34
C. M. Viallet
References
1. E. Brezin, V. Kazakov, Exactly solvable field theories of closed strings. Phys. Lett. B 236(2),
144–150 (1990)
2. D. Gross, A. Migdal, Non perturbative two-dimensional quantum gravity. Phys. Rev. Lett.
64(2), 127–130 (1990)
3. A.S. Fokas, A.R. Its, A.V. Kitaev, Discrete Painlevé equations and their appearance in quantum
gravity. Commun. Math. Phys 142, 313–344 (1991)
4. R.J. Baxter, The inversion relation method for some two-dimensional exactly solved models in
lattice statistics. J. Stat. Phys. 28, 1–41 (1982)
5. M.P. Bellon, J.-M. Maillard, C.-M. Viallet, Infinite discrete symmetry group for the YangBaxter equations: spin models. Phys. Lett. A A 157, 343–353 (1991)
6. M.P. Bellon, J.-M. Maillard, C.-M. Viallet, Infinite discrete symmetry group for the YangBaxter equations: vertex models. Phys. Lett. A B 260, 87–100 (1991)
7. V.G. Drinfeld, On some unsolved problems in quantum group theory, in “Quantum Groups”,
vol. 1510 of Lecture Notes in Math (Springer, 1992), pp. 1–8
8. M. Hénon, A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50,
69–77 (1976)
9. J. Hietarinta, N. Joshi, F.W. Nijhoff, Discrete Systems and Integrability. Cambridge Texts in
Applied Mathematics (Cambridge University Press, 2016)
10. S. Butler, N. Joshi, An inverse scattering transform for the lattice potential KdV equation.
Inverse Problems 26, 115012 (2010)
11. F. Nijhoff, Lax pair for the Adler (lattice Krichever-Novikov) system. Phys. Lett. A 297, 49–58
(2002). arXiv:nlin.SI/0110027
12. A.I. Bobenko, Yu.B. Suris, Integrable systems on quad-graphs. Int. Math. Res. Notices 11,
573–611 (2002). arXiv:nlin/0110004
13. V.E. Adler, A.I. Bobenko, Yu.B. Suris, Classification of integrable equations on quadgraphs. The consistency approach.
Commun. Math. Phys. 233(3), 513–543 (2003).
arXiv:nlin.SI/0202024
14. V.E. Adler, A.I. Bobenko, Yu.B. Suris, Discrete nonlinear hyperbolic equations. Classification
of integrable cases. Funct. Anal. Appl. 43, 3–17 (2009). arXiv:0705.1663
15. C.-M. Viallet, Integrable lattice maps: Q V , a rational version of Q 4 . Glasgow Math. J. 51 A,
157–163 (2009). arXiv:0802.0294
16. A.V. Mikhailov, J.P. Wang, A new recursion operator for Adler’s equation in the Viallet form.
Physics Letters A 375, 3960–3963 (2011). arXiv:1105.1269
17. P.J. Olver, Evolution equations possessing an infinitely many symmetries. J. Math. Phys. 18(6),
1212–1215 (1977)
18. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Inverse scattering transform-Fourier analysis
for nonlinear problems. Stud. Appl. Math. 53(4), 249–315 (1974)
19. S. Maeda, The similarity method for difference equations. IMA J. Appl. Math. 38, 129 (1987)
20. D. Levi, P. Winternitz, Continuous symmetries of difference equations. J. Phys. A Math. Gen.
39, R1–R63 (2006)
21. D. Levi, R.I. Yamilov, Conditions for the existence of higher symmetries of evolutionary
equations on the lattice. J. Math. Phys. 38, 6648–6674 (1997)
22. D. Levi, R.I. Yamilov, The generalized symmetry method for discrete equations. J. Phys. A
Math. Theor. 42, 454012 (2009)
23. D. Levi, R.I. Yamilov, Generalized symmetry integrability test for discrete equations on the
square lattice. J. Phys. A Math. Theor. 44, 145207 (2011)
24. G.R.W. Quispel, H.W. Capel, R. Sahadevan, Continuous symmetries of differential-difference
equations: the Kac-van Moerbeke equation and the Painlevé reduction. Phys. Lett. A(170),
379–383 (1992)
25. D. Levi, L. Vinet, P. Winternitz, Lie group formalism for difference equations. J. Phys. A Math.
Gen. 30(2), 633–649 (1997)
C. M. Viallet
References
1. E. Brezin, V. Kazakov, Exactly solvable field theories of closed strings. Phys. Lett. B 236(2),
144–150 (1990)
2. D. Gross, A. Migdal, Non perturbative two-dimensional quantum gravity. Phys. Rev. Lett.
64(2), 127–130 (1990)
3. A.S. Fokas, A.R. Its, A.V. Kitaev, Discrete Painlevé equations and their appearance in quantum
gravity. Commun. Math. Phys 142, 313–344 (1991)
4. R.J. Baxter, The inversion relation method for some two-dimensional exactly solved models in
lattice statistics. J. Stat. Phys. 28, 1–41 (1982)
5. M.P. Bellon, J.-M. Maillard, C.-M. Viallet, Infinite discrete symmetry group for the YangBaxter equations: spin models. Phys. Lett. A A 157, 343–353 (1991)
6. M.P. Bellon, J.-M. Maillard, C.-M. Viallet, Infinite discrete symmetry group for the YangBaxter equations: vertex models. Phys. Lett. A B 260, 87–100 (1991)
7. V.G. Drinfeld, On some unsolved problems in quantum group theory, in “Quantum Groups”,
vol. 1510 of Lecture Notes in Math (Springer, 1992), pp. 1–8
8. M. Hénon, A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50,
69–77 (1976)
9. J. Hietarinta, N. Joshi, F.W. Nijhoff, Discrete Systems and Integrability. Cambridge Texts in
Applied Mathematics (Cambridge University Press, 2016)
10. S. Butler, N. Joshi, An inverse scattering transform for the lattice potential KdV equation.
Inverse Problems 26, 115012 (2010)
11. F. Nijhoff, Lax pair for the Adler (lattice Krichever-Novikov) system. Phys. Lett. A 297, 49–58
(2002). arXiv:nlin.SI/0110027
12. A.I. Bobenko, Yu.B. Suris, Integrable systems on quad-graphs. Int. Math. Res. Notices 11,
573–611 (2002). arXiv:nlin/0110004
13. V.E. Adler, A.I. Bobenko, Yu.B. Suris, Classification of integrable equations on quadgraphs. The consistency approach.
Commun. Math. Phys. 233(3), 513–543 (2003).
arXiv:nlin.SI/0202024
14. V.E. Adler, A.I. Bobenko, Yu.B. Suris, Discrete nonlinear hyperbolic equations. Classification
of integrable cases. Funct. Anal. Appl. 43, 3–17 (2009). arXiv:0705.1663
15. C.-M. Viallet, Integrable lattice maps: Q V , a rational version of Q 4 . Glasgow Math. J. 51 A,
157–163 (2009). arXiv:0802.0294
16. A.V. Mikhailov, J.P. Wang, A new recursion operator for Adler’s equation in the Viallet form.
Physics Letters A 375, 3960–3963 (2011). arXiv:1105.1269
17. P.J. Olver, Evolution equations possessing an infinitely many symmetries. J. Math. Phys. 18(6),
1212–1215 (1977)
18. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Inverse scattering transform-Fourier analysis
for nonlinear problems. Stud. Appl. Math. 53(4), 249–315 (1974)
19. S. Maeda, The similarity method for difference equations. IMA J. Appl. Math. 38, 129 (1987)
20. D. Levi, P. Winternitz, Continuous symmetries of difference equations. J. Phys. A Math. Gen.
39, R1–R63 (2006)
21. D. Levi, R.I. Yamilov, Conditions for the existence of higher symmetries of evolutionary
equations on the lattice. J. Math. Phys. 38, 6648–6674 (1997)
22. D. Levi, R.I. Yamilov, The generalized symmetry method for discrete equations. J. Phys. A
Math. Theor. 42, 454012 (2009)
23. D. Levi, R.I. Yamilov, Generalized symmetry integrability test for discrete equations on the
square lattice. J. Phys. A Math. Theor. 44, 145207 (2011)
24. G.R.W. Quispel, H.W. Capel, R. Sahadevan, Continuous symmetries of differential-difference
equations: the Kac-van Moerbeke equation and the Painlevé reduction. Phys. Lett. A(170),
379–383 (1992)
25. D. Levi, L. Vinet, P. Winternitz, Lie group formalism for difference equations. J. Phys. A Math.
Gen. 30(2), 633–649 (1997)