integrable nonlinear partial differential and differential-difference equations; classification of integrable equations; higher (generalized) symmetries and conservation laws; Hamiltonian and Lagrangian structure; transformation theory for integrable equations; Miura, Backlund and Schlesinger tranformations.
UDC:
517.9
Subject:
The classification problem has been solved for classes of integrable (more precisely, possessing an infinite hierarchy of higher symmetries and conservation laws) equations including the differential-difference Volterra and Toda equations and also (with A. B. Shabat and A. V. Mikhailov) for a class which contains the nonlinear Schrodinger equation. The notion of a quasi-local function has been introduced (with A. V. Mikhailov) which has allowed to generalize the Symmetry Approach to the classification of integrable equations for the case of 1+2 dimensional equations. A number of papers is devoted to the transformation theory for integrable equations. In particular, a scheme of the construction of modified equations together with corresponding Miura transformations has been presented which does not use $L-A$ pairs, but only uses Miura transformations.
Main publications:
D. Levi, R. I. Yamilov, “The generalized symmetry method for discrete equations”, J. Phys. A, Math. Theor., 42:45 (2009), 18 , IOP Publishing, Bristol
R. Yamilov, “Symmetries as integrability criteria for differential difference equations”, J. Phys. A, Math. Gen., 39:45 (2006), r541–r623 , IOP Publishing Ltd., Bristol, UK
V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, Theoret. and Math. Phys., 125:3 (2000), 1603–1661
A. V. Mikhajlov, A. B. Shabat, R. I. Yamilov, “Extension of the module of invertible transformations. Classification of integrable systems”, Commun. Math. Phys., 115:1 (1988), 1–19 , Springer, Berlin/Heidelberg
A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems”, Russian Math. Surveys, 42:4 (1987), 1–63
A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems”, Russian Math. Surveys, 42:4 (1987), 1–63
2.
V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, Theoret. and Math. Phys., 125:3 (2000), 1603–1661
3.
D. Levi, R. Yamilov, “Conditions for the existence of higher symmetries of evolutionary equations on the lattice”, J. Math. Phys., 38:12 (1997), 6648–6674 , American Institute of Physics (AIP), Woodbury, NY
R. Yamilov, “Symmetries as integrability criteria for differential difference equations”, J. Phys. A, Math. Gen., 39:45 (2006), r541–r623 , IOP Publishing Ltd., Bristol, UK
A. V. Mikhajlov, A. B. Shabat, R. I. Yamilov, “Extension of the module of invertible transformations. Classification of integrable systems”, Commun. Math. Phys., 115:1 (1988), 1–19 , Springer, Berlin/Heidelberg
V. E. Adler, S. I. Svinolupov, R. I. Yamilov, “Multi-component Volterra and Toda type integrable equations”, Phys. Lett., A, 254:1–2 (1999), 24–36 , Elsevier (North-Holland), Amsterdam
I. Yu. Cherdantsev, R. I. Yamilov, “Master symmetries for differential-difference equations of the Volterra type”, Physica D, 87:1–4 (1995), 140–144 , Elsevier (North-Holland), Amsterdam
R. I. Yamilov, “Construction scheme for discrete Miura transformations”, J. Phys. A, Math. Gen., 27:20 (1994), 6839–6851 , IOP Publishing Ltd., Bristol, UK
A. B. Shabat, R. I. Yamilov, “To a transformation theory of two-dimensional integrable systems”, Phys. Lett., A, 227:1–2 (1997), 15–23 , Elsevier (North-Holland), Amsterdam
V. E. Adler, R. I. Yamilov, “Explicit auto-transformations of integrable chains”, J. Phys. A, Math. Gen., 27:2 (1994), 477–492 , IOP Publishing Ltd., Bristol, UK
A. N. Leznov, A. B. Shabat, R. I. Yamilov, “Canonical transformations generated by shifts in nonlinear lattices”, Phys. Lett. A, 174:5–6 (1993), 397–402
A. V. Mikhailov, R. I. Yamilov, “Towards classification of $(2+1)$-dimensional integrable equations. Integrability conditions. I”, J. Phys. A, Math. Gen., 31:31 (1998), 6707–6715 , IOP Publishing Ltd., Bristol, UK
R. N. Garifullin, R. I. Yamilov, “Generalized symmetry classification of discrete equations of a class depending on twelve parameters”, J. Phys. A, Math. Theor., 45:34 (2012), 23 , IOP Publishing, Bristol
R. I. Yamilov, “Invertible changes of variables generated by Bäcklund transformations”, Theoret. and Math. Phys., 85:2 (1990), 1269–1275
20.
D. Levi, R. I. Yamilov, “Generalized symmetry integrability test for discrete equations on the square lattice”, J. Phys. A, Math. Theor., 44:14 (2011), 22 , IOP Publishing, Bristol
R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Discrete equation on a square lattice with a nonstandard structure of generalized symmetries”, Theoret. and Math. Phys., 180:1 (2014), 765–780
22.
R. I. Yamilov, “On the construction of Miura type transformations by others of this kind”, Phys. Lett. A, 173:1 (1993), 53–57
R. N. Garifullin, R. I. Yamilov, D. Levi, “Classification of five-point differential-difference equations”, J. Phys. A, Math. Theor., 50:12 (2017), 125201 (27pp)
D. Levi, P. Winternitz, R. I. Yamilov, “Lie point symmetries of differential-difference equations”, J. Phys. A, Math. Theor., 43:29 (2010), 14 , IOP Publishing, Bristol
A. V. Mikhailov, R. I. Yamilov, “On integrable two-dimensional generalizations of nonlinear Schrödinger type equations”, Physics Letters, Section A: General, Atomic and Solid State Physics, 230:5–6 (1997), 295–300 , Elsevier (North-Holland), Amsterdam
Decio Levi, Pavel Winternitz, Ravil I. Yamilov, “Symmetries of the Continuous and Discrete Krichever–Novikov Equation”, SIGMA, 7 (2011), 97 , 16 pp., arXiv: 1110.5021
R. Yamilov, D. Levi, “Integrability conditions for $n$ and $t$ dependent dynamical lattice equations”, J. Nonlinear Math. Phys., 11:1 (2004), 75–101 , Taylor & Francis, Abingdon, Oxfordshire; Atlantis Press, Paris
S. I. Svinolupov, R. I. Yamilov, “Explicit Bäcklund transformations for multifield Schrödinger equations. Jordan generalizations of the Toda chain”, Theoret. and Math. Phys., 98:2 (1994), 139–146
29.
R. N. Garifullin, R. I. Yamilov and D. Levi, “Classification of five-point differential-difference equations II”, J. Phys. A: Math. Theor, 51:6 (2018), 065204 , 16 pp.
Giorgio Gubbiotti, Christian Scimiterna, Ravil I. Yamilov, “Darboux Integrability of Trapezoidal $H^{4}$ and $H^{6}$ Families of Lattice Equations II: General Solutions”, SIGMA, 14 (2018), 8 , 51 pp.
G. Gubbiotti, R. I. Yamilov, “Darboux integrability of trapezoidal $H^4$ and $H^4$ families of lattice equations I: first integrals”, J. Phys. A: Math. Theor., 50:34 (2017), 345205 , 26 pp.
R. N. Garifullin, R. I. Yamilov, “Examples of Darboux integrable discrete equations possessing first integrals of an arbitrarily high minimal order”, Ufimsk. matem. zhurn., 4:3 (2012), 177–183
R. N. Garifullin, R. I. Yamilov, “On integrability of a discrete analogue of Kaup–Kupershmidt equation”, Ufa Math. Journal, 9:3 (2017), 158–164
35.
R. N. Garifullin, R. I. Yamilov, D. Levi, “Non-invertible transformations of differential-difference equations”, J. Phys. A, Math. Theor., 49:37 (2016), 23 pp , IOP Publishing, Bristol
R. N. Garifullin, R. I. Yamilov, “Integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations”, Journal of Physics: Conference Series, 621:1 (2015), 012005
R. N. Garifullin, G. Gubbiotti, R. I. Yamilov, “Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations”, Journal of Nonlinear Mathematical Physics, 26:3 (2019), 333-357 , arXiv: 1810.11184
R. N. Garifullin, I. T. Habibullin, R. I. Yamilov, “Peculiar symmetry structure of some known discrete nonautonomous equations”, J. Phys. A, Math. Theor., 48:23 (2015), 27 , IOP Publishing, Bristol
D. Levi, R. Yamilov, “On the integrability of a new discrete nonlinear Schrödinger equation”, J. Phys. A, Math. Gen., 34:41 (2001), l553–l562 , IOP Publishing Ltd., Bristol, UK
R. N. Garifullin, R. I. Yamilov, “An unusual series of autonomous discrete integrable equations on a square lattice”, Theoret. and Math. Phys., 200:1 (2019), 966–984
41.
D. Levi, R. Yamilov, “Dilation symmetries and equations on the lattice”, J. Phys. A, Math. Gen., 32:47 (1999), 8317–8323 , IOP Publishing Ltd., Bristol, UK
R. N. Garifullin , R. I. Yamilov, “On series of Darboux integrable discrete equations on square lattice”, Ufa Math. J., 11:3 (2019), 99–108
43.
R. I. Yamilov, “Integrability conditions for an analogue of the relativistic Toda chain”, Theoret. and Math. Phys., 151:1 (2007), 492–504
44.
R. I. Yamilov, “Relativistic Toda Chains and Schlesinger Transformations”, Theoret. and Math. Phys., 139:2 (2004), 623–635
45.
D. Levi, R. Yamilov, “Non-point integrable symmetries for equations on the lattice”, J. Phys. A, Math. Gen., 33:26 (2000), 4809–4823 , IOP Publishing Ltd., Bristol, UK
R. N. Garifullin, R. I. Yamilov, “Modified series of integrable discrete equations on a quadratic lattice with a nonstandard symmetry structure”, Theoret. and Math. Phys., 205:1 (2020), 1265–1279
47.
Rustem N. Garifullin, Ravil I. Yamilov, “Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation”, SIGMA, 15 (2019), 62 , 15 pp., arXiv: 1903.11893
R. N. Garifullin, R. I. Yamilov, “On the Integrability of a Lattice Equation with Two Continuum Limits”, J. Math. Sci. (N. Y.), 252:2 (2021), 283–289
49.
A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “On an extension of the module of invertible transformations”, Dokl. Math., 36:1 (1988), 60–63
50.
D. Levi, R. I. Yamilov, “Generalized Lie symmetries for difference equations”, Symmetries and integrability of difference equations. Based upon lectures delivered during the summer school, Montreal, Canada, June 8–21, 2008, Cambridge: Cambridge University Press, 2011, 160–190
51.
D. Levi, R. I. Yamilov, “Integrability test for discrete equations via generalized symmetries”, Aip Conference Proceedings, 1323, no. 1, AMER INST PHYSICS, 2010, 203
52.
D. Levi, R. Yamilov, “Conditions for the existence of higher symmetries and nonlinear evolutionary equations on the lattice”, Algebraic methods in physics. A symposium for the 60th birthdays of Ji\ví Patera and Pavel Winternitz. Centre de Recherches Mathématiques (CRM), Montréal, Canada, January 1997, Springer, New York, 2001, 135–148
53.
I. T. Habibullin, V. V. Sokolov, R. I. Yamilov, “Multi-component integrable systems and nonassociative structures”, Nonlinear physics: theory and experiment. Nature, structure and properties of nonlinear phenomena. Proceedings of the workshop, Lecce, Italy, June 29–July 7, 1995, World Scientific, Singapore, 1996, 139–168
54.
I. Cherdantsev, R. Yamilov, “Local master symmetries of differential-difference equations”, Symmetries and integrability of difference equations. Papers from the workshop, May 22–29, 1994, Estérel, Canada, American Mathematical Society, Providence, RI, 1996, 51–61
55.
A. V. Mikhajlov, A. B. Shabat, R. I. Yamilov, “On extending the module of invertible transformations”, Sov. Math., Dokl., 36:1 (1987), 60–63 , American Mathematical Society, Providence, RI
56.
S. I. Svinolupov, V. V. Sokolov, R. I. Yamilov, “On Bäcklund transformations for integrable evolution equations”, Sov. Math., Dokl., 28 (1983), 165–168 , American Mathematical Society, Providence, RI
57.
R. I. Yamilov, “On the classification of discrete equations”, 1982, Integrable systems, Work Collect., Ufa 1982, 95-114 (1982).
58.
R. I. Yamilov, “On conservation laws for the difference Korteweg-de Vries equation”, Din. Splosh. Sredy, 44 (1980), 164–173 , Russian Academy of Sciences - RAS (Rossiĭskaya Akademiya Nauk - RAN), Siberian Branch (Sibirskoe Otdelenie), Institute of Hydrodynamics named after M. A. Lavrent'eva (Institut Gidrodinamiki Im. M. A. Lavrent'eva), Novosibirsk
59.
S. I. Svinolupov, V. V. Sokolov, R. I. Yamilov, “On Bäcklund transformations for integrable evolution equations”, Dokl. Akad. Nauk SSSR, 271:4 (1983), 802–805