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

SIGMA 6 (2010), 092, 14 pages      arXiv:1010.0361

On Non-Point Invertible Transformations of Difference and Differential-Difference Equations

Sergey Ya. Startsev
Ufa Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Str., Ufa, 450077, Russia

Received October 04, 2010, in final form December 03, 2010; Published online December 11, 2010

Non-point invertible transformations are completely described for difference equations on the quad-graph and for their differential-difference analogues. As an illustration, these transformations are used to construct new examples of integrable equations and autotransformations of the Hietarinta equation.

Key words: non-point transformation; Darboux integrability; discrete Liouville equation; higher symmetry.

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  1. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equation on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
  2. Adler V.E., Startsev S.Ya., Discrete analogues of the Liouville equation, Teoret. Mat. Fiz. 121 (1999), 271-284 (English transl.: Theoret. and Math. Phys. 121 (1999), 1484-1495), solv-int/9902016.
  3. Calogero F., Why are certain nonlinear PDEs both widely applicable and integrable?, in What is integrability?, Editor V.E. Zakharov, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 1-62.
  4. Habibullin I.T., Characteristic algebras of fully discrete hyperbolic type equations, SIGMA 1 (2005), 023, 9 pages, nlin.SI/0506027.
  5. Habibullin I.T., Zheltukhina N., Pekcan A., Complete list of Darboux integrable chains of the form t1x=tx+d(t,t1), J. Math. Phys. 50 (2009), 102710, 23 pages, arXiv:0907.3785.
  6. Hietarinta J., A new two-dimensional lattice model that is 'consistent around a cube', J. Phys. A: Math. Gen. 37 (2004), L67-L73, nlin.SI/0311034.
  7. Hirota R., Nonlinear partial difference equations. III. Discrete sine-Gordon equation, J. Phys. Soc. Japan 43 (1977), 2079-2086.
  8. Hirota R., Nonlinear partial difference equations. V. Nonlinear equations reducible to linear equations, J. Phys. Soc. Japan 46 (1979), 312-319.
  9. Hirota R., Discrete two-dimensional Toda molecule equation, J. Phys. Soc. Japan 56 (1987), 4285-4288.
  10. Levi R., Yamilov R.I., The generalized symmetry method for discrete equation, J. Phys. A: Math. Theor. 42 (2009), 454012, 18 pages, arXiv:0902.4421.
  11. Mikhailov A.V., Wang J.P., Xenitidis P.D., Recursion operators, conservation laws and integrability conditions for difference equations, arXiv:1004.5346.
  12. Nijhoff W.F., Quispel G.R.W., Capel H.W., Linearization of nonlinear differential-difference equations, Phys. Lett. A 95 (1983), 273-276.
  13. Nijhoff F.W., Capel H.W., The discrete Korteweg-de Vries equation, Acta Appl. Math. 39 (1995), 133-158.
  14. Orfanidis S.J., Discrete sine-Gordon equations, Phys. Rev. D 18 (1978), 3822-3827.
  15. Ramani A., Joshi N., Grammaticos B., Tamizhmani N., Deconstructing an integrable lattice equation, J. Phys. A: Math. Gen. 39 (2006), L145-L149.
  16. Sokolov V.V., Svinolupov S.I., On nonclassical invertible transformation of hyperbolic equations, European J. Appl. Math. 6 (1995), 145-156.
  17. Yamilov R.I., Invertible changes of variables generated by Bäcklund transformations, Teoret. Mat. Fiz. 85 (1990), 368-375 (English transl.: Theoret. and Math. Phys. 85 (1991), 1269-1275).
  18. Yamilov R.I., Construction scheme for discrete Miura transformation, J. Phys. A: Math. Gen. 27 (1994), 6839-6851.
  19. Zhiber A.V., Sokolov V.V., Startsev S.Ya., On nonlinear Darboux-integrable hyperbolic equations, Dokl. Ross. Akad. Nauk 343 (1995), 746-748 (English transl.: Doklady Math. 52 (1996), 128-130).
  20. Zhiber A.V., Sokolov V.V., Exactly integrable hyperbolic equations of Liouville type, Usp. Mat. Nauk 56 (2001), no. 1, 63-106 (English transl.: Russ. Math. Surv. 56 (2001), no. 1, 61-101).

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