The Liouville integrability of integrable couplings of Volterra lattice equation

A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.     

New exact solutions to the nonautonomous Liouville equation

We obtain new formulas for the exact analytic solutions to the nonautonomous elliptic Liouville equation in the two-dimensional coordinate space with the free function dependent specially on an arbitrary harmonic function. We present new exact solutions to the wave Liouville equation with two arbitrary functions, providing original formulas for the general solution for the classical (autonomous) and wave Liouville equations. Some equivalence transformations are presented for the elliptic Liouville equation depending on conjugate harmonic functions. In particular, we indicate a transformation that reduces the equation under study to an autonomous form.     

Hamiltonian description of singular solutions of the Liouville equation

Theoretical and Mathematical Physics -     

Exact solutions of the Liouville equation in multidimensional spaces

Exact solutions of the multidimensional Liouville equation, which are sought in the class of functional forms of the degree n equal to the coordinate space dimension, are constructed based on a special representation of the Laplace and D'Alembert equations. The corresponding Liouville equation solutions are completely described in the coordinate space dimensions d=3,4. For d>4, the general solution form and the method for obtaining algebraic equations for the coefficients of functional n-forms is presented. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 3–19, July, 1999.     

Solutions of the Knizhnik—Zamolodchikov trigonometric equation

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 69–71, October–December, 1991.     

Energy identities and blow-up analysis for solutions of the super Liouville equation

In this paper, we study the super Liouville equations, a natural generalization of the Liouville equation. We establish energy identities and a precise blow-up analysis for solutions of the super Liouville equations.     

Classification of constant solutions of the associative Yang-Baxter equation on Mat3

We find all nonequivalent constant solutions of the classical associative Yang-Baxter equation for Mat ₃ . New examples found in the classification yield the corresponding Poisson brackets on traces, double Poisson brackets on a free associative algebra with three generators, and anti-Frobenius associative algebras.     

Commutator identities on associative algebras and the integrability of nonlinear evolution equations

We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations. In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear integrable equation itself and its Lax pair from such a commutator identity. The problem of constructing new integrable nonlinear evolution equations thus reduces to the problem of constructing commutator identities on associative algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 477–491, March, 2008.     

The tetrahedron equation and algebraic geometry

The tetrahedron equation arises as a generalization of the famous Yang-Baxter equation to the2+1-dimensional quantum field theory and three-dimensional statistical mechanics. Not much is known about its solutions. In the present paper, a systematic method of constructing nontrivial solutions to the tetrahedron equation with spin-like variables on the links is described. The essence of this method is the use of the so-called tetrahedral Zamolodchikov algebras. Bibliography:12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 137–149. Translated by I. G. Korepanov.     

Complete integrability of the Kadomtsev-Petviashvili equation

The total hierarchy of the Kadomtsev-Petviashvili (KP) equation is transformed to the system of linear partial differential equations with constant coefficients. The complete integrability of the KP equation is proved by using this linear system. The existence and uniqueness theorem of the Cauchy problem of the KP hierarchy is obtained.     

Full symmetry groups, Painlevé integrability and exact solutions of the nonisospectral BKP equation

Based on the generalized symmetry group method presented by Lou and Ma [Lou and Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A: Math. Gen. 38 (2005) L129], firstly, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. Secondly, the nonisospectral BKP can be proved to be Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification, some solutions are obtained by using the standard truncated Painlevé expansion. Finally, based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed.     

Discrete analogues of the Liouville equation

The notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables. The sequence of Laplace invariants satisfies the discrete analogue of the two-dimensional Toda lattice. We prove that terminating this sequence by zeros is a necessary condition for the existence of integrals of the equation under consideration. We present formulas for the higher symmetries of equations possessing such integrals. We give examples of difference analogues of the Liouville equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 271–284, November, 1999.     

Note on some ideals of associative rings

It is well known that being an ideal (left ideal, right ideal) of a ring is not a transitive relation. Nevertheless in some cases the transitive property does hold. Systematic studies of this subject were started in [11,13]. In this paper we continue these studies.