Journal of Nonlinear Mathematical Physics

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905 articles

A note on q-Bernoulli numbers and polynomials

Taekyun KIM, A S HEGAZI, M MANSOUR
Pages: 315 - 322
Recently, B. A. Kupershmidt have constructed a reflection symmetries of q-Bernoulli polynomials (see [9]). In this paper we give another construction of a q-Bernoulli polynomials, which form Barnes' multiple Bernoulli polynomials at q = 1, cf. [1, 13,14]. By using q-Volkenborn integration, we can also...

Chevalley's theorem for the complex crystallographic groups

Joseph BERNSTEIN, Ossip SCHWARZMAN
Pages: 323 - 351
We prove that, for the irreducible complex crystallographic Coxeter group W, the following conditions are equivalent: a) W is generated by reflections; b) the analytic variety X/W is isomorphic to a weighted projective space. The result is of interest, for example, in application to topological conformal...

Bihamiltonian Equations on Polynomial Virasoro

Paolo CASATI, Giovanni ORTENZI
Pages: 352 - 364
We present and study bihamiltonian equations of Euler type which include a n

Feynman-Jackson integrals

Rafael DIAZ, Eddy PARIGUAN
Pages: 365 - 376
We introduce perturbative Feynman integrals in the context of q-calculus generalizing the Gaussian q-integrals introduced by Diaz and Teruel. We provide analytic as well as combinatorial interpretations for the Feynman-Jackson integrals.

Nonlocal Symmetries and the Complete Symmetry Group of 1 + 1 Evolution Equations

S.M. MYENI, P.G.L. LEACH
Pages: 377 - 392
The complete symmetry group of a 1 + 1 linear evolution equation has been demon- strated to be represented by the six-dimensional Lie algebra of point symmetries sl(2, R)?s W , where W is the three-dimensional Heisenberg-Weyl algebra. The infinite number of solution symmetries does not play a role in...

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

A.M. GRUNDLAND, B. HUARD
Pages: 393 - 419
In this paper we employ a “direct method” to construct rank-k solutions, express- ible in Riemann invariants, to hyperbolic system of first order quasilinear differential equations in many dimensions. The most important feature of our approach is the analysis of group invariance properties of these solutions...

Vectorial Regularization and Temporal Means in Keplerian Motion

Daniel CONDURACHE, Vladimir MARTINUSI
Pages: 420 - 440
We study the well-known Kepler’s problem by introducing a new vectorial regularization. This helps deduce Kepler’s equations by a simple and unified method. Some integral temporal means are also obtained by means of this regularization. The vectorial eccentricity plays a fundamental part in this approach.

Wave Breaking in a Class of Nonlocal Dispersive Wave Equations

Hailiang LIU
Pages: 441 - 466
The Korteweg de Vries (KdV) equation is well known as an approximation model for small amplitude and long waves in different physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. In this work we consider a class of nonlocal dispersive wave equations which also...

On a graded q-differential algebra

Viktor ABRAMOV
Pages: 1 - 8
Given an associative unital ZN -graded algebra over the complex numbers we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-differential d of the graded q-differential algebra is a homogeneous endomorphism of degree 1 satisfying...

Geometric approach to BRST-symmetry and ZN-generalization of superconnection

V ABRAMOV, O LIIVAPUU
Pages: 9 - 20
We propose a geometric approach to the BRST-symmetries of the Lagrangian of a topological quantum field theory on a four dimensional manifold based on the formalism of superconnections. Making use of a graded q-differential algebra, where q is a primitive N-th root of unity, we also propose a notion...

Non-coordinate case of graded differential algebra with ternary differential

Nadezda BAZUNOVA
Pages: 21 - 26
In this article, we generalize a construction of graded q-differential algebra with ternary differential satisfying the property d3 = 0 and q-Leibniz rule on the non-coordinate case, that is on the case where the differentials of generators of underlying algebra do not coincide with generators of bimodule...

On the matrix 3 × 3 exact solvable models of the type G2

C. BURDIK, S. POSTA, O NAVRATIL
Pages: 27 - 36
We study the exact solvable 3 × 3 matrix model of the type G2. We apply the construction similar to that one, which give the 2 × 2 matrix model. But in the studied case the construction does not give symmetric matrix potential. We conceive that the exact solvable 3 × 3 matrix potential model of the type...

SO(2) and Hamilton-Dirac mechanics

Cestmir BURDIK, Eugen PAAL, Juri VIRKEPU
Pages: 37 - 43
Canonical formalism for plane rotations is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide...

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

Jurgen FUCHS
Pages: 44 - 54
The graphical description of morphisms in rigid monoidal categories, in particular in ribbon categories, is summarized. It is illustrated with various examples of algebraic structures in such categories, like algebras, (weak) bi-algebras, Frobenius algebras, and modules and bimodules. Nakayama automorphisms...

Description of a Class of 2-Groups

Tatjana GRAMUSHNJAK, Peeter PUUSEMP
Pages: 55 - 65
Let n be an integer such that n 3 and Cm denote a cyclic group of order m . It is proved that there exist exactly 17 non-isomorphic groups of order 22n+1 which can be represented as a semidirect product (C2n × C2n ) C2. These groups are given by generators and defining relations.

Rewriting in Operads and PROPs

Lars HELLSTROM
Pages: 66 - 75
This paper is an informal collection of observations on how established rewriting techniques can be applied to or need to be adapted for use in non-associative algebras, operads, and PROPs.

On Some Almost Quadratic Algebras Coming from Twisted Derivations

Daniel LARSSON, Gunnar SIGURDSSON, Sergei D SILVESTROV
Pages: 76 - 87
This paper explores the quasi-deformation scheme devised in [1, 3] as applied to the simple Lie algebra sl2(F) for specific choices of the involved parameters and underlying algebras. One of the main points of this method is that the quasi-deformed algebra comes endowed with a canonical twisted Jacobi...

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.

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.

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...

Bosonic Realizations of the Colour Heisenberg Lie Algebra

Gunnar SIGURDSSON, Sergei D SILVESTROV
Pages: 110 - 128
We describe realizations of the colour analogue of the Heisenberg Lie algebra by power series in non-commuting indeterminates satisfying Heisenberg's canonical commutation relations of quantum mechanics. The obtained formulas are used to construct new operator representations of the colour Heisenberg...

Classical quasi-trigonometric r-matrices of Cremmer-Gervais type and their quantization

Julia Y.-MAGNUSSON, Maxim SAMSONOV, Alexander STOLIN
Pages: 129 - 136
We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.

Note on the evolution of compactly supported initial data under the Camassa-Holm flow

Enrique LOUBET
Pages: 158 - 162
We clarify and extend some remarks raised in [5] [Constantin A, J. Math. Phys. 46 (2005), 023506] about the evolution of compactly supported initial data under the Camassa-Holm flow.

Symmetries and invariants for the 2D-Ricci flow model

Rodica CIMPOIASU, Radu CONSTANTINESCU
Pages: 285 - 292
The paper investigates some special Lie type symmetries and associated invariant quantities which appear in the case of the 2D Ricci flow equation in conformal gauge. Starting from the invariants some simple classes of solutions will be determined.

Interpolation of entire functions, product formula for basic sine function

Fethi BOUZEFFOUR
Pages: 293 - 301
We solve the problem of constructing entire functions where ln M(r; f) grows like ln2 r from their values at q-n , for 0 < q < 1. As application we give a product formula for the basic sine function.

Complex crystallographic Coxeter groups and affine root systems

Joseph BERNSTEIN, Ossip SCHWARZMAN
Pages: 163 - 182
We classify (up to an isomorphism in the category of affine groups) the complex crystallographic groups generated by reflections and such that d, its linear part, is a Coxeter group, i.e., d is generated by "real" reflections of order 2.

The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources

Ting XIAO, Yunbo ZENG
Pages: 193 - 204
For the first time we show that the quasiclassical limit of the symmetry constraint of the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP...

Hamiltonian formalism of the Landau-Lifschitz equation for a spin chain with full anisotropy

Nian-Ning HUANG, Hao CAI, Tian YAN, Fan-Rong XU
Pages: 302 - 314
The Hamiltonian formalism of the Landau-Lifschitz equation for a spin chain with full anisotropy is formulated completely, which constructs a stable base for further investigations.

Massless Pseudo-scalar Fields and Solution of the Federbush Model

S E KORENBLIT, V V SEMENOV
Pages: 271 - 284
The formal Heisenberg equations of the Federbush model are linearized and then are directly integrated applying the method of dynamical mappings. The fundamental role of two-dimensional free massless pseudo-scalar fields is revealed for this procedure together with their locality condition taken into...

New Cellular Automata associated with the Schroedinger Discrete Spectral Problem

M BRUSCHI
Pages: 205 - 210
New Cellular Automata associated with the Schroedinger discrete spectral problem are derived. These Cellular Automata possess an infinite (countable) set of constants of motion.

Boundary conditions and Conserved densities for potential Zabolotskaya-Khokhlov equation

V ROSENHAUS
Pages: 255 - 270
We study local conservation laws and corresponding boundary conditions for the ptential Zabolotskaya-Khokhlov equation in (3+1)-dimensional case. We analyze an infinite Lie point symmetry group of the equation, and generate a finite number of conserved quantities corresponding to infinite symmetries...

New solvable many-body problems in the plane

F CALOGERO, J-P FRANCOISE
Pages: 231 - 254
We revisit an integrable (indeed, superintegrable and solvable) many-body model itroduced almost two decades ago by Gibbons and Hermsen and by Wojciechowski, and we modify it so that its generic solutions are all isochronous (namely, completely periodic with fixed period). We then show how this model...

Application of Lie group analysis to a core group model for sexually transmitted diseases

M EDWARDS, M C NUCCI
Pages: 211 - 230
Lie group analysis is applied to a core group model for sexually transmitted disease formulated by Hadeler and Castillo-Chavez [Hadeler K P and Castillo-Chavez C, A core group model for disease transmission, Math. Biosci. 128 (1995), 41­55]. Several instances of integrability even linearity are found...

A Note on Surface Profiles for Symmetric Gravity Waves with Vorticity

Mats EHRNSTROM
Pages: 1 - 8
We consider a nontrivial symmetric periodic gravity wave on a current with nondcreasing vorticity. It is shown that if the surface profile is monotone between trough and crest, it is in fact strictly monotone. The result is valid for both finite and infinite depth.

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...

The Transition of 2-Dimensional Solitons to 1-Dimensional Ones on Hexagonal Lattices

Betti HARTMANN, Wojtek J ZAKRZEWSKI
Pages: 111 - 116
We study solitons arising in a system describing the interaction of a two-dimensional discrete hexagonal lattice with an additional electron field (or, in general, an exciton field). We assume that this interaction is electron-phonon-like. In our previous paper [4] we have studied the existence of two-dimensional...

Extensions of 1-Dimensional Polytropic Gas Dynamics

Boris A KUPERSHMIDT
Pages: 145 - 157
1-dimensional polytropic gas dynamics is integrable for trivial reasons, having 2 < 3 components. It is realized as a subsystem of two different integrable systems: an infinite-component hydrodynamic chain of Lax type, and a 3-component system not of Lax type.

Construction of Special Solutions for Nonintegrable Systems

Sergey Yu VERNOV
Pages: 50 - 63
The Painlev´e test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonmetric ones. The standard methods for constructing the elliptic solutions consist of two independent steps: transformation of a...

Vortex Line Representation for the Hydrodynamic Type Equations

E A KUZNETSOV
Pages: 64 - 80
In this paper we give a brief review of the recent results obtained by the author and his co-authors for description of three-dimensional vortical incompressible flows in the hydrodynamic type systems. For such flows we introduce a new mixed LagrangiaEulerian description - the so called vortex line representation...

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,...

Gauge Transformation and Reciprocal Link for (2+1)-Dimensional Integrable Field Systems

Blazej M SZABLIKOWSKI
Pages: 117 - 128
Appropriate restrictions of Lax operators which allows to construction of (2+1dimensional integrable field systems, coming from centrally extended algebra of pseuddifferential operators, are reviewed. The gauge transformation and the reciprocal link between three classes of Lax hierarchies are established.

Deforming the Lie Superalgebra of Contact Vector Fields on S1|1 Inside the Lie Superalgebra of Superpseudodifferential Operators on S1|1

N BEN FRAJ, S OMRI
Pages: 19 - 33
We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie supealgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced...

Properties of the Dominant Behaviour of Quadratic Systems

A MAHARAJ, P G L LEACH
Pages: 129 - 144
We study the dominant terms of systems of Lotka-Volterra-type which arise in the the mathematical modelling of the evolution of many divers natural systems from the viewpoint of both symmetry and singularity analyses. The connections between an increase in the amount of symmetry possessed by the system...

A Note on q-Bernoulli Numbers and Polynomials

A S HEGAZI, M MANSOUR
Pages: 9 - 18
In this paper, we define a new q-analogy of the Bernoulli polynomials and the Bernoulli numbers and we deduced some important relations of them. Also, we dduced a q-analogy of the Euler-Maclaurin formulas. Finally, we present a relation between the q-gamma function and the q-Bernoulli polynomials.

Regular algebras of dimension 2, the generalized eigenvalue problem and Padé interpolation

Alexei ZHEDANOV
Pages: 333 - 356
We consider the generalized eigenvalue problem A = B for two operators A, B. Self-similar closure of this problem under a simplest Darboux transformation gives rise to two possible types of regular algebras of dimension 2 with generators A, B. Realiztion of the operators A, B by tri-diagonal operators...

Finite reductions of the two dimensional Toda chain

E V GUDKOVA
Pages: 197 - 205
The problem of the classification of integrable truncations of the Toda chain is dicussed. A new example of the cutting off constraint is found.

Bilinear recurrences and addition formulae for hyperelliptic sigma functions

Harry W BRADEN, Victor Z ENOLSKII, Andrew N W HONE
Pages: 46 - 62
The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous...

A novel approach to the theory of Padé approximants

Christopher ATHORNE
Pages: 15 - 27
By associating polynomials and power series expansions with sln(C) modules we dscribe the theory of Padé approximants in terms of tensor products of representations and interpret their recurrence relations algebraically. The treatment links with the theory of Hirota derivatives and discrete integrable...