Journal of Nonlinear Mathematical Physics

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902 articles
Research Article

The Rayleigh Problem for a Third Grade Electrically Conducting Fluid in a Magnetic Field

Tasawar Hayat, Herman Mambili-Mamboundou, Ebrahim Momoniat, Fazal M. Mahomed
Pages: 77 - 90
The influence of a magnetic field on the flow of an incompressible third grade electrically conducting fluid bounded by a rigid plate is investigated. The flow is induced due by the motion of a plate in its own plane with an arbitrary velocity. The solution of the equations of conservation of mass and...
Research Article

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.
Research Article

The Symmetry Reduction of Nonlinear Equations of the Type □u + F(u, u1)u0 = 0 to Ordinary Differential Equations

Leonid BARANNYK, Halyna LAHNO
Pages: 78 - 88
The reduction of two nonlinear equations of the type □u+F(u, u1)u0 = 0 with respect to all rank three subalgebras of a subdirect sum of the extended Euclidean algebras A ~E(1) and A ~E(3) is carried out. Some new invariant exact solutions of these equations are obtained.
Research Article

A Remark on Nonlocal Symmetries for the Calogero­Degasperis­Ibragimov­Shabat Equation

Artur SERGYEYEV, Jan A SANDERS
Pages: 78 - 85
We consider the Calogero­Degasperis­Ibragimov­Shabat (CDIS) equation and find the complete set of its nonlocal symmetries depending on the local variables and on the integral of the only local conserved density of the equation in question. The Lie algebra of these symmetries turns out to be a central...
Research Article

Amplitude-Dependent Oscillations in Gases

O L de LANGE, J PIERRUS
Pages: 79 - 81
We consider the following question: Suppose part of the boundary of a cavity contaiing a gas is set into oscillation, the damping in the boundary being small. What is the nature of the oscillations in the gas? We treat the low-frequency limit (wavelength much greater than dimensions of the cavity). Experiment...
Research Article

Symmetry Analysis and Solutions for a Generalization of a Family of BBM Equations

M.S. Bruzon, M. L. Gandarias, J. C. Camacho
Pages: 81 - 90
In this paper, the family of BBM equation with strong nonlinear dispersive B(m,n) is considered. We apply the classical Lie method of infinitesimals. The symmetry reductions are derived from the optimal system of subalgebras and lead to systems of ordinary differential equations. We obtain for special...
Research Article

The Generalized Version of Dressing Method with Applications to AKNS Equations

Junyi ZHU, Xianguo GENG
Pages: 81 - 89
The generalized dressing method is extended to variable-coefficient AKNS equations, including a variable-coefficient coupled nonlinear Schr¨odinger equation and a variablcoefficient coupled mKdV equation. A general variable-coefficient KP equation is proposed and decomposed into the two 1+1 dimensional...
Research Article

Inhomogeneous Burgers Lattices

S DE LILLO, V V KONOTOP
Pages: 82 - 87
We study statistical properties of inhomogeneous Burgers lattices which are solved by the discrete Cole­Hopf transformation. Using exact solutions we investigate effect of various kinds of noise on the dynamics of solutions.
Research Article

A note on the relationship between rational and trigonometric solutions of the WDVV equations

Andrew RILEY, Ian A B STRACHAN
Pages: 82 - 94
Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds....
Review Article

Unified approach to Miura, Bäcklund and Darboux Transformations for Nonlinear Partial Differential Equations

P.G. ESTÉVEZ, E. CONDE, P.R. GORDOA
Pages: 82 - 114
This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlevé Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura,...
Research Article

Statistical mechanics of a Class of Anyonic Systems. The Rigorous Approach

Roman GIELERAK, Robert RALOWSKI
Pages: 85 - 91
A class of involutive Wick algebras (called anyonic-type Wick algebras) is selected and some its elementary properties are described. In particular, the Fock representtions of the selected anyonic-type commutation relations are described. For the class of so-called r-yonic systems the question of the...
Research Article

Symmetry Analysis of Nonlinear PDE with A "Mathematica" Program SYMMAN

Evgenii M. VOROB'EV
Pages: 85 - 89
Computer-aided symbolic and graphic computation allows to make significantly easier both theoretical and applied symmetry analysis of PDE. This idea is illustrated by applying a special "Mathematica" package for obtaining conditional symmetries of the nonlinear wave equation ut = (u ux)x invariant or...
Research Article

Distinguishing Three-Dimensional Lens Spaces L(7, 1) and L(7, 2) by Means of Classical Pentagon Equation

I G KOREPANOV, E V MARTYUSHEV
Pages: 86 - 98
We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7, 1) and L(7, 2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e. algebraic relation corresponding to a...
Research Article

Lower Bounds for the Spinless Salpeter Equation

Fabian BRAU
Pages: 86 - 96
We obtain lower bounds on the ground state energy, in one and three dimensions, for the spinless Salpeter equation (Schrödinger equation with a relativistic kinetic energy operator) applicable to potentials for which the attractive parts are in Lp (Rn ) for some p > n (n = 1 or 3). An extension to confining...
Research Article

On a q-Difference Painlevé III Equation: I. Derivation, Symmetry and Riccati Type Solutions

Kenji KAJIWARA, Kinji KIMURA
Pages: 86 - 102
A q-difference analogue of the Painlevé III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.
Research Article

On Gerstner's Water Wave

David Henry
Pages: 87 - 95
We present a simple approach showing that Gerstner’s flow is dynamically possible: each particle moves on a circle, but the particles never collide and fill out the entire region below the surface wave.
Research Article

Note on operadic non-associative deformations

Eugen PAAL
Pages: 87 - 92
Deformation equation of a non-associative deformation in operad is proposed. Its integrability condition (the Bianchi identity) is considered. Algebraic meaning of the latter is explained.
Research Article

Exact Solutions of DNLS and Derivative Reaction-Diffuson Systems

Jyh-Hao LEE, Yen-Ching LEE, Chien-Chih LIN
Pages: 87 - 98
In this paper, we obtain some exact solutions of Derivative Reaction-Diffusion (DRD) system and, as by-products, we also show some exact solutions of DNLS via Hirota bilinearization method. At first, we review some results about two by two AKNS-ZS system, then introduce Hirota bilinearization method...
Research Article

Asymptotic Lattices and W-Congruences in Integrable Discrete Geometry

Adam DOLIWA
Pages: 88 - 92
The asymptotic lattices and their transformations are included into the theory of quadrilateral lattices.
Research Article

Solitons and Deformed Lattices I

Betti HARTMANN, Wojtek J ZAKRZEWSKI
Pages: 88 - 104
We study a model describing some aspects of the dynamics of biopolymers. The models involve either one or two finite chains with a number N of sites that represent the "units" of a biophysical system. The mechanical degrees of freedom of these chains are coupled to the internal degrees of freedom through...
Research Article

Symmetries of Vector Exterior Differential Systems and the Inverse Problem in Second-Order Ostrograds'kii Mechanics

R.Ya. MATSYUK
Pages: 89 - 97
Symmetries for variational problems are considered as symmetries of vector-valued exterior differential systems. This approach is applied to equations for the classical spinning particle.
Research Article

Parasupersymmetries and Non-Lie Constants of Motion for Two-Particle Equations

Violeta TRETYNYK
Pages: 90 - 95
We search for hidden symmetries of two-particle equations with oscillator-equivalent potential proposed by Moshinsky with collaborators. We proved that these equations admit hidden symmetries and parasupersymmetries which enable easily to find the Hamiltonian spectra using algebraic methods.
Short Communication

Non­Lie Ansatzes for Nonlinear Heat Equations

Ivan TSYFRA
Pages: 90 - 93
Operators of non­local symmetry are used to construct exact solutions of nonlinear heat equations. A method for finding of new classes of ansatzes reducing nonlinear wave equations to a systems of ordinary differential equations was suggested in [1]. This approach is based on non­local symmetry of differential...
Research Article

Symbolic Software for the Painlevé Test of Nonlinear Ordinary and Partial Differential Equations

Douglas BALDWIN, Willy HEREMAN
Pages: 90 - 110
The automation of the traditional Painlev´e test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of nonlinear ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants). Except where limited by memory,...
Research Article

Peristaltic MHD Flow of Third Grade Fluid with an Endoscope and Variable Viscosity

T Hayat, Ebrahim Momoniat, Fazal M Mahomed
Pages: 91 - 104
This investigation deals with the mechanism of peristaltic transport of a non-Newtonian, incompressible and electrically conducting fluid with variable viscosity and an endoscope effects. The magnetic Reynolds number is taken to be small. The mechanical properties of the material are represented by the...
Research Article

Coupling Nonlinear Sigma-Matter to Yang-Mills Fields: Symmetry Breaking Patterns

M. Calixto, V. Aldaya, F. F. Lopez-Ruiz, E. Sanchez-Sastre
Pages: 91 - 101
We extend the traditional formulation of Gauge Field Theory by incorporating the (non- Abelian) gauge group parameters (traditionally simple spectators) as new dynamical (nonlinearsigma- model-type) fields. These new fields interact with the usual Yang-Mills fields through a generalized minimal coupling...
Research Article

Group Invariant Solution and Conservation Law for a Free Laminar Two-Dimensional Jet

D P MASON
Pages: 92 - 101
A group invariant solution for a steady two-dimensional jet is derived by considering a linear combination of the Lie point symmetries of Prandtl's boundary layer equations for the jet. Only two Lie point symmetries contribute to the solution and the ratio of the constants in the linear combination is...
Research Article

Quantization of Soliton Cellular Automata

Demosthenes ELLINAS, Elena P. PAPADOPOULOU, Yiannis G. SARIDAKIS
Pages: 93 - 99
A method of quantization of classical soliton cellular automata (QSCA) is put forward that provides a description of their time evolution operator by means of quantum cicuits that involve quantum gates from which the associated Hamiltonian describing a quantum chain model is constructed. The intrinsic...
Research Article

Essential Spectrum Due to Singularity

Pavel KURASOV, Serguei NABOKO
Pages: 93 - 106
It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential expression
Research Article

Groups of Order Less Than 32 and Their Endomorphism Semigroups

Peeter PUUSEMP
Pages: 93 - 101
It is proved that among the finite groups of order less than 32 only the tetrahedral group and the binary tetrahedral group are not determined by their endomorphism semigroups in the class of all groups.
Review Article

Particles and Strings in a 2 + 1-D Integrable Quantum Model

I.G. KOREPANOV
Pages: 94 - 119
We give a review of some recent work on generalization of the Bethe ansatz in the case of 2 + 1-dimensional models of quantum field theory. As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization of the famous Yang­Baxter equation. We construct...
Research Article

A geometric interpretation of the complex tensor Riccati equation for Gaussian beams

M F DAHL
Pages: 95 - 111
We study the complex Riccati tensor equation DcG + GCG - R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space....
Research Article

Burchnall-Chaundy Theory for q-Difference Operators and q-Deformed Heisenberg Algebras

Daniel LARSSON, Sergei D SILVESTROV
Pages: 95 - 106
This paper is devoted to an extension of Burchnall-Chaundy theory on the inteplay between algebraic geometry and commuting differential operators to the case of q-difference operators.
Research Article

Bispectrality for deformed Calogero­Moser­Sutherland systems

Misha FEIGIN
Pages: 95 - 136
We prove bispectral duality for the generalized Calogero­Moser­Sutherland systems related to configurations An,2(m), Cn(l, m). The trigonometric axiomatics of the Baker­Akhiezer function is modified, the dual difference operators of rational Madonald type and the Baker­Akhiezer functions related to both...
Research Article

Lie Algebras of Approximate Symmetries

Rafail K. GAZIZOV
Pages: 96 - 101
Properties of approximate symmetries of equations with a small parameter are discussed. It turns out that approximate symmetries form an approximate Lie algebra. A concept of approximate invariants is introduced and the algorithm of their calculating is proposed.
Research Article

Global Dissipative Solutions of the Generalized Camassa-Holm Equation

Octavian G Mustafa
Pages: 96 - 115
A new approach to the analysis of wave-breaking solutions to the generalized Camassa-Holm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of dissipative solutions is also built. The solutions have non-increasing H1 energy...
Research Article

Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

J CHAVARRIGA, I A GARCIA
Pages: 96 - 105
The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G2 cosmologies. By using Darboux's theory in order to study ordinary differential equations...
Research Article

New Solvable Nonlinear Matrix Evolution Equations

M BRUSCHI
Pages: 97 - 105
We introduce an extension of the factorization-decomposition technique that allows us to manufacture new solvable nonlinear matrix evolution equations. Several examples of such equations are reported.
Research Article

Conditional Symmetry and Exact Solutions of a Nonlinear Galilei-Invariant Spinor Equation

Andrey ANDREYTSEV
Pages: 98 - 101
Reduction of a nonlinear system of differential equations for spinor field is studied. The ansatzes obtained are shown to correspond to operators of conditional symmetry of these equations.
Research Article

Hamiltonian Structure and Linear Stability of Solitary Waves of the Green-Naghdi Equations

Yi A LI
Pages: 99 - 105
We investigate linear stability of solitary waves of a Hamiltonian system. Unlike weakly nonlinear water wave models, the physical system considered here is nonlinearly dispersive, and contains nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue...
Research Article

Periodic Motions Galore: How to Modify Nonlinear Evolution Equations so that They Feature a Lot of Periodic Solutions

F CALOGERO, J-P FRANCOISE
Pages: 99 - 125
A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot ­ or, in some cases, only ­ periodic solutions. Several examples (ODEs and PDEs) are exhibited.
Research Article

Dynamical Correlation Functions for an Impenetrable Bose Gas with Neumann or Dirichlet Boundary Conditions

Takeo KOJIMA
Pages: 99 - 119
We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions (x1, 0) (x2, t) ±,T . We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case x1 = 0, we express...
Research Article

Brownian Motion on a Smash Line

Demosthenes ELLINAS, Ioannis TSOHANTJIS
Pages: 100 - 105
Brownian motion on a smash line algebra (a smash or braided version of the algebra resulting by tensoring the real line and the generalized paragrassmann line algebras), is constructed by means of its Hopf algebraic structure. Further, statistical moments, non stationary generalizations and its diffusion...
Research Article

Symbolic Dynamics and Chaotic Synchronization in Coupled Duffing Oscillators

Acilina Caneco, Clara Gracio, J. Leonel Rocha
Pages: 102 - 111
In this work we discuss the complete synchronization of two identical double-well Duffing oscillators unidirectionally coupled, from the point of view of symbolic dynamics. Working with Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the...
Research Article

Total Differentiation Under Jet Composition

Maido RAHULA, Vitali RETSNOI
Pages: 102 - 109
Total differentiation operators as linear vector fields, their flows, invariants and symmetries form the geometry of jet space. In the jet space the dragging of tensor fields obeys the exponential law. The composition of smooth maps induces a composition of jets in corresponding jet spaces. The prolonged...
Research Article

Graph Expansions and Graphical Enumeration Applied to Semiclassical Propagator Expansions

S.A. FULLING
Pages: 102 - 110
In recent years T.A. Osborn and his coworkers at the University of Manitoba have extensively developed the well known connected graph expansion and applied it to a wide variety of problems in semiclassical approximation to quantum dynamics [2, 5, 7, 19, 21, 22, 26, 27]. The work I am reporting on attempts...
Research Article

Symmetry Reduction of Poincaré-Invariant Nonlinear Wave Equations

A.F. BARANNYK, Yu.D. MOSKALENKO
Pages: 102 - 106
Reduction of multidimensional Poincaré-invariant equations to ordinary differential equations and 2-dimensional equations is considered.