# Journal of Nonlinear Mathematical Physics

1499 articles

**Research Article**

## The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

Robin S. Johnson

Pages: 72 - 92

In this contribution, we describe the simplest, classical problem in water waves, and use this as a vehicle to outline the techniques that we adopt to analyse this particular approach to the derivation of soliton-type equations. The surprise, perhaps, is that such an apparently transparent set of equations...

**Research Article**

## Integrable Models for Shallow Water with Energy Dependent Spectral Problems

Rossen Ivanov, Tony Lyons

Pages: 72 - 88

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse...

**Research Article**

## The Structure of Gelfand-Levitan-Marhenko Type Equations for Delsarte Transmutation Operators of Linear Multidimensional Differential Operators and Operator Pencils. Part 1.

Jolanta Golenia, Anatoliy K. Prykarpatsky, Yarema A. Prykarpatsky

Pages: 73 - 87

An analog of Gelfand-Levitan-Marchenko integral equations for multi- dimensional Delsarte transmutation operators is constructed by means of studying their differentiageometric structure based on the classical Lagrange identity for a formally conjugated pair of differential operators. An extension of...

**Research Article**

## Symmetries and Integrating Factors

P.G.L. Leach, S.É. Bouquet

Pages: 73 - 91

Cheb-Terrab and Roche (J. Sym. Comp. 27 (1999), 501519) presented what they termed a systematic algorithm for the construction of integrating factors for second order ordinary differential equations. They showed that there were instances of odinary differential equations without Lie point symmetries...

**Research Article**

## q-Probability: I. Basic Discrete Distributions

Boris A. Kupershmidt

Pages: 73 - 93

For basic discrete probability distributions, - Bernoulli, Pascal, Poisson, hypergemetric, contagious, and uniform, - q-analogs are proposed.

**Research Article**

## Symmetry Reduction for Equation 2u + (u2 1 + u2 2 + u2 3)1/2 u0 = 0

L.F. Barannyk, H.O. Lahno

Pages: 73 - 89

The subalgebras of the invariance algebra of equation 2u+(u2

**Research Article**

## Nonlinear-Integral-Equation Construction of Orthogonal Polynomials

Carl M. Bender, E. Ben-Naim

Pages: 73 - 80

The nonlinear integral equation P(x) = dyw(y)P(y)P(x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions Pn(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations...

**Research Article**

## Complete Specification of Some Partial Differential Equations That Arise in Financial Mathematics

S. Dimas, K. Andriopoulos, D. Tsoubelis, P. G. L. Leach

Pages: 73 - 92

We consider some well-known partial differential equations that arise in Financial Mathematics, namely the Black–Scholes–Merton, Longstaff, Vasicek, Cox–Ingersoll–Ross and Heath equations. Our central aim is to discover any underlying connections taking into account the Lie remarkability property of...

**Research Article**

## A family of integrable evolution equations of third order

Matthew Babela, Alexandre Odesskii

Pages: 73 - 78

We construct a family of integrable equations of the form vt = f(v; vx; vxx; vxxx) such that f is a transcendental function in v; vx; vxx. This family is related to the Krichever-Novikov equation by a differential substitution. Our construction of integrable equations and the corresponding differential...

**Research Article**

## Functional Equations and the Generalised Elliptic Genus

H.W. Braden, K.E. Feldman

Pages: 74 - 85

We give a new derivation and characterisation of the generalised elliptic genus of Krichever-Höhn by means of a functional equation.

**Research Article**

## On the Fluid Motion in Standing Waves

Mats Ehrnstrom, Erik Wahlen

Pages: 74 - 86

This paper concerns linear standing gravity water waves on finite depth. We obtain qualitative and quantitative understanding of the particle paths within the wave.

**Research Article**

## Geometry of Real Forms of the Complex Neumann System

Tina Novak

Pages: 74 - 91

In the paper, we study real forms of the complex generic Neumann system. We prove that the real forms are completely integrable Hamiltonian systems. The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax pair (Lℂ(λ), Mℂ(λ)) of 2 × 2...

**Research Article**

## Integrable Substructure in a Korteweg Capillarity Model. A Karman-Tsien Type Constitutive Relation

Colin Rogers

Pages: 74 - 88

A classical Korteweg capillarity system with a Karman-Tsien type (κ, ρ) constitutive relation is shown, via a Madelung transformation and use of invariants of motion, to admit integrable Hamiltonian subsystems.

**Research Article**

## Integrability Conditions for n and t Dependent Dynamical Lattice Equations

R. Yamilov, D. Levi

Pages: 75 - 101

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

**Research Article**

## Rational Solutions of an Extended Lotka-Volterra Equation

X.B. Hu, P.A. Clarkson

Pages: 75 - 83

A series of rational solutions are presented for an extended Lotka-Volterra eqution. These rational solutions are obtained by using Hirota's bilinear formalism and Bäcklund transformation. The crucial step is the use of nonlinear superposition fomula. The so-called extended Lotka-Volterra equation is...

**Research Article**

## Symmetry reduction and exact solutions of the Navier-Stokes equations. I

Wilhelm Fushchych, Roman Popowych

Pages: 75 - 113

Ansatzes for the Navier-Stokes field are described. These ansatzes reduce the Navier-Stokes equations to system of differential equations in three, two, and one independent variables. The large sets of exact solutions of the Navier-Stokes equations are constructed.

**Research Article**

## Exact solutions and Riccati-type first integrals

J. Mendoza, C. Muriel

Pages: 75 - 89

The λ-symmetry approach is applied to a family of second-order ODEs whose algebra of Lie point symmetries is insufficient to integrate them. The general solution and two functionally independent first integrals of a subclass of the studied equations can be expressed in terms of a fundamental set of solutions...

**Research Article**

## Folding Transformations and HKY Mappings

B. Grammaticos, A. Ramani, R. Willox

Pages: 75 - 85

We present a new method for the derivation of mappings of HKY type. These are second-order mappings which do not have a biquadratic invariant like the QRT mappings, but rather an invariant of degree higher than two in at least one of the variables. Our method is based on folding transformations which...

**Research Article**

## 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...

**Research Article**

## The Asymptotic Behaviors of Solutions to the Perturbed Riemann Problem near the Singular Curve for the Chromatography System*

Chun Shen

Pages: 76 - 101

The Riemann problem for a simplified chromatography system is considered and the global Riemann solutions are constructed in all kinds of situations. In particular, the zero rarefaction wave, the zero shock wave and the zero delta shock wave are discovered in the Riemann solutions in some limit situations,...

**Research Article**

## Integrable flows and Bäcklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)

Yuri N. Fedorov

Pages: 77 - 94

We show that the m-dimensional EulerManakov top on so (m) can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety ¯V(k, m), and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic...

**Research Article**

## The Discrete Nonlinear Schrödinger Equation and its Lie Symmetry Reductions

R. Hernández Heredero, D. Levi

Pages: 77 - 94

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

**Research Article**

## The Numerical Study of the Solution of the 4 0 Model

S. Gladkoff, A. Alaie, Y. Sansonnet, M. Manolessou

Pages: 77 - 85

We present a numerical study of the nonlinear system of 4 0 equations of motion. The solution is obtained iteratively, starting from a precise point-sequence of the appropriate Banach space, for small values of the coupling constant. The numerical results are in perfect agreement with the main theoretical...

**Research Article**

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

Tasawar Hayat, Herman Mambili-Mamboundou, Ebrahim Momoniat, Fazal M. Mahomed

Pages: 77 - 90

The influence of a magnetic field on the flow of an incompressible third grade electrically conducting fluid bounded by a rigid plate is investigated. The flow is induced due by the motion of a plate in its own plane with an arbitrary velocity. The solution of the equations of conservation of mass and...

**Research Article**

## The Converse Problem for the Multipotentialisation of Evolution Equations and Systems

Norbert Euler, Marianna Euler

Pages: 77 - 105

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension...

**Research Article**

## Group Analysis and Heir-Equations for Thin Liquid Films

S. Martini, N. Ciccoli, M. C. Nucci

Pages: 77 - 92

Lie group analysis is applied to a mathematical model for thin liquid films, namely a nonlinear fourth order partial differential equation in two independent variables. A three-dimensional Lie symmetry algebra is found and reductions to fourth order ordinary differential equations are obtained by using...

**Research Article**

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

Marek Czachor

Pages: 78 - 84

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

**Research Article**

## A Remark on Nonlocal Symmetries for the CalogeroDegasperisIbragimovShabat Equation

Artur Sergyeyev, Jan A. Sanders

Pages: 78 - 85

We consider the CalogeroDegasperisIbragimovShabat (CDIS) equation and find the complete set of its nonlocal symmetries depending on the local variables and on the integral of the only local conserved density of the equation in question. The Lie algebra of these symmetries turns out to be a central...

**Research Article**

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

Leonid Barannyk, Halyna Lahno

Pages: 78 - 88

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

**Research Article**

## Method of Generating *N*-dimensional Isochronous Nonsingular Hamiltonian Systems

A. Durga Devi, R. Gladwin Pradeep, V.K. Chandrasekar, M. Lakshmanan

Pages: 78 - 93

In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that a class of singular Hamiltonian systems obtained through the Ω-modified procedure is equivalent to constrained Newtonian systems. Even though such systems admit isochronous...

**Research Article**

## Amplitude-Dependent Oscillations in Gases

O.L. de Lange, J. Pierrus

Pages: 79 - 81

We consider the following question: Suppose part of the boundary of a cavity contaiing a gas is set into oscillation, the damping in the boundary being small. What is the nature of the oscillations in the gas? We treat the low-frequency limit (wavelength much greater than dimensions of the cavity). Experiment...

**Research Article**

## Additional symmetries and string equations of the noncommutative B and C type KP hierarchies

Qiufang Liu, Chuanzhong Li

Pages: 79 - 92

In this paper, we construct the noncommutative B and C type KP hierarchies using pseudo-differential operators and reducing conditions. Further a series of additional flows of the noncommutative B and C type KP hierarchies will be defined and the additional symmetries constitute the B and C type infinite...

**Research Article**

## The Generalized Version of Dressing Method with Applications to AKNS Equations

Junyi Zhu, Xianguo Geng

Pages: 81 - 89

The generalized dressing method is extended to variable-coefficient AKNS equations, including a variable-coefficient coupled nonlinear Schr¨odinger equation and a variablcoefficient coupled mKdV equation. A general variable-coefficient KP equation is proposed and decomposed into the two 1+1 dimensional...

**Research Article**

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

M.S. Bruzon, M. L. Gandarias, J. C. Camacho

Pages: 81 - 90

In this paper, the family of BBM equation with strong nonlinear dispersive B(m,n) is considered. We apply the classical Lie method of infinitesimals. The symmetry reductions are derived from the optimal system of subalgebras and lead to systems of ordinary differential equations. We obtain for special...

**Research Article**

## Nambu Bracket Formulation of Nonlinear Biochemical Reactions Beyond Elementary Mass Action Kinetics

T. D. Frank

Pages: 81 - 97

We develop a Nambu bracket formulation for a wide class of nonlinear biochemical reactions by exploiting previous work that focused on elementary biochemical mass action reactions. To this end, we consider general reaction mechanisms including for example enzyme kinetics. Furthermore, we go beyond elementary...

**Research Article**

## Inhomogeneous Burgers Lattices

S. De Lillo, V.V. Konotop

Pages: 82 - 87

We study statistical properties of inhomogeneous Burgers lattices which are solved by the discrete ColeHopf transformation. Using exact solutions we investigate effect of various kinds of noise on the dynamics of solutions.

**Review Article**

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

P.G. Estévez, E. Conde, P.R. Gordoa

Pages: 82 - 114

This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlevé Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura,...

**Research Article**

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

Andrew Riley, Ian A.B. Strachan

Pages: 82 - 94

Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds....

**Research Article**

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

Roman Gielerak, Robert Ralowski

Pages: 85 - 91

A class of involutive Wick algebras (called anyonic-type Wick algebras) is selected and some its elementary properties are described. In particular, the Fock representtions of the selected anyonic-type commutation relations are described. For the class of so-called r-yonic systems the question of the...

**Research Article**

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

Evgenii M. Vorob'ev

Pages: 85 - 89

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

**Research Article**

## Properties of the series solution for Painlevé I

A.N.W. Hone, O. Ragnisco, F. Zullo

Pages: 85 - 100

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions...

**Research Article**

## Lower Bounds for the Spinless Salpeter Equation

Fabian Brau

Pages: 86 - 96

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

**Research Article**

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

Kenji Kajiwara, Kinji Kimura

Pages: 86 - 102

A q-difference analogue of the Painlevé III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.

**Research Article**

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

I.G. Korepanov, E.V. Martyushev

Pages: 86 - 98

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7, 1) and L(7, 2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e. algebraic relation corresponding to a...

**Research Article**

## Rosenhain-Thomae formulae for higher genera hyperelliptic curves

Keno Eilers

Pages: 86 - 105

Rosenhain's famous formula expresses the periods of first kind integrals of genus two hyperelliptic curves in terms of θ-constants. In this paper we generalize the Rosenhain formula to higher genera hyperelliptic curves by means of the second Thomae formula for derivative non-singular θ-constants.

**Research Article**

## Note on operadic non-associative deformations

Eugen Paal

Pages: 87 - 92

Deformation equation of a non-associative deformation in operad is proposed. Its integrability condition (the Bianchi identity) is considered. Algebraic meaning of the latter is explained.

**Research Article**

## Exact Solutions of DNLS and Derivative Reaction-Diffuson Systems

Jyh-Hao Lee, Yen-Ching Lee, Chien-Chih Lin

Pages: 87 - 98

In this paper, we obtain some exact solutions of Derivative Reaction-Diffusion (DRD) system and, as by-products, we also show some exact solutions of DNLS via Hirota bilinearization method. At first, we review some results about two by two AKNS-ZS system, then introduce Hirota bilinearization method...

**Research Article**

## 10-Commutators, 13-commutators and odd derivations

Askar Dzhumadil'daev

Pages: 87 - 103

**Research Article**

## On Gerstner's Water Wave

David Henry

Pages: 87 - 95

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

**Research Article**

## Invariants of Lie Algebras Extended Over Commutative Algebras Without Unit

Pasha Zusmanovich

Pages: 87 - 102

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated...

**Research Article**

## Hypergeometric Solutions to an Ultradiscrete Painlevé Equation

Christopher M. Ormerod

Pages: 87 - 102

We show that an ultradiscrete analogue of the third Painlevé equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework we may...

**Research Article**

## A Route to Routh — The Classical Setting

Ladislav Adamec

Pages: 87 - 107

There is a well known principle in classical mechanic stating that a variational problem independent of a configuration space variable w (so called cyclic variable), but dependent on its velocity w′ can be expressed without both w and w′. This principle is known as the Routh reduction.
In this paper,...

**Research Article**

## Solitons and Deformed Lattices I

Betti Hartmann, Wojtek J. Zakrzewski

Pages: 88 - 104

We study a model describing some aspects of the dynamics of biopolymers. The models involve either one or two finite chains with a number N of sites that represent the "units" of a biophysical system. The mechanical degrees of freedom of these chains are coupled to the internal degrees of freedom through...

**Research Article**

## Asymptotic Lattices and W-Congruences in Integrable Discrete Geometry

Adam Doliwa

Pages: 88 - 92

The asymptotic lattices and their transformations are included into the theory of quadrilateral lattices.

**Research Article**

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

R.Ya. Matsyuk

Pages: 89 - 97

Symmetries for variational problems are considered as symmetries of vector-valued exterior differential systems. This approach is applied to equations for the classical spinning particle.

**Research Article**

## On Periodic Water Waves with Coriolis Effects and Isobaric Streamlines

Anca-Voichita Matioc, Bogdan-Vasile Matioc

Pages: 89 - 103

In this paper we prove that solutions of the f-plane approximation for equatorial geophysical deep water waves, which have the property that the pressure is constant along the streamlines and do not possess stagnation points, are Gerstner-type waves. Furthermore, for waves traveling over a flat bed,...

**Research Article**

## On nonlinear coherent states properties for electron-phonon dynamics

Isiaka Aremua, Mahouton Norbert Hounkonnou, Ezinvi Baloïtcha

Pages: 89 - 119

This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to an electron-phonon model in the condensed matter physics, obeying a f-deformed Heisenberg algebra. The existence and properties...

**Research Article**

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

Douglas Baldwin, Willy Hereman

Pages: 90 - 110

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

**Research Article**

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

Violeta Tretynyk

Pages: 90 - 95

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

**Short Communication**

## NonLie Ansatzes for Nonlinear Heat Equations

Ivan Tsyfra

Pages: 90 - 93

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

**Research Article**

## On Lie-point symmetries for Ito stochastic differential equations

G. Gaeta, C. Lunini

Pages: 90 - 102

In the deterministic realm, both differential equations and symmetry generators are geometrical objects, and behave properly under changes of coordinates; actually this property is essential to make symmetry analysis independent of the choice of coordinates and applicable. When trying to extend symmetry...

**Research Article**

## Symmetries, Conservation Laws, Invariant Solutions and Difference Schemes of the One-dimensional Green-Naghdi Equations

V.A. Dorodnitsyn, E.I. Kaptsov, S.V. Meleshko

Pages: 90 - 107

The paper is devoted to the Lie group properties of the one-dimensional Green-Naghdi equations describing the behavior of fluid flow over uneven bottom topography. The bottom topography is incorporated into the Green-Naghdi equations in two ways: in the classical Green-Naghdi form and in the approximated...

**Research Article**

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

T. Hayat, Ebrahim Momoniat, Fazal M. Mahomed

Pages: 91 - 104

This investigation deals with the mechanism of peristaltic transport of a non-Newtonian, incompressible and electrically conducting fluid with variable viscosity and an endoscope effects. The magnetic Reynolds number is taken to be small. The mechanical properties of the material are represented by the...

**Research Article**

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

M. Calixto, V. Aldaya, F. F. Lopez-Ruiz, E. Sanchez-Sastre

Pages: 91 - 101

We extend the traditional formulation of Gauge Field Theory by incorporating the (non- Abelian) gauge group parameters (traditionally simple spectators) as new dynamical (nonlinearsigma- model-type) fields. These new fields interact with the usual Yang-Mills fields through a generalized minimal coupling...

**Research Article**

## A 3-Lie algebra and the dKP Hierarchy

Min-Ru Chen, Ying Chen, Zhao-Wen Yan, Jian-Qin Mei, Xiao-Li Wang

Pages: 91 - 97

In terms of a 3-Lie algebra and the classical Poisson bracket {Bn,L} of the dKP hierarchy, a special 3-bracket {Bm,Bn,L} is proposed. When m = 0 or m = 1, the 3-lax equation ∂L∂t={Bm,Bn,L} is the dKP hierarchy and the corresponding proof is given. Meanwhile, for the generalized case (m,n), the generalized...

**Research Article**

## When is a sum of projections equal to a scalar operator?

Yurii S. Samoǐlenko

Pages: 92 - 103

**Research Article**

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

D.P. Mason

Pages: 92 - 101

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

**Research Article**

## The Ablowitz–Ladik hierarchy integrability analysis revisited: the vertex operator solution representation structure

Yarema A. Prykarpatskyy

Pages: 92 - 107

A regular gradient-holonomic approach to studying the Lax type integrability of the Ablowitz–Ladik hierarchy of nonlinear Lax type integrable discrete dynamical systems in the vertex operator representation is presented. The relationship to the Lie-algebraic integrability scheme is analyzed and the connection...

**Research Article**

## Groups of Order Less Than 32 and Their Endomorphism Semigroups

Peeter Puusemp

Pages: 93 - 101

It is proved that among the finite groups of order less than 32 only the tetrahedral group and the binary tetrahedral group are not determined by their endomorphism semigroups in the class of all groups.

**Research Article**

## Essential Spectrum Due to Singularity

Pavel Kurasov, Serguei Naboko

Pages: 93 - 106

It is proven that the essential spectrum of any self-adjoint operator associated with the matrix differential expression

**Research Article**

## Quantization of Soliton Cellular Automata

Demosthenes Ellinas, Elena P. Papadopoulou, Yiannis G. Saridakis

Pages: 93 - 99

A method of quantization of classical soliton cellular automata (QSCA) is put forward that provides a description of their time evolution operator by means of quantum cicuits that involve quantum gates from which the associated Hamiltonian describing a quantum chain model is constructed. The intrinsic...

**Research Article**

## Multipotentialisations and Iterating-Solution Formulae: The Krichever–Novikov Equation

Norbert Euler, Marianna Euler

Pages: 93 - 106

We derive solution-formulae for the Krichever–Novikov equation by a systematic multipotentialisation of the equation. The formulae are achieved due to the connections of the Krichever–Novikov equations to certain symmetry-integrable 3rd-order evolution equations which admit autopotentialisations.

**Research Article**

## A Mean-Field Version of the SSB Model For X-Chromosome Inactivation

Giuseppe Gaeta

Pages: 93 - 103

Nicodemi and Prisco recently proposed a model for X-chromosome inactivation in mammals, explaining this phenomenon in terms of a spontaneous symmetry-breaking mechanism [Phys. Rev. Lett. 99 (2007) 108104]. Here we provide a mean-field version of their model.

**Research Article**

## Nonlocal symmetries and conservation laws of the Sinh-Gordon equation

Xiao-yan Tang, Zu-feng Liang

Pages: 93 - 106

Nonlocal symmetries of the (1+1)-dimensional Sinh-Gordon (ShG) equation are obtained by requiring it, together with its Bäcklund transformation (BT), to be form invariant under the infinitesimal transformation. Naturally, the spectrum parameter in the BT enters the nonlocal symmetries, and thus through...

**Review Article**

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

I.G. Korepanov

Pages: 94 - 119

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

**Research Article**

## Bäcklund Transformation, Lax Pair and Solitons of the (2+1)-dimensional Davey-Stewartson-like Equations with Variable Coefficients for the Electrostatic Wave Packets

Hui-Ping Zhou, Bo Tian, Hui-Xia Mo, Min Li, Pan Wang

Pages: 94 - 105

The (2+1)-dimensional Davey-Stewartson-like equations with variable coefficients have the applications in the ultra-relativistic degenerate dense plasmas and Bose-Einstein condensates. Via the Bell polynomials and symbolic computation, the bilinear form, Bäcklund transformation and Lax pair for such...

**Research Article**

## Bispectrality for deformed CalogeroMoserSutherland systems

Misha Feigin

Pages: 95 - 136

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

**Research Article**

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

Daniel Larsson, Sergei D. Silvestrov

Pages: 95 - 106

This paper is devoted to an extension of Burchnall-Chaundy theory on the inteplay between algebraic geometry and commuting differential operators to the case of q-difference operators.

**Research Article**

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

M.F. Dahl

Pages: 95 - 111

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

**Research Article**

## Interrelations of discrete Painlevé equations through limiting procedures

A. Ramani, B. Grammaticos, T. Tamizhmani

Pages: 95 - 105

We study the discrete Painlevé equations associated to the E7(1) affine Weyl group which can be obtained by the implementation of a special limits of E8(1)-associated equations. This study is motivated by the existence of two E7(1)-associated discrete both having a double ternary dependence in their...

**Research Article**

## Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

J. Chavarriga, I.A. Garcia

Pages: 96 - 105

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G2 cosmologies. By using Darboux's theory in order to study ordinary differential equations...

**Research Article**

## Lie Algebras of Approximate Symmetries

Rafail K. Gazizov

Pages: 96 - 101

Properties of approximate symmetries of equations with a small parameter are discussed. It turns out that approximate symmetries form an approximate Lie algebra. A concept of approximate invariants is introduced and the algorithm of their calculating is proposed.

**Research Article**

## Global Dissipative Solutions of the Generalized Camassa-Holm Equation

Octavian G. Mustafa

Pages: 96 - 115

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

**Research Article**

## New Solvable Nonlinear Matrix Evolution Equations

M. Bruschi

Pages: 97 - 105

We introduce an extension of the factorization-decomposition technique that allows us to manufacture new solvable nonlinear matrix evolution equations. Several examples of such equations are reported.

**Research Article**

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

Andrey Andreytsev

Pages: 98 - 101

Reduction of a nonlinear system of differential equations for spinor field is studied. The ansatzes obtained are shown to correspond to operators of conditional symmetry of these equations.

**Research Article**

## Neumann-Bogolyubov-Rosochatius Oscillatory Dynamical Systems and their Integrability Via Dual Moment Maps. Part I

A. Prykarpatsky, O. Hentosh, M. Kopych, R. Samuliak

Pages: 98 - 113

**Research Article**

## On a canonical nine-body problem. Integrable Ermakov decomposition via reciprocal transformations

Colin Rogers

Pages: 98 - 106

Here, a recently introduced nine-body problem is shown to be decomposable via a novel class of reciprocal transformations into a set of integrable Ermakov systems. This Ermakov decomposition is exploited to construct more general integrable nine-body systems in which the canonical nine-body system is...

**Research Article**

## Steady Internal Water Waves with a Critical Layer Bounded by the Wave Surface

Anca-Voichita Matioc

Pages: 98 - 118

In this paper we construct small amplitude periodic internal waves traveling at the boundary region between two rotational and homogeneous fluids with different densities. Within a period, the waves we obtain have the property that the gradient of the stream function associated to the fluid beneath the...

**Research Article**

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

Yi A. Li

Pages: 99 - 105

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

**Research Article**

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

F. Calogero, J-P Françoise

Pages: 99 - 125

A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot or, in some cases, only periodic solutions. Several examples (ODEs and PDEs) are exhibited.

**Research Article**

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

Takeo Kojima

Pages: 99 - 119

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

**Research Article**

## Brownian Motion on a Smash Line

Demosthenes Ellinas, Ioannis Tsohantjis

Pages: 100 - 105

Brownian motion on a smash line algebra (a smash or braided version of the algebra resulting by tensoring the real line and the generalized paragrassmann line algebras), is constructed by means of its Hopf algebraic structure. Further, statistical moments, non stationary generalizations and its diffusion...

**Research Article**

## The space of initial conditions and the property of an almost good reduction in discrete Painlevé II equations over finite fields

Masataka Kanki, Jun Mada, Tetsuji Tokihiro

Pages: 101 - 109

We investigate the discrete Painlevé equations (dPII and qPII) over finite fields. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. Then we treat them over local fields and observe that they have a property that is similar to...

**Research Article**

## Total Differentiation Under Jet Composition

Maido Rahula, Vitali Retsnoi

Pages: 102 - 109

Total differentiation operators as linear vector fields, their flows, invariants and symmetries form the geometry of jet space. In the jet space the dragging of tensor fields obeys the exponential law. The composition of smooth maps induces a composition of jets in corresponding jet spaces. The prolonged...

**Research Article**

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

Francesco Calogero

Pages: 102 - 112

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

**Research Article**

## Approximate Waiting-Time for a Thin Liquid Drop Spreading under Gravity

E. Momoniat

Pages: 102 - 109

The method of multiple scales is used to introduce a small-time scale into the nolinear diffusion equation modelling the spreading of a thin liquid drop under gravity. The Lie group method is used to analyse the resulting system. An approximate group invariant solution and an approximation to the waiting-time...

**Research Article**

## Symmetry Reduction of Poincaré-Invariant Nonlinear Wave Equations

A.F. Barannyk, Yu.D. Moskalenko

Pages: 102 - 106

Reduction of multidimensional Poincaré-invariant equations to ordinary differential equations and 2-dimensional equations is considered.

**Research Article**

## Graph Expansions and Graphical Enumeration Applied to Semiclassical Propagator Expansions

S.A. Fulling

Pages: 102 - 110

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

**Research Article**

## Symbolic Dynamics and Chaotic Synchronization in Coupled Duffing Oscillators

Acilina Caneco, Clara Gracio, J. Leonel Rocha

Pages: 102 - 111

In this work we discuss the complete synchronization of two identical double-well Duffing oscillators unidirectionally coupled, from the point of view of symbolic dynamics. Working with Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the...

**Research Article**

## On Frequencies of Small Oscillations of Some Dynamical Systems Associated with Root Systems

A.M. Perelomov

Pages: 103 - 109

In the paper by F Calogero and the author [Commun. Math. Phys. 59 (1978), 109