Volume 4, Issue 3-4, September 1995

Pages: 251 - 260

In this paper, we bring out the Lie symmetries and associated similarity reductions
of the recently proposed (2+1) dimensional long dispersive wave equation. We point
out that the integrable system admits an infinite-dimensional symmetry algebra along
with Kac-Moody-Virasoro-type subalgebras. We also...

Pages: 261 - 270

I analyze the one-dimensional, cubic Schrödinger equation with a nonlinearity constructed from the current density rather than, as is usual, from the charge density.
A soliton solution is found, where the soliton moves only in one direction. Relation
to the higher-dimensional ChernSimons theory is...

Pages: 271 - 277

It is shown that every point transformation group whose prolonged orbit dimensions
pseudo-stabilize at order 0 is equivalent, under a change of variables, to the elementary
similarity group consisting of translations and dilatations.

Pages: 278 - 286

We prove a generalization to the case of s × s matrix linear differential operators
of the classical theorem of E. Cotton giving necessary and sufficient conditions for
equivalence of eigenvalue problems for scalar linear differential operators. The conditions for equivalence to a matrix Schrödinger...

Pages: 287 - 292

An analogue of the Holstein-Primakoff and Dyson realizations for the Lie superalgebra
sl(1/n) is written down. Expressions are the same as for the Lie algebra sl(n + 1),
however in the latter, Bose operators have to be replaced with Fermi operators.

Pages: 293 - 309

By starting from known graded Lie algebras, including Virasoro algebras, new kinds
of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified
KP equations with m arbitrary time-dependent coefficients...

Pages: 310 - 337

In this paper, we consider a general anharmonic oscillator of the form ¨x + f1(t) x +
f2(t)x+f3(t)xn
= 0, with n Q. We seek the most general conditions on the functions
f1, f2 and f3, by which the equation may be integrable, as well as conditions for the
existence of Lie point symmetries. Time-dependent...

Pages: 338 - 349

In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable
under the perturbation given by the interaction of the oscillator with the quantum
field. A general mathematical structure underlying this...

Pages: 350 - 357

An inclined periodic soliton solution can be expressed as imbricate series of rational
soliton solutions. A convenient form of the imbrication is given by using the bilinear form. A lattice soliton solution which propagaties in any direction can be also
constructed by doubly imbricating rational solitons.

Pages: 358 - 363

A weakly nonlinear quasiconservative Duffing oscillator under quasiperiodic forcing is
studied with the help of an analytic expression for the complex Poincare mapping.
This mapping is then used to analyze the quasiperiodic response of the oscillator and
the different zones of various periodicity....

Pages: 364 - 376

The variational bicomplex of forms invariant under the symmetry algebra of the potential Kadomtsev-Petviashvili equation is described and the cohomology of the associated Euler-Lagrange complex is computed. The results are applied to a characterization problem of the Kadomtsev-Petviashvili equation by...

Pages: 377 - 382

The solution of the three-dimensional free Schrödinger equation due to W.M. Shtelen
based on the invariance of this equation under the Lorentz Lie algebra so(1,3) of nonlocal transformations is considered. Various properties of this solution are examined,
including its extension to n 3 spatial dimensions...

Pages: 383 - 391

Self-dual Yang-Mills fields with values in a Lie superalgebra on the four-dimensional
Euclidean space and pseudo-Euclidean space of signature (2,2) can be reduced by
subgroups of the corresponding conformal group to integrable systems with anticommuting degrees of freedom. Examples of reductions are...

Pages: 392 - 400

A general procedure for construction of conformally invariant Ansätze for the Maxwell
field is suggested. Ansätze invariant with respect to inequivalent three-parameter
subgroups of the conformal group are constructed.

Pages: 401 - 408

First, we determine the radial Schrödinger equation in D-dimensional curved spaces
when central problems are considered. Second, we develop the so-called factorization
method on the basis of supersymmetric arguments for solving such radial equations
when D = 1, 2, 3-harmonic oscillator and D = 3-hydrogen...

Pages: 409 - 417

The substantiation of the algorithm for classifying subalgebras of the Poincaré algebra
AP(1, n) up to P(1, n)-conjugacy is completed

Pages: 418 - 425

We show that Maxwell's equations have a generalization associated with the propertime of the source and a new invariance group which leaves this variable fixed for all
observers. We show that the second postulate (of Einstein) depends on the anthropocentric view that the only clock to use is the proper-clock...

Pages: 426 - 435

We have constructed new realizations of the Galilei group and its natural extensions
by Lie vector fields. These realizations together with the ones obtained by Fushchych
& Cherniha (Ukr. Math. J., 1989, 41, N 10, 1161; N 12, 1456) and Rideau &
Winternitz (J. Math. Phys., 1993, 34, 558) give a complete...

Pages: 436 - 444

New algebras of symmetries of the Dirac equation are presented, which are formed by
linear and antilinear firstorder differential operators. These symmetries are applied
to decouple the Dirac equation for a charged particle interacting with an external field.

Pages: 445 - 454

Subalgebras of the Lie algebra AC(2, 2) of the group C(2, 2), which is the group
of conformal transformations of the pseudo-Euclidean space R2,2, are studied. All
subalgebras of the algebra AC(2, 2) are splitted into three classes, each of those is
characterized by the isotropic rank 0, 1, or 3. We...

Pages: 455 - 469

The Moser type reductions of modified Boussinessq and super-Korteweg-de Vries equations upon the finite-dimensional invariant subspaces of solutions are considered. For
the Hamiltonian and Liouville integrable finite-dimensional dynamical systems concerned with the invariant subspaces, the Lax representations...

Pages: 470 - 479

We study symmetry properties of the heat equation with convection term (the equation
of convection diffusion) and the Schrödinger equation with convection term. We also
investigate the symmetry of systems of these equations with additional conditions for
potentials. The obtained results are applied...

Pages: 480 - 481

We show that the free Schrödinger equation admits Lorentz space-time transformations when corresponding transformations of the -function are nonlocal. Some consequences of this symmetry are discussed.

Pages: 482 - 491

Solutions invariant under subalgebras of the affine algebra AIGL(3, R) are found.

Pages: 492 - 503

Exact solutions of Heisenberg equations and two-particle eigenvalue problems for the
nonrelativistic four-fermion interaction and N, model are obtained in the framework
of a dynamical mapping method. Equivalence of different dynamical mappings is
shown.

Pages: 504 - 515

Stationary kinetic equations for a quark plasma (QP) in the Abelian dominance approximation are reduced to the nonlinear system of A2-periodic Toda chains (with
elliptic operator). Using solutions of this system, which are found with the help of
the first integrals and Hirota's method, such characteristics...

Pages: 516 - 524

Representations of the q-deformed Euclidean algebra Uq(iso3), which at q 1 gives the
universal enveloping algebra U(iso3) of the Lie algebra iso3 of the Euclidean Lie group
ISO(3), are studied. Explicit formulas for operators of irreducible -representations
defined by two parameters R and s 1
2...