vi q p vi+z2 vi 1 1 p k 1 p k + vi p k + vi+1 2 p + vi + vi+1 2/ i 2 q p q k + vi+z2 q k + vi 2 q + vi + vi+z2 2/ i 2 k 2 p 2 q 2 /p 2 vi/p vi+1/p k 2 p k q/p k p 2 q 2 p 2 vi+z2/p vi/q q p p p k q/p k k 2 p 2 k 2 1 p 2 k 2 k 2 1 q p p q 2 1 p 2 1 p 2 q 2 p 1 q 2 1 p 2 1 q 2 p 2 p 1 pvi+1/k vi+1+vi pv i/k vi+vi+1 p p qvi/k vi+vi+z 2 qv i+z 2 /k vi+z 2 +vi q q k pδpk vi+1+kvi k pδpk v i+kvi+1 kp kp k qδqk vi+kvi+z 2 k qδqk v i+z 2 +kvi kq kq p 2 k 2 vi+1+pk 2 1vi p2 k 2 vi+pk 2 1vi+1 k 2 1p 2 1 k 2 1p 2 1 q 2 k 2 vi+qk 2 1vi+z 2 q2 k 2 vi+z 2 +qk 2 1vi k 2 1q 2 1 k 2 1q 2 1 pk 2 1vi+p 2 k 2 vi+1 pk2 1vi+1+p 2 k 2 vi k 2 1p 2 1 k 2 1p 2 1 q 2 k 2 vi+qk 2 1vi+z 2 q2 k 2 vi+z 2 +qk 2 1vi k 2 1q 2 1 k 2 1q 2 1 δp+v ivi+1 1/ 2 p i δq+v ivi+z 2 q 1/ 2 i vi vi+1 δ p vi vi+z 2 +δq q δp 2 δp 2 2v i 2vi+1+vi vi+1 2 p 2 vi vi+1+pδvi vi+1 pδ q 2 vi vi+z 2 δqvi vi+z 2 +δq 1/ 2 p i δq 2 δq 2 2v i 2vi+z 2 +vi vi+z 2 2 q 1/ 2 i pvi+1 vi p 2 1 qui vi+z 2 q 2 1 δ 2 1 p pv ivi+11 p 2 +4p 2 v i 2 v2 i+1 p 2 14p δ 2 1 q qv ivi+z 2 1 q 2 +4q 2 v i 2 v2 i+z 2 4qq 2 1 p pvi+1 vipvi vi+1 q qvi+z 2 viqvi vi+z 2 1 4p 2 i 1 4q 2 i Table 1 Reduced Lax pais fo equations in the ABS list

19 Closed-fom expessions fo integals 19 5 Discussion 51 Compaing the two sets of integals of the mkdv equation We have obtained two sets of integals fo the mkdv equation in equation 38 and 48, which ae expessed in tems of multi-sums of poducts, Theta and Psi, espectively This is because we can expand the tace of the monodomy matix eithe in powes of k o in powes of λ 1 k 2 Theefoe, we have TL z 2 +1/2 i k 2i I i Equating the coefficient of k 2, we have z 2 +1/2 i I 1 Ĩ i i In paticula we have and i I α 2 + α 3 z 2 +1/2 i 1 k 2 i Ĩ i I z2 +1/2 1 z 2+1/2 Ĩ z2 +1/2 z 2 +1/2 i Ĩ i The late equation means that the set of non-constant integals {Ĩ} is not functionally independent Explicitly taking z 2 3, we have and I α 2 + α 3, v I 1 α 1 + v v 3 + v 3 + v 2 + v 1v 2 + v 1 v1 + α 2 + v v 1 + v v 2 v 1 v 2 v 1 v v v 3 v 3 v 3 v 2 v 3 v 2 v3 + α 3 + v 2v 3 + v 2, v 1 v v 1 v I 2 2α 1, Ĩ α v v 2 + v v 3 v 1 v 2 + v 3 v 1 + v 2 v + v 1v 2 v v 3 + v 1 v 3 + α v 1 v 3 + v v 1 v 2 v 3 + v v 2 + α v 3 v 1 + v 2v 3 v v 1 + v 2 v, Ĩ 1 α v + v v 3 + v 3 + v 2 + v 1v 2 + v 1 v1 α 2 + v v 1 + v v 2 v 1 v 2 v 1 v v v 3 v 3 v 3 v 2 v 3 v 2 v3 α 3 + v 2v 3 + v 2, v 1 v v 1 v Ĩ 2 2α 1 It seems that the set of non-constant integals I expessed in Theta is simple and shote than the integals Ĩ expessed in Psi Howeve, it is inteesting to know that with ou Maple pogamme, it took much moe time to calculate the integals in tems of Theta than in tems of Phi

20 Closed-fom expessions fo integals 2 Similaly, fo the sine-godon mapping the set of non-constant integals expessed in tems of Psi is not functionally independent and also longe than the one expessed in tems of Theta 52 Futue wok We have pesented a tool to obtain closed-fom expessions fo integals in tems of multi-sums of poducts, Theta and Psi This is a fist step to pove the integability of a discete map in the Liouville-Anold sense [1], [12] a map has sufficiently many functionally independent integals in involution The ecusive fomulas fo the multi-sums of poducts make it possible fo us to pove functional independence and involutivity which we hope to publish elsewhee [13, 14] We have given closed-fom expessions fo integals of all equations in the ABS list but Q 4 It would be woth studying this exceptional case, as Q 4 is the most geneal equation in the ABS list In this pape, we have consideed 1, z 2 taveling wave eductions It would be inteesting to study moe geneal taveling wave eductions, cf [8, 11] Anothe diection of inteest is studying moe geneal equations, and systems of equations, which ae not necessay on defined on elementay squaes, cf [4, 5] Acknowledgments This eseach has been funded by the Austalian Reseach Council though the Cente of Excellence fo Mathematics and Statistics of Complex Systems Dinh T Tan acknowledges the suppot of two scholaships, one fom La Tobe Univesity and the othe fom the Endeavou IPRS pogamme Thanks to Pof Willy Heeman fo a vey useful collaboation on the topic of Lax pais fo equations on the ABS list Appendix A: Poving Lemma 4 Hee we give a poof of Lemma 4 Poof We wite L i kj + Y i as in 14 So we have Y i+k Y i+k 1 Y i vi v i+k+1 v i+k+1 v i Using the Vieta fomula 9b and the above fomula, we obtain Z a,b 1 a i 1 <i 2 <<i Y b 1 b 1 Y i+1jy i 1 Y i 1 +1J Y i1 +1JY i1 1 Y a a i 1 <i 2 <<i b 1 vi+1 v b v b v ii+1 J vi 1 +1 v i v i v i 1 +1 J vi1 +1 v i2 v i2 v i1 +1 J va v i1 v i1 v a

21 Closed-fom expessions fo integals 21 If is even and using the popeties JY Y 1 J and J 2 I, then we have Z a,b 1 a i 1 <i 2 <<i b 1 vi vi+1 v b v b v ii+1 a i 1 <i 2 <<i b 1 va vb v i v i+1 v i 1 v i 1 +1 va v b 1, v i 1 +1 v i 1 +1 v i v i 2 v i 2 +1 v i 2 v i2 +1 v i 3 v i 3 +1 v i1 v i1 +1 v b v a 1,1 Similaly, if is odd, then we have Z a,b 1 a i 1 <i 2 <<i b 1 J J vb v i+1 v ii+1 v b v a v b 1, vi v av b 1,1 1 v av b 1,1 v a v b 1, v i v i v i 1 +1 vi2 v i1 +1 v i1 +1 v i2 va v i1 v i1 v a v b v a v i 1 v i 1 +1 v i v i+1 v i 3 v i 3 +1 v i 2 v i 2 +1 v i 1 v i1 +1 v i2 v i2 +1 vi2 v i1 +1 v i1 +1 v i2 va v i1 v i1 v a Appendix B: Diect poof of invaiance of integals of the mkdv equation in tems of Psi Poposition 9 Hee we give a poof of Poposition 9 Poof Applying a shift opeato on the integal Ĩ, we obtain SĨ α 1 Ψ 2,z α 2 v 1 + α 1v 1 v z2 +1 Ψ 2,z α 3 v z α 1v 1 v z2 +1 v 2 α 1 v 1 v z2 +1Ψ 3,z α 2v 1 v z2 + α 3v z2 +1 v 2 + α 1v 1 v z2 +1 v 2 v z2 v z2 Ψ 3,z α 1 Ψ 2,z 2 1 Now we wite SĨ Ĩ A + B z 2 i1 v 1 i whee A α 1 Ψ 2,z α 2 v 1 + α 1v 1 v z2 +1 v 2 Ψ 2,z α 1 v 1 v z2 +1Ψ 3,z vz α 1 Ψ 1,z α 3 v z2 + α 1v v z2 v Ψ2,z 2 2 z2 1 1 α 1 v v z2 Ψ 2,z v 1 B α 3 v z α 1v 1 v z2 +1 v z2 Ψ 3,z α 2v 1 v z2 + α 3v z2 +1 v 2 + α 1v 1 v z2 +1 v 2 v z2 + α 1 Ψ 2,z 2 1 α 2 v + α 1v v z2 v 1 Ψ 1,z α 2v + α 3v z2 v z2 1 v 1 + α 1v v z2 + α v 1 v z2 1Ψ 1,z v 1 z 2 +1 i1 v 1 z 2 +1 v 1 i

22 Closed-fom expessions fo integals 22 Using popeties 52 and 53, we have Ψ 2,z 2 1 v z2 v z2 +1 Ψ 1,z v z2 1 v v 1 1 v + 1 v 2 Ψ 2,z Ψ 2,z v z2 v z2 +1Ψ 2,z v z2 +1 v v 1 Ψ 3,z Substituting these fomulas into A, we obtain A Ψ2,z F mkdv v v z2 +1 Applying the popeties 52 and 53, we have Ψ 3,z 2 1 Ψ 1,z v + 1 v 2 Ψ 2,z v z2 1 Ψ1,z 2 1 v v Ψ 1,z 2 2 v z2 +1 Substituting these fomulas into B, we get 1 B Ψ 1,z 2 1 v 1 Ψ 2,z 2 1 v v 1 v z2 v z2 +1 This poves the statement Ψ1,z 2 1 v z2 v z2 +1 v z2 Ψ 1,z 2 2 F mkdv Refeences [1] Adle V, Bobenko A and Suis Y 23 Classification of Integable Equations on quad-gaphs The Consistency Appoach Commun Math Phys [2] Nijhoff F N 22 Lax pai fo the Adle lattice Kicheve-Novikov system Phys Lett A [3] Nijhoff F N and Walke A J 21 The discete and continuous Painlevé hieachy and the Ganie systems Glasgow Math JA [4] Van de Kamp P H and Quispel G R W Initial value poblems fo lattice equations, discete taveling waves in pepeation [5] Van de Kamp P H and Quispel G R W The staicase method in pepeation [6] Van de Kamp P H, Rojas O and Quispel G R W 27 Closed-fom expessions fo integals of sine-godon and MKDV maps JPhys A: Math Gen [7] Papageogiou V G, Nijhoff F W and Capel H W 199 Integable mappings and nonlinea integable lattice equations Phys Lett A [8] Quispel G R W, Capel H W, Papageogiou V G and Nijhoff F W 1991 Integable mappings deived fom soliton equations Physica A [9] Quispel G R W, Capel H W and Robets J A G 25 Duality fo discete integable systems, JPhys A: Math Gen [1] Robets J A G and Quispel G R W 28 Symplectic stuctues fo diffeence equations including integable maps in pepeation [11] Rojas O, Van de Kamp P H and Quispel G R W Lax epesentations fo integable maps, in pepeation [12] Veselov A P 1991 Integable maps, Russian Math Suveys [13] Tan D T, Van de Kamp P H and Quispel G R W Involutivity of integals of sine-godon, mkdv and pkdv maps, in pepeation [14] Tan D T, Van de Kamp P H and Quispel G R W Functional independence of of integals of sine-godon, mkdv and pkdv maps, in pepeation