Journal of Nonlinear Mathematical Physics

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905 articles

Action-Angle Analysis of Some Geometric Models of Internal Degrees of Freedom

Barbara GOLUBOWSKA
Pages: 138 - 144
We derive and discuss equations of motion of infinitesimal affinely-rigid body moving in Riemannian spaces. There is no concept of extended rigid and affinely rigid body in a general Riemannian space. Therefore the gyroscopes with affine degrees of freedom are described as moving bases attached to the...

A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

Agata BEZUBIK, Agata DBROWSKA, Aleksander STRASBURGER
Pages: 167 - 173
This article summarizes a new, direct approach to the determination of the expansion into spherical harmonics of the exponential ei(x|y) with x, y Rd . It is elementary in the sense that it is based on direct computations with the canonical decomposition of homogeneous polynomials into harmonic components...

Nonlinear Wave Equation in Special Coordinates

Alexander SHERMENEV
Pages: 110 - 115
Some classical types of nonlinear periodic wave motion are studied in special coodinates. In the case of cylinder coordinates, the usual perturbation techniques leads to the overdetermined systems of linear algebraic equations for unknown coefficients whose compatibility is key step of the investigation....

Canonical Analysis of Symmetry Enhancement with Gauged Grassmannian Model

Sang-Ok HAHN, Phillial OH, Cheonsoo PARK, Sunyoung SHIN
Pages: 185 - 190
We study the Hamiltonian structure of the gauge symmetry breaking and enhancment. After giving a general discussion of these phenomena in terms of the constrained phase space, we perform a canonical analysis of the Grassmannian nonlinear sigma model coupled with Chern-Simons term, which contains a free...

Poisson configuration spaces, von Neumann algebras, and harmonic forms

Alexei DALETSKII
Pages: 179 - 184
We give a short review of recent results on L2 -cohomology of infinite configuration spaces equipped with Poisson measures.

The Jungle Book updated

Bogdan MIELNIK
Pages: 228 - 236
Data from many sources indicate that the Earth ecological crisis might not wait till distant future. To avert it, some difficult truth must be accepted and adequate steps taken. One of them is the strict protection of the world forests, even at the cost of the short term economic growth.

Green function for Klein-Gordon-Dirac equation

Vasyl KOVALCHUK
Pages: 72 - 77
The Green function for Klein-Gordon-Dirac equation is obtained. The case with the dominating Klein-Gordon term is considered. There seems to be a formal analogy between our problem and a certain problem for a 4-dimensional particle moving in the external field. The explicit relations between the wave...

Quantization of the planar affinely-rigid body

Agnieszka MARTENS
Pages: 145 - 150
This paper is a continuation of [1] where the classical model was analyzed. Discussed are some quantization problems of two-dimensional affinely rigid body with the double dynamical isotropy. Considered are highly symmetric models for which the variables can be separated. Some explicit solutions are...

Limit Spectra of Random Gram Matrices

V.I. SERDOBOLSKII
Pages: 116 - 121
Solutions to basic non-linear limit spectral equation for matrices RT R of increasing dmension are investigated, where R are rectangular random matrices with independent normal entries. The analytical properties of limiting normed trace for the resolvent of RT R are investigated, boundaries of limit...

EPR-B correlations: non-locality or geometry?

A F KRACKLAUER
Pages: 104 - 109
A photoelectron-by-photoelectron classical simulation of EPR-B correlations is dscribed. It is shown that this model can be made compatible with Bell's renowned "no-go" theorem by restricting the source to that which produces only what is known as paired photons.

Scattering and Spectral Singularities for some Dissipative Operators of Mathematical Physics

S. A. STEPIN
Pages: 194 - 203
Analogies in the spectral study of dissipative Schrödinger operator and Boltmann transport operator are analyzed. Scattering theory technique together with functional model approach are applied to construct spectral representtions for these operators.

Geometry of differential operators, odd Laplacians, and homotopy algebras

Hovhannes KHUDAVERDIAN, Theodore VORONOV
Pages: 217 - 227
We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction.

Reducible representations of CAR and CCR with possible applications to field quantization

Marek CZACHOR
Pages: 78 - 84
Reducible representations of CAR and CCR are applied to second quantization of Dirac and Maxwell fields. The resulting field operators are indeed operators and not operator-valued distributions. Examples show that the formalism may lead to a finite quantum field theory.

von Neumann Quantization of Aharonov-Bohm Operator with Interaction: Scattering Theory, Spectral and Resonance Properties

Gilbert HONNOUVO, Mahouton Norbert HOUNKONNOU, Gabriel Yves Hugues
Pages: 66 - 71
Using the theory of self-adjoint extensions, we study the interaction model formally given by the Hamiltonian H + V (r), where H is the Aharonov-Bohm Hamiltonian and V (r) is the -type interaction potential on the cylinder of radius R . We give the mathematical definition of the model, the self-adjointness...

A Note on Fermionic Flows of the N=(1|1) Supersymmetric Toda Lattice Hierarchy

O LECHTENFELD, A S SORIN
Pages: 294 - 296
We extend the Sato equations of the N=(1|1) supersymmetric Toda lattice hierachy by two new infinite series of fermionic flows and demonstrate that the algebra of the flows of the extended hierarchy is the Borel subalgebra of the N=(2|2) loop superalgebra.

Replicator - Mutator Evolutionary Dynamics

Vasyl V GAFIYCHUK, Anatoliy K PRYKARPATSKY
Pages: 350 - 360
We consider the general properties of the quasispecies dynamical system from the standpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system have been studied. Mathematical modeling of the system under consideration has been performed.

A Family of Linearizations of Autonomous Ordinary Differential Equations with Scalar Nonlinearity

Fethi BELKHOUCHE
Pages: 276 - 288
This paper deals with a method for the linearization of nonlinear autonomous diferential equations with a scalar nonlinearity. The method consists of a family of approximations which are time independent, but depend on the initial state. The family of linearizations can be used to approximate the derivative...

The Kac Construction of the Centre of U(g) for Lie Superalgebras

Maria GORELIK
Pages: 325 - 349
In 1984, Victor Kac [8] suggested an approach to a description of central elements of a completion of U(g) for any Kac-Moody Lie algebra g. The method is based on a recursive procedure. Each step is reduced to a system of linear equations over a certain subalgebra of meromorphic functions on the Cartan...

Sundman Symmetries of Nonlinear Second-Order and Third-Order Ordinary Differential Equations

Norbert EULER, Marianna EULER
Pages: 399 - 421
We investigate the Sundman symmetries of second-order and third-order nonlinear odinary differential equations. These symmetries, which are in general nonlocal tranformations, arise from generalised Sundman transformations of autonomous equations. We show that these transformations and symmetries can...

Normal Forms for Coupled Takens-Bogdanov Systems

David Mumo MALONZA
Pages: 376 - 398
The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module. In this paper Groebner basis methods are used to determine a Groebner basis...

Superalgebras for the 3D Harmonic Oscillator and Morse Quantum Potentials

H FAKHRI
Pages: 361 - 375
In addition to obtaining supersymmetric structure related to the partner Hamiltonans, we get another supersymmetric structure via factorization method for both the 3D harmonic oscillator and Morse quantum potentials. These two supersymmetries induce also an additional supersymmetric structure involving...

A Holomorphic Point of View about Geodesic Completeness

Claudio MENEGHINI
Pages: 297 - 324
We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with their complex counterparts, and of Clifton-Pohl torus, to show that...

The Influence of Quantum Field Fluctuations on Chaotic Dynamics of Yang-Mills System II. The Role of the Centrifugal Term

V I KUVSHINOV, A V KUZMIN, V A PIATROU
Pages: 289 - 293
We have considered SU(2) U(1) gauge field theory describing electroweak interations. We have demonstrated that centrifugal term in model Hamiltonian increases the region of regular dynamics of Yang-Mills and Higgs fields system at low densities of energy. Also we have found analytically the approximate...

A Variational Approach to the Stability of Periodic Peakons

Jonatan LENELLS
Pages: 151 - 163
The peakons are peaked traveling wave solutions of an integrable shallow water eqution. We present a variational proof of their stability.

Applications of Nambu Mechanics to Systems of Hydrodynamical Type II

Partha GUHA
Pages: 223 - 232
In this paper we further investigate some applications of Nambu mechanics in hydrdynamical systems. Using the Euler equations for a rotating rigid body Névir and Blender [J. Phys. A 26 (1993), L1189­L1193] had demonstrated the connection btween Nambu mechanics and noncanonical Hamiltonian mechanics....

Self-Invariant Contact Symmetries

Peter E HYDON
Pages: 233 - 242
Every smooth second-order scalar ordinary differential equation (ODE) that is solved for the highest derivative has an infinite-dimensional Lie group of contact symmetries. However, symmetries other than point symmetries are generally difficult to find and use. This paper deals with a class of one-parameter...

The Krichever Map and Automorphic Line Bundles

Min Ho LEE
Pages: 199 - 207
A solution of the KP-hierarchy can be given by the -function or the Baker function associated to an element of the Grassmannian Gr(L2 (S1 )) consisting of some subspaces of the space L2 (S1 ) of square-integrable functions on the unit circle S1 . The Krichever map associates an element W Gr(L2 (S1 ))...

A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions

P G ESTÉVEZ, J PRADA
Pages: 164 - 179
The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation...

Geometrical Formulation of the Conformal Ward Identity

Mohamed KACHKACHI
Pages: 141 - 150
In this paper we use deep ideas in complex geometry that proved to be very powerful in unveiling the Polyakov measure on the moduli space of Riemann surfaces and lead to obtain the partition function of perturbative string theory for 2, 3, 4 loops. Indeed a geometrical interpretation of the conformal...

Stable Equilibria to Parabolic Systems in Unbounded Domains

Joachim ESCHER, Zhaoyang YIN
Pages: 243 - 255
We investigate weakly coupled semilinear parabolic systems in unbounded domains in R2 or R3 with polynomial nonlinearities. Three sufficient conditions are presented to ensure the stability of the zero solution with respect to non-negative H2 -perturbations.

Integral Equation Approach for the Propagation of TE-Waves in a Nonlinear Dielectric Cylindrical Waveguide

Y SMIRNOV, H W SCHÜRMANN, Y SHESTOPALOV
Pages: 256 - 268
We consider the propagation of TE-polarized electromagnetic waves in cylindrical dielectric waveguides of circular cross section filled with lossless, nonmagnetic, and isotropic medium exhibiting a local Kerr-type dielectric nonlinearity. We look for axially-symmetric solutions and reduce the problem...

Two New Classes of Isochronous Hamiltonian Systems

Francesco CALOGERO
Pages: 208 - 222
An isochronous dynamical system is characterized by the existence of an open domain of initial data such that all motions evolving from it are completely periodic with a fixed period (independent of the initial data). Taking advantage of a recently introduced trick, two new Hamiltonian classes of such...

Soliton Collisions and Ghost Particle Radiation

Hiêú D NGUYÊÑ
Pages: 180 - 198
This paper investigates the nature of particle collisions for three-soliton solutions of the Korteweg-de Vries (KdV) equation by describing mathematically the interaction of soliton particles and generation of ghost particle radiation. In particular, it is proven that a collision between any two soliton...

The Lie Algebra sl(2, R) and so-called Kepler-Ermakov Systems

P G L LEACH, Ayse Karasu Kalkanli
Pages: 269 - 275
A recent paper by Karasu (Kalkanli) and Yildirim (Journal of Nonlinear Mathematical Physics 9 (2002) 475-482) presented a study of the Kepler-Ermakov system in the context of determining the form of an arbitrary function in the system which was compatible with the presence of the sl(2, R) algebra characteristic...

On Conditionally Invariant Solutions of Magnetohydrodynamic Equations. Multiple Waves.

A M GRUNDLAND, P PICARD
Pages: 47 - 74
We present a version of the conditional symmetry method in order to obtain multiple wave solutions expressed in terms of Riemann invariants. We construct an abelian distribution of vector fields which are symmetries of the original system of PDEs subjected to certain first order differential constraints....

Competing Species: Integrability and Stability

PGL LEACH, J MIRITZIS
Pages: 123 - 133
We examine the classical model of two competing species for integrability in terms of analytic functions by means of the Painlevé analysis. We find that the governing equations are integrable for certain values of the essential parameters of the system. We find that, for all integrable cases with the...

On the Quantization of Yet Another Two Nonlinear Harmonic Oscillators

Francesco CALOGERO
Pages: 1 - 6
In two previous papers the quantization was discussed of three one-degree-of-freedom Hamiltonians featuring a constant c, the value of which does not influence at all the corresponding classical dynamics (which is characterized by isochronous solutions, all of them periodic with period 2: "nonlinear...

Solution of the Goldfish N-Body Problem in the Plane with (Only) Nearest-Neighbor Coupling Constants All Equal to Minus One Half

Francesco CALOGERO
Pages: 102 - 112
The (Hamiltonian, rotation- and translation-invariant) "goldfish" N-body problem in the plane is characterized by the Newtonian equations of motion ¨zn - i zn = 2 N m=1,m=n an,m zn zm (zn - zm) -1 , written here in their complex version, entailing the identification of the real "physical" plane with...

Replicator Dynamics and Mathematical Description of Multi-Agent Interaction in Complex Systems

Vasyl V GAFIYCHUK, Anatoliy K PRYKARPATSKY
Pages: 113 - 122
We consider the general properties of the replicator dynamical system from the stanpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. A Lyaponuv function for the investigation of the evolution of the system has been proposed. The...

The Heun Equation and the Calogero-Moser-Sutherland System III: The Finite-Gap Property and the Monodromy

Kouichi TAKEMURA
Pages: 21 - 46
A new approach to the finite-gap property for the Heun equation is constructed. The relationship between the finite-dimensional invariant space and the spectral curve is clarified. The monodromies are calculated and are expressed as hyperelliptic integrals. Applications to the spectral problem for the...

Reduction of Order for Systems of Ordinary Differential Equations

C WAFO SOH, F M MAHOMED
Pages: 13 - 20
The classical reduction of order for scalar ordinary differential equations (ODEs) fails for a system of ODEs. We prove a constructive result for the reduction of order for a system of ODEs that admits a solvable Lie algebra of point symmetries. Applications are given for the case of a system of two...

Integrability Conditions for n and t Dependent Dynamical Lattice Equations

R YAMILOV, D LEVI
Pages: 75 - 101
Conditions necessary for the existence of local higher order generalized symmetries and conservation laws are derived for a class of dynamical lattice equations with explicit dependence on the spatial discrete variable and on time. We explain how to use the obtained conditions for checking a given equation....

µ-Holomorphic Projective Connections and Conformal Covariance

Mohamed KACHKACHI
Pages: 7 - 12
At the quantum level of a bidimensional conformal model, the conformal symmtry is broken by the diffeomorphism anomaly and the conformal covariance is not maintained. Here we interpret geometrically this conformal covariance as an exact holomorphy condition on a two-dimensional Riemann surface on which...

A Two-Phase Free Boundary Problem for the Nonlinear Heat Equation

S DE LILLO, M C SALVATORI
Pages: 134 - 140
A two-phase free boundary problem associated with nonlinear heat conduction is cosidered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution...

A New Discrete Hénon-Heiles System

Alan K COMMON, Andrew N W HONE, Micheline MUSETTE
Pages: 27 - 40
By considering the Darboux transformation for the third order Lax operator of the Sawada-Kotera hierarchy, we obtain a discrete third order linear equation as well as a discrete analogue of the Gambier 5 equation. As an application of this result, we consider the stationary reduction of the fifth order...

Lax Matrices for Yang-Baxter Maps

Yuri B SURIS, Alexander P VESELOV
Pages: 107 - 118
It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quagraphs has been recently discovered by A. Bobenko and...

On the Transformations of the Sixth Painlevé Equation

Valery I GROMAK, Galina FILIPUK
Pages: 107 - 118
In this paper we investigate relations between different transformations of the slutions of the sixth Painlevé equation. We obtain nonlinear superposition formulas linking solutions by means of the Bäcklund transformation. Algebraic solutions are also studied with the help of the Bäcklund transformation.

The Discrete Nonlinear Schrödinger Equation and its Lie Symmetry Reductions

R HERNÁNDEZ HEREDERO, D LEVI
Pages: 77 - 94
The Lie algebra L(h) of symmetries of a discrete analogue of the non-linear Schrödinger equation (NLS) is studied. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symmtry algebra L(0) of the NLS equation. We use the lowest...

On a Certain Fractional q-Difference and its Eigen Function

Atsushi NAGAI
Pages: 133 - 142
A fractional q-difference operator is presented and its properties are investigated. Epecially, it is shown that this operator possesses an eigen function, which is regarded as a q-discrete analogue of the Mittag-Leffler function. An integrable nonlinear mapping with fractional q-difference is also presented.