Journal of Nonlinear Mathematical Physics

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1499 articles
Research Article

The number of independent traces and supertraces on symplectic reflection algebras

S.E. Konstein, I.V. Tyutin
Pages: 308 - 335
It is shown that A:= H1, η (G), the sympectic reflection algebra over ℂ, has TG independent traces, where TG is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ⊂ Sp(2N) ⊂ End(ℂ2N) generated by the system of symplectic reflections. Simultaneously, we show...
Research Article

Symmetry and Nonlocal Ansatzes for Nonlinear Heat Equations

Ivan Tsyfra
Pages: 312 - 318
Operators of nonlocal symmetry are used to construct exact solutions of nonlinear heat equations In [1] the idea of constructing nonlocal symmetry of differential equations was proposed. By using this symmetry, we have suggested a method for finding new classes of ansatzes reducing nonlinear wave equations...
Research Article

Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy

Yuqin Yao, Juhui Zhang, Runliang Lin, Xiaojun Liu, Yehui Huang
Pages: 309 - 323
In this paper, we discuss how to construct the bilinear identities for the wave functions of the (γn, σk)-KP hierarchy and its Hirota’s bilinear forms. First, based on the corresponding squared eigenfunction symmetry of the KP hierarchy, we prove that the wave functions of the (γn, σk)-KP hierarchy are...
Research Article

Birkhoff Strata of Sato Grassmannian and Algebraic Curves

Boris G. Konopelchenko, Giovanni Ortenzi
Pages: 309 - 347
Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum ΣS contains a subset Wŝ of points for which each fiber of the corresponding tautological subbundle TBWS is closed with respect to multiplication. Algebraically...
Research Article

A Local Equivariant Index Theorem for Sub-Signature Operators

Kaihua Bao, Jian Wang, Yong Wang
Pages: 309 - 320
In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes Weiping Zhang’s index theorem for sub-signature operators.
Research Article

Solving the Difference Initial-Boundary Value Problems by the Operator Exponential Method

I.M. Nefedov, I.A. Shereshevskii
Pages: 313 - 324
We suggest a modification of the operator exponential method for the numerical soling the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions as the peturbation of the same operator for periodic ones. We...
Research Article

Viewing the Efficiency of Chaos Control

Philippe Chanfreau, Hannu Lyyjynen
Pages: 314 - 331
This paper aims to cast some new light on controlling chaos using the OGY- and the Zero-Spectral-Radius methods. In deriving those methods we use a generalized procedure differing from the usual ones. This procedure allows us to conveniently treat maps to be controlled bringing the orbit to both various...
Research Article

Application of the Group-Theoretical Method to Physical Problems

Mina B. Abd-El-Malek
Pages: 314 - 330
The concept of the theory of continuous groups of transformations has attracted the attention of applied mathematicians and engineers to solve many physical problems in the engineering sciences. Three applications are presented in this paper. The first one is the problem of time-dependent vertical temperature...
Research Article

Diffraction of Electromagnetic Waves by a Layer Filled with a Kerr-Type Nonlinear Medium

Yury Shestopalov, Vasyl Yatsyk
Pages: 311 - 335
The diffraction of a plane wave by a transversely inhomogeneous isotropic nonmagnetic linearly polarized dielectric layer filled with a Kerr-type nonlinear medium is considered. The analytical and numerical solution techniques are developed. The diffraction problem is reduced to a singular boundary value...
Research Article

How to Extend any Dynamical System so That it Becomes Isochronous, Asymptotically Isochronous or Multi-Periodic

F. Calogero, F. Leyvraz
Pages: 311 - 338
We indicate how one can extend any dynamical system (namely, any system of nonlinearly coupled autonomous ordinary differential equations) so that the extended dynamical system thereby obtained is either isochronous or asymptotically isochronous or multi-periodic, namely its generic solutions are either...
Research Article

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...
Research Article

Isochronous Systems and Perturbation Theory

J-P Françoise
Pages: 315 - 326
This article displays examples of planar isochronous systems and discuss the new techniques found by F. Calogero with these examples. A sufficient condition is found to keep track of some periodic orbits for perturbations of isochronous systems.
Research Article

On a special two-dimensional lattice by Blaszak and Szum: pfaffianization and molecule solutions

Guo-Fu Yu, Chun-Xia Li, Jun-Xiao Zhao
Pages: 316 - 332
In this paper, we first present the Casorati and grammian determinant solutions to a special two-dimensional lattice by Blaszak and Szum. Then, by using the pfaffianiztion procedure of Hirota and Ohta, a new integrable coupled system is generated from the special lattice. Moreover, gram-type pfaffian...
Research Article

Conservation Laws and Non-Lie Symmetries for Linear PDEs

Anthony C.L. Ashton
Pages: 316 - 332
We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the...
Research Article

Moving Boundary Problems for Heterogeneous Media. Integrability via Conjugation of Reciprocal and Integral Transformations

Colin Rogers
Pages: 313 - 325
The combined action of reciprocal and integral-type transformations is here used to sequentially reduce to analytically tractable form a class of nonlinear moving boundary problems involving heterogeneity. Particular such Stefan problems arise in the description of the percolation of liquids through...
Research Article

Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations

H. A. Erbay, S. Erbay, A. Erkip
Pages: 314 - 322
In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the...
Research Article

q-Analogs of Classical 6-Periodicity: From Euler to Chebyshev

Boris A. Kupershmidt
Pages: 318 - 339
The sequence of period 6 starting with 1, 1, 0, -1, -1, 0 appears in many different disguises in mathematics. Various q-versions of this sequence are found, and their relations with Euler's pentagonal numbers theorem and Chebyshev polynomials are discussed. The motto on Cardinal Newman's tomb ought to...
Research Article

Hamiltonian Structures and Integrability of Frobenius Algebra-Valued (n, m)th KdV Hierarchy

Hai Zhang
Pages: 315 - 327
We introduce Frobenius algebra ℱ-valued (n, m)th KdV hierarchy and construct its bi-Hamiltonian structures by employing ℱ-valued pseudo-differential operators. As an illustrative example, the (1, 1)th 𝒵2-valued case is analyzed in detail. Its Hamiltonian structures and recursion operator are derived....
Research Article

Group Analysis of Nonlinear Heat-Conduction Problem for a Semi-Infinite Body

N.A. Badran, M.B. Abdelmalek
Pages: 319 - 328
The transformation group theoretic approach is applied to present an analysis of the nonlinear unsteady heat conduction problem in a semi­infinite body. The application of one­parameter group reduces the number of independent variables by one, and consequently the governing partial differential equation...
Research Article

Nonclassical Contact Symmetries and Charpit's Method of Compatibility

Daniel J. Arrigo
Pages: 321 - 329
Charpit's method of compatibility and the method of nonclassical contact symmetries for first order partial differential equation are considered. It is shown that these two methods are equivalent as Charpit's method leads to the determining equations arising from the method of nonclassical contact symmetries....
Short Communication

Second-order recursion operators of third-order evolution equations with fourth-order integrating factors

Marianna Euler, Norbert Euler
Pages: 321 - 323
We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit a fourth-order integrating factor. Under some assumptions we obtain the complete list of equations, one of which is a special case of the Schwarzian Korteweg-de Vries equation.
Research Article

Nonlinear Deterministic Equations in Biological Evolution

Kavita Jain, Sarada Seetharaman
Pages: 321 - 338
We review models of biological evolution in which the population frequency changes deterministically with time. If the population is self-replicating, although the equations for simple prototypes can be linearised, nonlinear equations arise in many complex situations. For sexual populations, even in...
Research Article

The First Cohomology of the Superconformal Algebra K(1|4)

Elena Poletaeva
Pages: 318 - 329
The infinitesimal deformations of the embedding of the Lie superalgebra of contact vector fields on the supercircle S1|4 into the Poisson superalgebra of symbols of pseudodifferential operators on S1|2 are explicitly calculated.
Research Article

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...
Research Article

Multiscale Expansion and Integrability Properties of the Lattice Potential KdV Equation

Rafael Hernandez Heredero, Decio Levi, Matteo Petrera, Christian Scimiterna
Pages: 323 - 333
We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schr¨odinger equation, the Lax pair gives at the same order...
Research Article

Tauberian Theorems In Quantum Calculus

Ahmed Fitouhi, Kamel Brahim
Pages: 324 - 340
In this paper we attempt to establish some tauberian theorems in quantum calculus. This constitutes the beginning of the study of the q-analogue of analytic theory of numbers which is the aim of a forthcoming paper.
Research Article

Continuous Correspondence of Conservation Laws of the Semi-discrete AKNS System

Wei Fu, Zhijun Qiao, Junwei Sun, Da-jun Zhang
Pages: 321 - 341
In this paper we investigate the semi-discrete Ablowitz–Kaup–Newell–Segur (sdAKNS) hierarchy, and specifically their Lax pairs and infinitely many conservation laws, as well as the corresponding continuum limits. The infinitely many conserved densities derived from the Ablowitz-Ladik spectral problem...
Research Article

Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model

Mingxuan Zhu, Lu Cao, Zaihong Jiang, Zhijun Qiao
Pages: 321 - 336
This paper devotes to present analysis work on the fifth order Camassa-Holm (FOCH) model which recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss...
Research Article

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...
Research Article

Higher Order Terms in Multiscale Expansions: A Linearized KdV Hierarchy

Hervé Leblond
Pages: 325 - 346
We consider a wide class of model equations, able to describe wave propagation in dispersive nonlinear media. The Korteweg-de Vries (KdV) equation is derived in this general frame under some conditions, the physical meanings of which are clarified. It is obtained as usual at leading order in some multiscale...
Research Article

Volume Preserving Multidimensional Integrable Systems and Nambu­Poisson Geometry

Partha Guha
Pages: 325 - 341
In this paper we study generalized classes of volume preserving multidimensional intgrable systems via Nambu­Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. All these dispersionless...
Research Article

New Type of Nonisospectral KP Equation with Self-Consistent Sources and its Bilinear Bäcklund Transformation

Ye-Peng Sun, Hon-Wah Tam
Pages: 323 - 336
A new type of the nonisospectral KP equation with self-consistent sources is constructed by using the source generation procedure. A new feature of the obtained nonisospectral system is that we allow y-dependence of the arbitrary constants in the determinantal solution for the nonisospectral KP equation....
Research Article

Tensor fields defined by Lax representations

Alexander V. Balandin
Pages: 323 - 334
In this paper, some properties of tensor fields constructed by the Lax representation of chiral-type systems are investigated.
Research Article

Dimension Increase and Splitting for Poincaré-Dulac Normal Forms

Giuseppe Gaeta, Sebastian Walcher
Pages: 327 - 342
Integration of nonlinear dynamical systems is usually seen as associated to a symmetry reduction, e.g. via momentum map. In Lax integrable systems, as pointed out by Kazhdan, Kostant and Sternberg in discussing the Calogero system, one proceeds in the opposite way, enlarging the nonlinear system to a...
Research Article

Induced Dynamics

A. K. Pogrebkov
Pages: 324 - 336
Construction of new integrable systems and methods of their investigation is one of the main directions of development of the modern mathematical physics. Here we present an approach based on the study of behavior of roots of functions of canonical variables with respect to a parameter of simultaneous...
Research Article

Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. I. Auxiliary System and Fundamental Properties

A. B. Yanovski, T. I. Valchev
Pages: 324 - 350
We consider an auxiliary spectral problem originally introduced by Gerdjikov, Mikhailov and Valchev (GMV system) and its modification called pseudo-Hermitian reduction which is extensively studied here for the first time. We describe the integrable hierarchies of both systems in a parallel way and construct...
Research Article

Symmetry Reduction and Exact Solutions of the Euler­Lagrange­Born­Infeld, Multidimensional Monge­Ampere and Eikonal Equations

Vasyl Fedorchuk
Pages: 329 - 333
Using the subgroup structure of the generalized Poincaré group P(1, 4), ansatzes which reduce the Euler­Lagrange­Born­Infeld, multidimensional Monge­Ampere and eikonal equations to differential equations with fewer independent variables have been constructed. Among these ansatzes there are ones which...
Research Article

The Stimulated Scattering of Solitons on a Resonance

Sergei Glebov, Oleg Kiselev
Pages: 330 - 341
We investigate a propagation of solitons for nonlinear Schrödinger equation under small driving force. The driving force passes through the resonance. The process of scattering on the resonance leads to changing of number of solitons. After the resonance the number of solitons depends on the amplitude...
Research Article

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Evgenii M. Vorob'ev
Pages: 330 - 335
Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form ut = (k(u) ux)x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit...
Letter to Editor

A Note on the Equivalence of Methods to finding Nonclassical Determining Equations

J. Goard
Pages: 327 - 332
In this note we prove that the method of Bîlã and Niesen to determine nonclassical determining equations is equivalent to that of Nucci’s method with heir-equations and thus in general is equivalent to using an appropriate form of generalised conditional symmetry.
Research Article

Some Homogenization and Corrector Results for Nonlinear Monotone Operators

Peter Wall
Pages: 331 - 348
This paper deals with the limit behaviour of the solutions of quasi-linear equations of the form - div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. The sequence (h) tends to 0 and the map a(x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions. It is proved...
Research Article

The Multiplication of Distributions in the Study of a Riemann Problem in Fluid Dynamics

C.O.R. Sarrico, A. Paiva
Pages: 328 - 345
The present paper concerns the study of a Riemann problem for the system ut+(12u2+φ(v))x=0,vt+(uv)x=0 , with a one dimensional space variable. We consider φ an entire function that takes real values on the real axis. Under certain conditions, this system provides solutions to the pressureless...
Research Article

A Note on the Third Family of N = 2 Supersymmetric KdV Hierarchies

F. Delduc, L. Gallot
Pages: 332 - 343
We propose a hamiltonian formulation of the N = 2 supersymmetric KP type hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic hamiltonian structure which allows for several reductions of the KP type hierarchy. In particular, the third family of N = 2 KdV hierarchies is recovered. We...
Research Article

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...
Research Article

A new extended q-deformed KP hierarchy

Runliang Lin, Xiaojun Liu, Yunbo Zeng
Pages: 333 - 347
A method is proposed in this paper to construct a new extended q-deformed KP (q-KP) hiearchy and its Lax representation. This new extended q-KP hierarchy contains two types of q-deformed KP equation with self-consistent sources, and its two kinds of reductions give the q-deformed Gelfand-Dickey hierarchy...
Research Article

Lagrangians for Biological Models

M. C. Nucci, K. M. Tamizhmani
Pages: 330 - 352
We show that a method presented in [S. L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974) 299–324] and later in [G. H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol. 44 (1982) 749–760] for finding Lagrangians of classic models...
Research Article

The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

P.G. Estévez, P.R. Gordoa
Pages: 334 - 355
We present in this paper the singular manifold method (SMM) derived from Painlevé analysis, as a helpful tool to obtain much of the characteristic features of nonlinear partial differential equations. As is well known, it provides in an algorithmic way the Lax pair and the Bäcklund transformation for...
Research Article

The Klein-Gordon Equation on the Half Line: a Riemann-Hilbert Approach

Beatrice Pelloni, Dimitrios A. Pinotsis
Pages: 334 - 344
We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems...
Research Article

Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

Rustem N. Garifullin, Giorgio Gubbiotti, Ravil I. Yamilov
Pages: 333 - 357
In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous...
Research Article

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...
Research Article

On a integrable deformations of Heisenberg supermagnetic model

Zhaowen Yan, Gegenhasi
Pages: 335 - 342
The Heisenberg supermagnet model which is the supersymmetric generalization of the Heisenberg ferromagnet model is an important integrable system. We consider the deformations of Heisenberg supermagnet model under the two constraint 1. S2 = S for S ∈ USPL(2/1)/S(L(1/1) × U(1)) and 2. S2 = 3S − 2I S ∈...
Research Article

On the Pseudo-Schrödinger Equation Approximation of the Transfer-Integral Operator for 1-Dimensional DNA Models

Marc Joyeux
Pages: 339 - 357
The Transfer-Integral (TI) operator is a powerful method to investigate the statistical physics of 1-dimensional models, like those used to describe DNA denaturation. At the cost of a certain number of approximations, the TI equation can be reduced to a Pseudo–Schrödinger Equation (PSE), according to...
Research Article

Periodic orbits associated to Hamiltonian functions of degree four

Dante Carrasco-Olivera, Marco Uribe, Claudio Vidal
Pages: 336 - 356
We consider the Hamiltonian polynomial function H of degree fourth given by either H(x,y,px,py)=12(px2+py2)+12(x2+y2)+V3(x,y)+V4(x,y) , or H(x,y,px,py)=12(-px2+py2)+12(-x2+y2)+V3(x,y)+V4(x,y) , where V3 (x, y) and V4 (x, y) are homogeneous polynomials of degree three and four, respectively....
Research Article

Generalisations of the Laplace­Runge­Lenz Vector

P.G.L. Leach, G.P. Flessas
Pages: 340 - 423
The characteristic feature of the Kepler Problem is the existence of the so-called Laplace­Runge­Lenz vector which enables a very simple discussion of the properties of the orbit for the problem. It is found that there are many classes of problems, some closely related to the Kepler Problem and others...
Research Article

Coisotropic quasi-periodic motions near the relative equilibrium of a Hamiltonian system

Ihor Parasyuk
Pages: 340 - 357
We consider the Hamiltonian system which is invariant under locally Hamiltonian (non-Poissonian) action of torus. We show that when a certain set of conditions is satisfied the majority of motions in a sufficiently small neighbourhood of system's relative equilibrium are quasi-periodic and cover coisotropic...
Research Article

Classification of 3D Consistent Quad-Equations

Raphael Boll
Pages: 337 - 365
We consider 3D consistent systems of six possibly different quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of these equations to the whole lattice ℤ3 is possible by reflecting...
Research Article

Magnetohydrodynamics's Type Equations Over Clifford Algebras

Igor Kondrashuk, Eduardo A. Notte-Cuello, Marko A. Rojas-Medar
Pages: 337 - 347
We study a system of equations modeling the stationary motion of incompressible electrical conducting fluid. Based on methods of Clifford analysis, we rewrite the system of magnetohydrodynamics fluid in the hypercomplex formulation and represent its solution in Clifford operator terms.
Review Article

Linearizable boundary value problems for the elliptic sine-Gordon and the elliptic Ernst equations

Jonatan Lenells, Athanassios S. Fokas
Pages: 337 - 356
By employing a novel generalization of the inverse scattering transform method known as the unified transform or Fokas method, it can be shown that the solution of certain physically significant boundary value problems for the elliptic sine-Gordon equation, as well as for the elliptic version of the...
Research Article

On the Generalized KdV Hierarchy and Boussinesq Hierarchy with Lax Triple

Xiaoli Wang, Jian-Qin Mei
Pages: 337 - 343
Based on the Nambu 3-bracket and the operators of the KP hierarchy, we propose the generalized Lax equation of the Lax triple. Under the operator constraints, we construct the generalized KdV hierarchy and Boussinesq hierarchy. Moreover, we present the exact solutions of some nonlinear evolution equations.
Research Article

The Method of an Exact Linearization of n-order Ordinary Differential Equations

L.M. Berkovich
Pages: 341 - 350
Necessary and sufficient conditions are found that the n-order nonlinear and nonautonomous ordinary differential equation could be transformed into a linear equation with constant coefficients with the help, generally speaking, nonlocal transformation of dependent and independent variables. These conditions...
Research Article

Fractional q-Calculus on a time scale

Ferhan M. Atici, Paul W. Eloe
Pages: 341 - 352
The study of fractional q-calculus in this paper serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale Tt0 = {t : t = t0 q n , n a nonnegative integer } ∪ {0}, where t0 ∈ R and 0
Research Article

Compactly Supported Solutions of the Camassa-Holm Equation

David Henry
Pages: 342 - 347
We give a simple proof that for any non-zero initial data, the solution of the CamassHolm equation loses instantly the property of being compactly supported.
Research Article

A Tree of Linearisable Second-Order Evolution Equations by Generalised Hodograph Transformations

Norbert Euler, Marianna Euler
Pages: 342 - 362
We present a list of (1+1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution equation. The list includes autonomous and nonautonomous equations.
Research Article

Darboux Polynomials for Lotka–Volterra Systems in Three Dimensions

Yiannis T. Christodoulides, Pantelis A. Damianou
Pages: 339 - 354
We consider Lotka–Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding...
Research Article

Factorization of the Loop Algebras and Compatible Lie Brackets

I.Z. Golubchik, V.V. Sokolov
Pages: 343 - 350
It is shown that any decomposition of the loop algebra over a simple Lie algebra into a direct sum of the Taylor series and a complementary subalgebra is defined by a pair of compatible Lie brackets.
Research Article

Coadjoint Poisson Actions of Poisson-Lie Groups

Boris A. Kupershmidt, Ognyan S. Stoyanov
Pages: 344 - 354
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.
Research Article

On Differential Operators on Sequence Spaces

M. Maldonado, J. Prada, M.J. Senosiain
Pages: 345 - 352
Two differential operators T1 and T2 on a space are said to be equivalent if there is an isomorphism S from onto such that ST1 = T2 S. The notion was first introduced by Delsarte in 1938 [2] where T1 and T2 are differential operators of second order and a space of functions of one variable defined for...
Research Article

Massey products, A-algebras, differential equations, and Chekanov homology

Lida Mendoza, Enrique G. Reyes
Pages: 342 - 360
We consider (classical and generalized) Massey products on the Chekanov homology of a Legendrian knot, and we prove that they are invariant under Legendrian isotopies. We also construct a minimal A∞-algebra structure on the Chekanov algebra of a Legendrian knot, we prove that this structure is invariant...
Research Article

On a connection between formulas about q–gamma functions

Wolfram Koepf, Predrag M. Rajković, Sladjana D. Marinković
Pages: 343 - 350
In this short communication, we want to pay attention to a few wrong formulas which are unfortunately cited and used in a dozen papers afterwards. We prove that the provided relations and asymptotic expansion about the q-gamma function are not correct. This is illustrated by numerous concrete counterexamples....
Research Article

Poisson Cohomology of SU(2)-Covariant "Necklace" Poisson Structures on S2

Dmitry Roytenberg
Pages: 347 - 356
We compute the Poisson cohomology of the one-parameter family of SU(2)-covariant Poisson structures on the homogeneous space S2 = CP1 = SU(2)/U(1), where SU(2) is endowed with its standard Poisson­Lie group structure, thus extending the result of Ginzburg [2] on the Bruhat­Poisson structure which is...
Research Article

Symmetries of Modules of Differential Operators

H. Gargoubi, P. Mathonet, V. Ovsienko
Pages: 348 - 380
Let F(S1 ) be the space of tensor densities of degree (or weight) on the circle S1 . The space Dk ,µ(S1 ) of k-th order linear differential operators from F(S1 ) to Fµ(S1 ) is a natural module over Diff(S1 ), the diffeomorphism group of S1 . We determine the algebra of symmetries of the modules Dk ,µ(S1...
Short Communication

A Hidden Group Structure for the Integrals of the Benney System

Boris A. Kupershmidt
Pages: 348 - 352
The integrals of the Benney system are shown to possess a group structure. The KP hierarchy breaks the group law down.
Research Article

On the Analytical Approach to the N-Fold Bäcklund Transformation of Davey-Stewartson Equation

S.K. Paul, A. ROY Chowdhury
Pages: 349 - 356
N-fold Bäcklund transformation for the Davey-Stewartson equation is constructed by using the analytic structure of the Lax eigenfunction in the complex eigenvalue plane. Explicit formulae can be obtained for a specified value of N. Lastly it is shown how generalized soliton solutions are generated from...
Research Article

Explicit integration of a generic Hénon-Heiles system with quartic potential

Nicola Sottocornola
Pages: 346 - 355
There are seven time independent, integrable, Hénon-Heiles systems: three with cubic and four with quartic potential. The cubic and one of the quartic cases have been separated in the last decades. The other three cases 1:6:1, 1:6:8 and 1:12:16 have resisted several attempts in the last years. In this...
Research Article

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.
Research Article

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.
Research Article

Periods of the Goldfish Many-Body Problem

David Gomez-Ullate, Matteo Sommacal
Pages: 351 - 362
Calogero's goldfish N-body problem describes the motion of N point particles subject to mutual interaction with velocity-dependent forces under the action of a constant magnetic field transverse to the plane of motion. When all coupling constants are equal to one, the model has the property that for...
Research Article

Group and Renormgroup Symmetry of Quasi-Chaplygin Media

Vladimir F. Kovalev
Pages: 351 - 356
Results of renormgroup analysis of a quasi-Chaplygin system of equations are presented. Lie-Bäcklund symmetries and corresponding invariant solutions for different "Chaplygin" functions are obtained. The algorithm of construction of a group on a solution (renormgroup) using two different approaches is...
Letter to Editor

Properties of the Zeros of the Sum of two Polynomials

Francesco Calogero
Pages: 348 - 354
Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.
Research Article

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
Research Article

Symmetry of osp(m|n) Spin Calogero–Sutherland Models

Kazuyuki Oshima
Pages: 349 - 356
We introduce osp(m|n) spin Calogero–Sutherland models and find that the models have the symmetry of osp(m|n) half-loop algebra or Yangian of osp(m|n) if and only if the coupling constant of the model equals to 2m−n−4.
Research Article

On Separation of Variables for Integrable Equations of Soliton Type

Julia Bernatska, Petro Holod
Pages: 353 - 374
We propose a general scheme for separation of variables in the integrable Hamilto- nian systems on orbits of the loop algebra sl(2, C) × P (λ, λ −1 ). In particular, we illus- trate the scheme by application to modified Korteweg—de Vries (MKdV), sin(sinh)- Gordon, nonlinear Schr ̈odinger, and...
Research Article

Where Do Braid Statistics and Discrete Motion Meet Each Other?

Luigi Martina, Alexander Protogenov, Valery Verbus
Pages: 353 - 361
We consider universal statistical properties of systems that are characterized by phase states with macroscopic degeneracy of the ground state. A possible topological order in such systems is described by non-linear discrete equations. We focus on the discrete equations which take place in the case of...
Research Article

Contiguity relations for discrete and ultradiscrete Painleve equations

A. Ramani, B. Grammaticos, R. Willox
Pages: 353 - 364
We show that the solutions of ultradiscrete Painleve equations satisfy contiguity relations just as their continuous and discrete counterparts. Our starting point are the relations for q-discrete Painleve equations which we then proceed to ultradiscretise. In this paper we obtain results for the one-parameter...
Letter to Editor

On nonlinearity in three-dimensional equatorial flows

David Henry
Pages: 351 - 357
We examine an aspect of the modelling of the underlying fluid motion in the equatorial region of the ocean. In particular, we assess whether nonlinearity is inherently vital in capturing the three-dimensional upwelling and downwelling phenomena. A recent applied mathematical approach has successfully...
Research Article

Semi-discrete hyperbolic equations admitting five dimensional characteristic x-ring

Kostyantyn Zheltukhin, Natalya Zheltukhina
Pages: 351 - 367
The necessary and sufficient conditions for a hyperbolic semi-discrete equation to have five dimensional characteristic x-ring are derived. For any given chain, the derived conditions are easily verifiable by straightforward calculations.
Research Article

Continuous and Discrete Transformations of a One-Dimensional Porous Medium Equation

Christodoulos Sophocleous
Pages: 355 - 364
We consider the one-dimensional porous medium equation ut = (un ux)x + µ x un ux. We derive point transformations of a general class that map this equation into itself or into equations of a similar class. In some cases this porous medium equation is connected with well known equations. With the introduction...
Research Article

On the Spectral Theory of Operator Pencils in a Hilbert Space

Roman I. Andrushkiw
Pages: 356 - 366
Consider the operator pencil L = A - B - 2 C, where A, B, and C are linear, in general unbounded and nonsymmetric, operators densely defined in a Hilbert space H. Sufficient conditions for the existence of the eigenvalues of L are investigated in the case when A, B and C are K-positive and K-symmetric...
Research Article

Functional Representation of the Negative AKNS Hierarchy

V. E. Vekslerchik
Pages: 353 - 372
This paper is devoted to the negative flows of the AKNS hierarchy. The main result of this work is the functional representation of the extended AKNS hierarchy, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed by means of the Miwa's...
Research Article

Solutions of the Generalized Weierstrass Representation in Four-Dimensional Euclidean Space

P. Bracken, A.M. Grundland
Pages: 357 - 381
Several classes of solutions of the generalized Weierstrass system, which induces costant mean curvature surfaces into four-dimensional Euclidean space are constructed. A gauge transformation allows us to simplify the system considered and derive fatorized classes of solutions. A reduction of the generalized...
Research Article

Fundamental Solution of the Volkov Problem (Characteristic Representation)

A.A. Borghardt, D.Ya. Karpenko
Pages: 357 - 363
The characteristic representation, or Goursat problem, for the Klein-Fock-Gordon equation with Volkov interaction [1] is regarded. It is shown that in this representation the explicit form of the Volkov propagator can be obtained. Using the characteristic representation technique, the Schwinger integral...
Research Article

Symmetries of the Fokker-Type Relativistic Mechanics in Various Forms of Dynamics

Roman Gaida, Volodymyr Tretyak
Pages: 357 - 371
The single-time nonlocal Lagrangians corresponding to the Fokker-type action integrals are obtained in arbitrary form of relativistic dynamics. The symmetry conditions for such Lagrangians under an arbitrary Lie group acting on the Minkowski space are formulated in various forms of dynamics. An explicit...
Research Article

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. This...
Research Article

Symmetry Classification for a Coupled Nonlinear Schrödinger Equations

M. Euler, N. Euler, W.W. Zachary, M.F. Mahmood, T.L. Gill
Pages: 358 - 379
Research Article

A New Class of Symmetry Reductions for Parameter Identification Problems

Nicoleta Bîlă, Jitse Niesen
Pages: 355 - 371
This paper introduces a new type of symmetry reductions called extended nonclassical symmetries that can be studied for parameter identification problems described by partial differential equations. Including the data function in the parameter space, we show that specific data and parameter classes that...
Research Article

Algebro-Geometric Solutions and Their Reductions for the Fokas-Lenells Hierarchy

Peng Zhao, Engui Fan, Yu Hou
Pages: 355 - 393
This paper is dedicated to provide theta function representations of algebro-geometric solutions for the Fokas- Lenells (FL) hierarchy through studying an algebro-geometric initial value problem. Further, we reduce these solutions into n-dark solutions through the degeneration of associated Riemann surfaces.
Research Article

Some Rigorous Results on the Eigen Quasispecies Model with a Periodically Moving Sharp-Peak Landscape

Armando G. M. Neves
Pages: 359 - 379
In this paper we prove some results and detail some calculations published in a previous paper by us on the Eigen model with a periodically moving sharp-peak landscape. The model is concerned with evolution of a virus population in a time-dependent environment mimicking interaction of the viruses with...