Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented

EPR study of superconductors

Historical review on the studies of the electron paramagnetic resonance in superconductors performed in the period from 1970 to 1990 at the Kazan Physical Technical Institute of the Russian Academy of Sciences (group of Dr. E. G. Kharakhash'yan) in collaboration with Kazan State University (group of Prof. B. I. Kochelaev) and with the Institute for Physical Problems of Russian Academy of Sciences (group of Prof. N. E. Alekseevskii) is presented. We have observed for first time electron paramagnetic resonance of impurities in a type II superconductor; found indication for a long-range exchange interaction between magnetic impurities arising due to the superconducting correlations; observed the magnetic ordering of impurities in the superconducting state; and, finally, we found one of the first evidences for heterogeneity of the 1:2:3 high-$T_c$ superconductor which is its natural propert

Classification of five-point differential-difference equations

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the discrete Sawada-Kotera equations. The resulting list contains 17 equations some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries.Comment: 29 page

Method for searching higher symmetries for quad graph equations

Generalized symmetry integrability test for discrete equations on the square lattice is studied. Integrability conditions are discussed. A method for searching higher symmetries (including non-autonomous ones) for quad graph equations is suggested based on characteristic vector fields.Comment: 16 pages, submitted to JP