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Comment to: “Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation” [Phys. Lett. A 338 (2005) 117]
Physics Letters A 350 (2006) 419–420 www.elsevier.com/locate/pla
Comment
Comment to: “Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation” [Phys. Lett. A 338 (2005) 117] A.K. Svinin Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences, P.O. Box 1233, 664033 Irkutsk, Russia Received 5 September 2005; accepted 26 October 2005 Available online 2 November 2005 Communicated by A.P. Fordy
In Ref. [1] the authors exhibit “new” integrable differentialdifference systems rn+1 rn + − rn sn , rnt = −rn rn−1 rn rn+1 rn snt = sn (1) − , rn−1 rn and rn , sn rn+1 rn = − rn sn+1 rn−1 sn−1
rnτ = − snτ
(2)
with n ∈ Z. This equations stem from suitable discrete zerocurvature equation1 d Un − EΓn[m] Un + Un Γn[m] = 0 dt
rn rn−1
,
cnτ =
cn cn − , dn dn+1
dnτ =
cn cn−1 − . dn+1 dn−1
(4)
The former is well-known infinite relativistic Toda lattice (RTL) in polynomial form. This system was introduced by Ruijsenaars [2] and then thoroughly studied by many authors from different points of view. For consulting, see, for example, Refs. [3–9]. The latter is the symmetry of RTL. It is common of knowledge that RTL (3) admits Hamiltonian form with standard symplectic structure w = n∈Z dpn ∧ dqn . To write it in this form one needs to use following ansatz: dn = epn ,
cn = epn +qn −qn+1 .
Then one has qnt = epn 1 + eqn −qn+1 , pnt = epn−1 +qn−1 −qn − epn +qn −qn+1 from which one obtains RTL in its exponential form qn+1,t eqn −qn+1 qn−1,t eqn−1 −qn qntt = qnt − . 1 + eqn−1 −qn 1 + eqn −qn+1
with some 2 × 2 matrices Un and Γn[m] . Let us introduce the variables cn and dn through cn =
and
dn = sn−1 .
As for symmetry (4), one can write 1 1 + eqn−1 −qn 1 + eqn −qn+1 . qnt
Then Eqs. (1) and (2) become
qnτ =
cnt = cn (cn−1 − cn+1 + dn − dn+1 ),
This symmetry is also well-known [10]. Moreover this kind of symmetries admit a whole class of RTL type systems.
dnt = dn (cn−1 − cn )
(3)
References DOI of original article: 10.1016/j.physleta.2005.02.021. E-mail address: [email protected] (A.K. Svinin). 1 Throughout this comment we try, as far as possible, to follow the notations of Ref. [1]. 0375-9601/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.10.081
[1] H.-X. Yang, X.-X. Xu, H.-Y. Ding, Phys. Lett. A 338 (2005) 117. [2] S.N. Ruijsenaars, Commun. Math. Phys. 133 (1990) 217. [3] M. Bruschi, O. Ragnisco, Phys. Lett. A 129 (1988) 21.
420
A.K. Svinin / Physics Letters A 350 (2006) 419–420
[4] M. Bruschi, O. Ragnisco, Phys. Lett. A 134 (1989) 365. [5] W. Oevel, B. Fuchssteiner, H. Zhang, O. Ragnisco, J. Math. Phys. 30 (1989) 2664. [6] Yu.B. Suris, Phys. Lett. A 145 (1990) 113. [7] Yu.B. Suris, Phys. Lett. A 180 (1993) 419.
[8] S. Cosentino, Inverse Problems 7 (1991) 535. [9] P. Damianou, J. Math. Phys. 35 (1994) 5511. [10] V.E. Adler, A.B. Shabat, R.I. Yamilov, Theor. Math. Phys. 125 (2000) 1603.
Comment
Comment to: “Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation” [Phys. Lett. A 338 (2005) 117] A.K. Svinin Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences, P.O. Box 1233, 664033 Irkutsk, Russia Received 5 September 2005; accepted 26 October 2005 Available online 2 November 2005 Communicated by A.P. Fordy
In Ref. [1] the authors exhibit “new” integrable differentialdifference systems rn+1 rn + − rn sn , rnt = −rn rn−1 rn rn+1 rn snt = sn (1) − , rn−1 rn and rn , sn rn+1 rn = − rn sn+1 rn−1 sn−1
rnτ = − snτ
(2)
with n ∈ Z. This equations stem from suitable discrete zerocurvature equation1 d Un − EΓn[m] Un + Un Γn[m] = 0 dt
rn rn−1
,
cnτ =
cn cn − , dn dn+1
dnτ =
cn cn−1 − . dn+1 dn−1
(4)
The former is well-known infinite relativistic Toda lattice (RTL) in polynomial form. This system was introduced by Ruijsenaars [2] and then thoroughly studied by many authors from different points of view. For consulting, see, for example, Refs. [3–9]. The latter is the symmetry of RTL. It is common of knowledge that RTL (3) admits Hamiltonian form with standard symplectic structure w = n∈Z dpn ∧ dqn . To write it in this form one needs to use following ansatz: dn = epn ,
cn = epn +qn −qn+1 .
Then one has qnt = epn 1 + eqn −qn+1 , pnt = epn−1 +qn−1 −qn − epn +qn −qn+1 from which one obtains RTL in its exponential form qn+1,t eqn −qn+1 qn−1,t eqn−1 −qn qntt = qnt − . 1 + eqn−1 −qn 1 + eqn −qn+1
with some 2 × 2 matrices Un and Γn[m] . Let us introduce the variables cn and dn through cn =
and
dn = sn−1 .
As for symmetry (4), one can write 1 1 + eqn−1 −qn 1 + eqn −qn+1 . qnt
Then Eqs. (1) and (2) become
qnτ =
cnt = cn (cn−1 − cn+1 + dn − dn+1 ),
This symmetry is also well-known [10]. Moreover this kind of symmetries admit a whole class of RTL type systems.
dnt = dn (cn−1 − cn )
(3)
References DOI of original article: 10.1016/j.physleta.2005.02.021. E-mail address: [email protected] (A.K. Svinin). 1 Throughout this comment we try, as far as possible, to follow the notations of Ref. [1]. 0375-9601/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.10.081
[1] H.-X. Yang, X.-X. Xu, H.-Y. Ding, Phys. Lett. A 338 (2005) 117. [2] S.N. Ruijsenaars, Commun. Math. Phys. 133 (1990) 217. [3] M. Bruschi, O. Ragnisco, Phys. Lett. A 129 (1988) 21.
420
A.K. Svinin / Physics Letters A 350 (2006) 419–420
[4] M. Bruschi, O. Ragnisco, Phys. Lett. A 134 (1989) 365. [5] W. Oevel, B. Fuchssteiner, H. Zhang, O. Ragnisco, J. Math. Phys. 30 (1989) 2664. [6] Yu.B. Suris, Phys. Lett. A 145 (1990) 113. [7] Yu.B. Suris, Phys. Lett. A 180 (1993) 419.
[8] S. Cosentino, Inverse Problems 7 (1991) 535. [9] P. Damianou, J. Math. Phys. 35 (1994) 5511. [10] V.E. Adler, A.B. Shabat, R.I. Yamilov, Theor. Math. Phys. 125 (2000) 1603.