Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 093, 17 pages      nlin.SI/0612060      https://doi.org/10.3842/SIGMA.2006.093
Contribution to the Vadim Kuznetsov Memorial Issue

On the One Class of Hyperbolic Systems

Vsevolod E. Adler and Alexey B. Shabat
L.D. Landau Institute for Theoretical Physics, 1A prosp. ak. Semenova, 142432 Chernogolovka, Russia

Received October 27, 2006; Published online December 27, 2006

Abstract
The classification problem is solved for some type of nonlinear lattices. These lattices are closely related to the lattices of Ruijsenaars-Toda type and define the Bäcklund auto-transformations for the class of two-component hyperbolic systems.

Key words: hyperbolic systems; Bäcklund transformations; Ruijsenaars-Toda lattice; discrete Toda lattice.

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References

  1. Ruijsenaars S.N.M., Relativistic Toda system, Preprint Stichting Centre for Mathematics and Computer Sciences, Amsterdam, 1986.
  2. Ruijsenaars S.N.M., Relativistic Toda systems, Comm. Math. Phys., 1990, V.133, 217-247.
  3. Bruschi M., Ragnisco O., On a new integrable Hamiltonian system with nearest-neighbour interaction, Inverse Problems, 1989, V.5, 983-998.
  4. Suris Yu.B., Discrete time generalized Toda lattices: complete integrability and relation with relativistic Toda lattices, Phys. Lett. A, 1990, V.145, 113-119.
  5. Adler V.E., Shabat A.B., On the one class of the Toda chains, Theor. Math. Phys., 1997, V.111, 323-334.
  6. Adler V.E., Shabat A.B., Generalized Legendre transformations, Theor. Math. Phys., 1997, V.112, 935-948.
  7. Adler V.E., Shabat A.B., First integrals of the generalized Toda lattices, Theor. Math. Phys., 1998, V.115, 349-358.
  8. Marikhin V.G., Shabat A.B., Integrable lattices, Theor. Math. Phys., 1999, V.118, 217-228.
  9. Adler V.E., Marikhin V.G., Shabat A.B., Canonical Bäcklund transformations and Lagrangian chains, Theor. Math. Phys., 2001, V.129, 163-183.
  10. Ragnisco O., Santini P.M., A unified algebraic approach to integral and discrete evolution equations, Inverse Problems, 1990, V.6, 441-452.
  11. Shabat A.B., Yamilov R.I., Symmetries of nonlinear lattices, Leningrad Math. J., 1991, V.2, 377-400.
  12. Adler V.E., Yamilov R.I., Explicit auto-transformations of integrable chains, J. Phys. A: Math. Gen., 1994, V.27, 477-492.
  13. Yamilov R.I., Symmetry approach to the classification from the point of view of the differential-difference equations. Theory of transformations, Doctoral Thesis, Ufa, 2000.
  14. Adler V.E., Shabat A.B., Yamilov R.I., Symmetry approach to the integrability problem, Theor. Math. Phys., 2000, V.125, 1603-1661.
  15. Adler V.E., Discretizations of the Landau-Lifshitz equation, Theor. Math. Phys., 2000, V.124, 897-908.
  16. Hirota R., Nonlinear partial difference equations. II. Discrete-time Toda equation, J. Phys. Soc. Japan, 1977, V.43, 2074-2078.
  17. Suris Yu.B., Bi-Hamiltonian structure of the qd algorithm and new discretizations of the Toda lattice, Phys. Lett. A, 1995, V.206, 153-161.
  18. Adler V.E., On the structure of the Bäcklund transformations for the relativistic lattices, J. Nonlinear Math. Phys., 2000, V.7, 34-56, nlin.SI/0001072.
  19. Adler V.E., Suris Yu.B., Q4: Integrable master equation related to an elliptic curve, Internat. Math. Res. Not., 2004, V.47, 2523-2553, nlin.SI/0309030.
  20. Zhiber A.V., Ibragimov N.H., Shabat A.B., Liouville type equations, DAN SSSR, 1979, V.249, 26-29.
  21. Sokolov V.V., Zhiber A.V., On the Darboux integrable hyperbolic equations, Phys. Lett. A, 1995, V.208, 303-308.
  22. Zhiber A.V., Sokolov V.V., Exactly solvable hyperbolic equations of the Liouville type, Uspekhi Mat. Nauk, 2001, V.56, N 1, 63-106.
  23. Yamilov R.I., On classification of discrete evolution equations, Uspekhi Mat. Nauk, 1983, V.38, N 6, 155-156.
  24. Adler V.E., Shabat A.B., Dressing chain for the acoustic spectral problem, Theor. Math. Phys., 2006, V.149, 1324-1337, nlin.SI/0604008.
  25. Nijhoff F., Hone A., Joshi N., On a Schwarzian PDE associated with the KdV hierarchy, Phys. Lett. A, 2000, V.267, 147-156, solv-int/9909026.
  26. Svinolupov S.I., Yamilov R.I., The multi-field Schrödinger lattices, Phys. Lett. A, 1991, V.160, 548-552.
  27. Merola I., Ragnisco O., Tu G.-Z., A novel hierarchy of integrable lattices, Inverse Problems, 1994, V.10, 1315-1334, solv-int/9401005.

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