Volume 2, Issue 3-4, September 1995

Pages: 201 - 215

Bäcklund transformations, which are relations among solutions of partial differential
equationsusually nonlinearhave been found and applied mainly for systems with two
independent variables. A few are known for equations like the Kadomtsev-Petviashvili
equation [1], which has three independent variables,...

Pages: 216 - 235

In this talk I am going to present a brief review of some key ideas and methods
which were given start and were developed in Kyiv, at the Institute of Mathematics
of National Academy of Sciences of Ukraine during recent years.

Pages: 236 - 246

The Program SUBMODELS [1] is aimed to exhaust all possibilities derived from the symmetry
of differential equations for construction of submodels (i.e., systems of equations of the reduced
dimension) which describe classes of exact solutions for initial equations. In the frame of this
Program, our...

Pages: 247 - 262

We consider a representation of canonical commutation relations (CCR) appearing in a
(non-Abelian) gauge theory on a non-simply connected region in the two-dimensional
Euclidean space. A necessary and sufficient condition for the representation to be
equivalent to the Schrödinger representation of...

Pages: 263 - 268

We investigate the structure of certain types of subalgebras of Galilei algebras and
the relationship between the conjugacies of these subalgebras under different groups
of automorphisms.

Pages: 269 - 277

We study symmetries of the real Maxwell-Bloch equations. We give a Lax pair, biHamiltonian formulations and we find a symplectic realization of the system. We have
also constructed a hierarchy of master symmetries which is used to generate nonlinear
Poisson brackets. In addition we have calculated...

Pages: 278 - 282

The goal of this paper is to describe some interesting phenomena which occur in Hamiltonian systems with symplectic (locally Hamiltonian) symmetries.

Pages: 292 - 300

Pages: 301 - 311

Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is
described.

Pages: 312 - 318

Operators of nonlocal symmetry are used to construct exact solutions of nonlinear
heat equations
In [1] the idea of constructing nonlocal symmetry of differential equations was proposed.
By using this symmetry, we have suggested a method for finding new classes of ansatzes
reducing nonlinear wave...

Pages: 319 - 328

The transformation group theoretic approach is applied to present an analysis of the
nonlinear unsteady heat conduction problem in a semiinfinite body. The application
of oneparameter group reduces the number of independent variables by one, and
consequently the governing partial differential equation...

Pages: 329 - 333

Using the subgroup structure of the generalized Poincaré group P(1, 4), ansatzes
which reduce the EulerLagrangeBornInfeld, multidimensional MongeAmpere and
eikonal equations to differential equations with fewer independent variables have been
constructed. Among these ansatzes there are ones which...

Pages: 334 - 355

We present in this paper the singular manifold method (SMM) derived from Painlevé
analysis, as a helpful tool to obtain much of the characteristic features of nonlinear
partial differential equations. As is well known, it provides in an algorithmic way the
Lax pair and the Bäcklund transformation...

Pages: 356 - 366

Consider the operator pencil L = A - B - 2
C, where A, B, and C are linear, in
general unbounded and nonsymmetric, operators densely defined in a Hilbert space H.
Sufficient conditions for the existence of the eigenvalues of L are investigated in the
case when A, B and C are K-positive and K-symmetric...

Pages: 367 - 373

The presentation of Lie (super)algebras by a finite set of generators and defining
relations is one of the most general mathematical and algorithmic schemes of their
analysis. It is very important, for instance, for investigation of the particular Lie
(super)algebras arising in different (super)symmetric...

Pages: 374 - 383

All systems of (n+1)-dimensional quasilinear evolutional second- order equations invariant under the chain of algebras AG(1.n) AG1(1.n) AG2(1.n) are described.
The obtained results are illustrated by examples of nonlinear Schrödinger equations.

Pages: 384 - 391

The two-parameter deformation of canonical commutation relations is discussed. The
self-adjointness property of the (p, q)-deformed position Q and momentum P operators
is investigated. The (p, q)-analog of two-dimensional conformal field theory based on
the (p, q)-deformation of the su(1, 1) subalgebra...

Pages: 392 - 397

Similarity reductions of the Zabolotskaya-Khokhlov equation with a dissipative term
to one-dimensional partial differential equations including Burgers' equation are investigated by means of Lie's method of infinitesimal transformation. Some similarity
solutions of the Z-K equation are obtained.

Pages: 398 - 404

The Painlevé-test has been applied to checking the integrability of nonlinear PDEs,
since similarity solutions of many soliton equations satisfy the Painlevé equation. As
is well known, such similarity solutions can be obtained by the infinitesimal transformation, that is, the classical similarity...

Pages: 405 - 415

Appeared more than one century ago, the classical Lie approach serves as a powerful tool
in investigations of symmetries of partial differential equations. In the last three decades
there appear essential generalizations of this approach. They are the modern version of
the Lie-Bäcklund symmetries...

Pages: 416 - 417

Consider equation

Pages: 418 - 424

The paper contains a symmetry classification of the onedimensional second order
equation of a hydrodynamical type L(Lu) + Lu = F(u), where L t + ux. Some
classes of exact solutions of this equation are given.