Journal of Nonlinear Mathematical Physics

Volume 4, Issue 3-4, September 1995

1. Invariance Analysis of the (2+1) Dimensional Long Dispersive Wave Equation

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...

2. A Nonrelativistic Chiral Soliton in One Dimension

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...

3. Pseudo-Stabilization of Prolonged Group Actions I. The Order Zero Case

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.

4. On the Equivalence of Matrix Differential Operators to Schrödinger Form

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...

5. An Analogue of Holstein-Primakoff and Dyson Realizations for Lie Superalgebras. The Lie Superalgebra sl(1/n)

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.

6. Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP Equations

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...

7. Transformation Properties of x'' + f_1(t)x' + f2(t)x + f3(t)x^n = 0

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...

8. A Class of Representations of the *-Algebra of the Canonical Commutation Relations over a Hilbert Space and Instability of Embedded Eigenvalues in Quantum Field Models

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...

9. Periodic Soliton Solutions as Imbricate Series of Rational Solitons: Solutions to the Kadomtsev-Petviashvili Equation with Positive Dispersion

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.

10. An Analytic Approach to Torus Bifurcation in a Quasiperiodically Forced Duffing Oscillator

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....

11. The Cohomology of the Variational Bicomplex Invariant under the Symmetry Algebra of the Potential Kadomtsev-Petviashvili Equation

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...

12. On Shtelen's Solution of the Free Linear Schrödinger Equation

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...

13. Lie Superalgebra Valued Self-Dual Yang-Mills Fields and Symmetry Reduction

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...

14. Conformally Invariant Ansätze for the Maxwell Field

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.

15. On Radial Schrödinger Equations in Curved Spaces and Their Spectra Through Nonlinear Constraints

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...

16. On Classification of Subalgebras of the Poincaré Algebra

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

17. Proper-Time Formulation of Classical Electrodynamics

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...

18. On New Representations of Galilei Groups

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...

19. Non-Lie and Discrete Symmetries of the Dirac Equation

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.

20. On Subalgebras of the Conformal Algebra AC(2,2)

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...

21. The Finite-Dimensional Moser Type Reduction of Modified Boussinesq and Super-Korteweg-de Vries Hamiltonian Systems via the Gradient-Holonomic Algorithm and Dual Moment Maps. Part I

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...

22. Symmetry of Equations with Convection Terms

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...

23. Lorentz Transformations for the Schrödinger Equation

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.

24. Invariant Solutions of a Nonlinear System of Differential Equations for Electromagnetic Field

Lyudmyla L. BARANNYK
Pages: 482 - 491
Solutions invariant under subalgebras of the affine algebra AIGL(3, R) are found.

25. A Dynamical Mapping Method in Nonrelativistic Models of Quantum Field Theory

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.

26. Exact Solutions of a Quark Plasma Equilibrium in the Abelian Dominance Approximation

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...

27. Representations of the Q-deformed Euclidean Algebra Uq(iso3) and Spectra of their Operators

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...