We draw attention to the connections recently established by others between the classical integrable KdV and KP hierarchies in 1+1 and 2+1 dimensions respectively and the matrix models which relate to the partition functions of 2-dimensional (1 + 1 dimensional) quantum gravity. The symmetries of the...

One of the most powerful methods for finding and solving integrable nonlinear partial differential equations is Hirota's bilinear method. The idea behind it is to make first a nonlinear change in the dependent variables after which multisoliton solutions of integrable systems can be expressed as polynomials...

The reduction by symmetry of the linear system of the self-dual Yang-Mills equations in four-dimensions under representatives of the conjugacy classes of subgroups of the connected part to the identity of the corresponding Euclidean group under itself is carried out. Only subgroups leading to systems...

Generalized self-duality equations for the supersymmetric Yang-Mills theory with a scalar multiplet are presented in terms of component fields and superfields as well.

Classical ideas and methods developed by Sophus Lie provide us with a powerful tool for constructing exact solutions of partial differential equations (PDE) (see, e.g., [1­4]). In the present paper we apply the above methods to obtain new explicit solutions of the nonlinear Yang-Mills equations (YME)....

We prove that two nonlinear generalizations of the nonlinear Schrödinger equation are invariant with respect to a Lie algebra that coincides with the invariance algebra of the Hamilton-Jacobi equation.

In this contribution we review and summarize recent articles on a family of nonlinear Schrödinger equations proposed by G.A. Goldin and one of us (HDD) [J. Phys. A. 27, 1994, 1771­1780], dealing with a gauge description of the family, a classification of its Lie symmetries in terms of gauge invariants...

We give ansatzes obtained from Lie symmetries of some hyperbolic equations which reduce these equations to the heat or Schrödinger equations. This enables us to construct new solutions of the hyperbolic equations using the Lie and conditional symmetries of the parabolic equations. Moreover, we note...

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form ut = (k(u) ux)x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations...

Necessary and sufficient conditions are found that the n-order nonlinear and nonautonomous ordinary differential equation could be transformed into a linear equation with constant coefficients with the help, generally speaking, nonlocal transformation of dependent and independent variables. These conditions...

Results of renormgroup analysis of a quasi-Chaplygin system of equations are presented. Lie-Bäcklund symmetries and corresponding invariant solutions for different "Chaplygin" functions are obtained. The algorithm of construction of a group on a solution (renormgroup) using two different approaches...

The single-time nonlocal Lagrangians corresponding to the Fokker-type action integrals are obtained in arbitrary form of relativistic dynamics. The symmetry conditions for such Lagrangians under an arbitrary Lie group acting on the Minkowski space are formulated in various forms of dynamics. An explicit...

A relativistic two-particle model with superposition of time-asymmetric scalar and vector interactions is proposed and its symmetries are considered. It is shown that first integrals of motion satisfy nonlinear Poisson-bracket relations which include the Poincaré algebra and one of the algebras so(1,3),...

We study a class of explicitly Poincare-invariant equations of motion (EMs) of two point bodies with a finite speed of propagation of interactions (combination of retarded and advanced ones) that may be considered as functional-differential equations or differential equations with deviating argument...

We study hidden symmetry of a two-particle system of equations for parastates. Invariance operators are described for various potentials. It is a well-known fact that the systems of partial differential equations have a hidden symmetry, which can not be observed in the classical approach of Lie [1].

In this paper we find the complete set of symmetry operators for the two-particle Breit equation in the class of first-order differential operators with matrix coefficients. A new integral of motion is obtained.

The infinite series of Lorentz and Poincaré-invariant nonlinear versions of the Maxwell equations are suggested. Some properties of these equations are considered.

Studied in this paper are real forms of the quantum algebra Uq(sl(3)). Integrable operator representations of -algebras are defined. Irreducible representations are classified up to a unitary equivalence.

A class of nonlinear wave equations is considered. Symmetry of these equations is extended using nonlocal transformations.

Group classification of the nonlinear wave equation is carried out and the conditional invariance of the wave equation with the nonlinearity F(u) = u is found.

New exact solutions are obtained for the systems of classical electrodynamics equations.

Some new symmetric integral operators with kernels involving the generalized Legendre's function of the first kind Pm,n k (z) are introduced. Some their applications are given.

Generators of multiparameter deformations Uq;s1,s2,...,sn-1 (gln) of the universal enveloping algebra U(gln) are realized bilinearly by means of an appropriately generalized form of anyonic oscillators (AOs). This modification takes into account the parameters s1, ..., sn-1 and yields usual AOs when...

Reductions and classes of new exact solutions are constructed for a class of Galileiinvariant heat equations.

We present symmetry classification of the polywave equation 2l u = F(u). It is established that the equation in question is invariant under the conformal group C(1, n) iff F(u) = eu , n + 1 = 2l or F(u) = u(n+1+2l)/(n+1-2l) , n + 1 = 2l. Symmetry reduction for the biwave equation 22 u = uk is carried...

A complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the Euler equations is constructed. Using some of them, the Euler equations are reduced to systems of partial differential equations in two independent variables which are integrated in quadratures.

In this paper we obtain the maximal Lie symmetry algebra of a system of PDEs. We reduce this system to a system of ODEs, using some rank three subalgebras of the finite-dimensional part of the symmetry algebra. The corresponding invariant solutions of the PDEs are obtained.

Differential forms are used for construction of nonlocal symmetries of partial differential equations with conservation laws. Every conservation law allows to introduce a nonlocal variable corresponding to a conserved quantity. A prolongation technique is suggested for action of symmetry operators...

Fundamental solutions (FS) with a given boundary condition on the characteristics of relativistic problems with axial symmetry are considered. This is so-called the Goursat problem (GP) or zero plane formalism in Dirac's terminology, or modification of the proper time method in the Fock-Nambu-Schwinger...

Asymptotic formulae for resolution of L-diagonal systems of ordinary differential equations with symmetrical matrices are derived.

We describe all systems of three equations of the form 2uj = Fj(u1, u2, u3), j = 1, 3 invariant under the extended Poincaré group. As a result, we have obtained ten classes of ~P(1, 3)-invariant nonlinear partial differential equations for real vector fields.

By means of splitting subgroups of the generalized Poincaré group P(1, 4), ansatzes which reduce the eikonal equation to differential equations with fewer independent variables have been constructed. The corresponding symmetry reduction has been done. By means of the solutions of the reduced equations...

We describe a class of generalized gas dynamics equations invariant under the extended Galilei algebra A ~G(1, n).