Volume 4, Issue 3-4, September 1995

M. SENTHIL VELAN, M. LAKSHMANAN

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...

R. JACKIW

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 indicated....

Peter OLVER

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.

F. FINKEL, N. KAMRAN

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...

T.D. PALEV

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.

Wen-Xiu MA, R.K. BULLOUGH, P.J. CAUDREY

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...

Norbert EULER

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...

Asao ARAI

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...

Masayoshi TAJIRI, Yosuke WATANABE

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.

K. CHOWDHURY, A. ROY CHOWDHURY

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. This...

Juha POHJANPELTO

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...

W.W. ZACHARY

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...

M. LEGARÉ

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...

Victor LAHNO

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.

J. BECKERS, N. DEBERGH

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...

Leonid F. BARANNYK

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

Tepper L. GILL, W.W. ZACHARY

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...

R.Z. ZHDANOV, W.I. FUSHCHYCH

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...

Jiri NIEDERLE, Anatolii NIKITIN

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.

A.F. BARANNYK

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 present...

A.K. PRYKARPATSKY, O.E. HENTOSH, D.L. BLACKMORE

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...

W.I. FUSHCHYCH, Z.I. SYMENOH

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 to...

Vladimir SHTELEN

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.

Lyudmyla L. BARANNYK

Pages: 482 - 491

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

A.N. VALL, S.E. KORENBLIT, V.M. LEVIANT, A.B. TANAEV.

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.

Yu.A. MARKOV, M.A. MARKOVA

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...

I.I. KACHURIK

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 Z are...