# Journal of Nonlinear Mathematical Physics

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 (*γ*_{n}, *σ*_{k})-KP Hierarchy

_{n}

_{k}

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**

## Discrete Symmetry and Its Use to Find Multi-Soliton Solutions of the Equations of Anisotropic Heisenberg Ferromagnets

N.A. Belov, A.N. Leznov, W.J. Zakrzewski

Pages: 319 - 329

**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 SchrÂ¨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**

## General Solution of the Universal Equation in n-Dimensional Space

D.B. Fairlie, A.N. Leznov

Pages: 333 - 339

**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**

## Conditional Symmetry and Exact Solutions of the Multidimensional Nonlinear d'Alembert Equation

A.F. Barannyk, Yu.D. Moskalenko

Pages: 336 - 340

**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 } âˆª {0}, where t0 âˆˆ 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) Ã— P (Î», Î» âˆ’1 ). In particular, we illus- trate the scheme by application to modified Kortewegâ€”de Vries (MKdV), sin(sinh)- Gordon, nonlinear Schr Ìˆodinger, 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...