# Journal of Nonlinear Mathematical Physics

1499 articles

**Research Article**

## Representations of the Infinite Unitary Group from Constrained Quantization

N.P. Landsman

Pages: 161 - 180

We attempt to reconstruct the irreducible unitary representations of the Banach Lie group U0(H) of all unitary operators U on a separable Hilbert space H for which U - I is compact, originally found by Kirillov and Ol'shanskii, through constrained quantization of its coadjoint orbits. For this purpose...

**Research Article**

## On Relativistic Mass Spectra of a Two-Particle System

Volodymyr Tretyak, Volodymyr Shpytko

Pages: 161 - 167

A relativistic two-particle system with time-asymmetric scalar and vector interactions in the two-dimensional space-time is considered within the frame of the front form of dynamics using the dynamical symmetry approach. The mass-shell equation may be represented in terms of the nonlinear canonical realization...

**Research Article**

## Geometrical Methods for Equations of Hydrodynamical Type

Joachim Escher, Boris Kolev

Pages: 161 - 178

We describe some recent results for a class of nonlinear hydrodynamical approximation models where the geometric approach gives insight into a variety of aspects. The main contribution concerns analytical results for Euler equations on the diffeomorphism group of the circle for which the inertia operator...

**Research Article**

## On Properties of Elliptic Jacobi Functions and Applications

A. Raouf Chouikha

Pages: 162 - 169

In this paper we are interested in developments of the elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53-57)...

**Research Article**

## Hamiltonian Formalism in Quantum Mechanics

Boris A. Kupershmidt

Pages: 162 - 180

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties...

**Research Article**

## On the propagation of vector ultra-short pulses

Monika Pietrzyk, I. Kanattsikov, Uwe Bandelow

Pages: 162 - 170

A two component vector generalization of the Sch ?afer-Wayne short pulse equation is derived. It describes propagation of ultra-short pulses in optical fibers with Kerr nonlinearity beyond the slowly varying envelope approximation and takes into account the effects of anisotropy and polarization. We...

**Research Article**

## Recursion Operators for KP, mKP and Harry Dym Hierarchies

Jipeng Cheng, Lihong Wang, Jingsong He

Pages: 161 - 178

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev–Petviashvili (KP), modified Kadomtsev–Petviashvili (mKP) and Harry Dym under n-reduction. This shows a new inherent relationship between them. To illustrate...

**Research Article**

## On Geodesic Completeness of Nondegenerate Submanifolds in Semi-Euclidean Spaces

Fazilet Erkekog˜lu

Pages: 161 - 168

In this paper, we study the geodesic completeness of nondegenerate submanifolds in semi-Euclidean spaces by extending the study of Beem and Ehrlich [1] to semi-Euclidean spaces. From the physical point of view, this extend may have a significance that a semi-Euclidean space contains more variety of Lorentzian...

**Research Article**

## Complex crystallographic Coxeter groups and affine root systems

Joseph Bernstein, Ossip Schwarzman

Pages: 163 - 182

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

**Research Article**

## Lie Group Analysis for Multi-Scale Plasma Dynamics

Vladimir F. Kovalev

Pages: 163 - 175

An application of approximate transformation groups to study dynamics of a system with distinct time scales is discussed. The utilization of the Krylov–Bogoliubov–Mitropolsky method of averaging to find solutions of the Lie equations is considered. Physical illustrations from the plasma kinetic theory...

**Research Article**

## Defining Relations of Almost Affine (Hyperbolic) Lie Superalgebras

Sofian Bouarroudj, Pavel Grozman, Dimitry Leites

Pages: 163 - 168

For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators.

**Research Article**

## SO(4)-symmetry of mechanical systems with 3 degrees of freedom

Sofiane Bouarroudj, Semyon E. Konstein

Pages: 162 - 169

We answered an old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra 𝔬(4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie...

**Research Article**

## A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions

P.G. Estévez, J. Prada

Pages: 164 - 179

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation...

**Research Article**

## Ghost Symmetries

Peter J. Olver, Jan A. Sanders, Jing Ping Wang

Pages: 164 - 172

We introduce the notion of a ghost characteristic for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for the charateristics of nonlocal vector fields.

**Research Article**

## Invariant Linear Spaces and Exact Solutions of Nonlinear Evolution Equations

Sergey R. Svirshchevskii

Pages: 164 - 169

The paper presents a survey of some new results concerning the approach to construction of explicit solutions for nonlinear evolution equations du/dt = F[u], proposed in [1, 2].

**Research Article**

## On a class of mappings between Riemannian manifolds

Thomas H. Otway

Pages: 164 - 173

Effects of geometric constraints on a steady flow potential are described by an elliptic- hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality.

**Research Article**

## Partial Noether Operators and First Integrals for a System with two Degrees of Freedom

I. Naeem, Fazal M. Mahomed

Pages: 165 - 178

We construct all partial Noether operators corresponding to a partial agrangian for a system with two degrees of freedom. Then all the first integrals are obtained explicitly by utilizing a Noether-like theorem with the help of the partial Noether operators. We show how the first integrals can be constructed...

**Research Article**

## Conditional Linearizability of Fourth-Order Semi-Linear Ordinary Differential Equations

F. M. Mahomed, Asghar Qadir

Pages: 165 - 178

By the use of geometric methods for linearizing systems of second-order cubically semi-linear ordinary differential equations and the conditional linearizability of third-order quintically semi-linear ordinary differential equations, we extend to the fourth-order by differentiating the third-order conditionally...

**Research Article**

## A Riemann–Hilbert approach to Painlevé IV

Marius van der Put, Jaap Top

Pages: 165 - 177

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevéequation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ℙ1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties...

**Research Article**

## The Hasse-Weil Bound and Integrability Detection in Rational Maps

John A.G. Roberts, Danesh Jogia, Franco Vivaldi

Pages: 166 - 180

We reduce planar measure-preserving rational maps over finite fields, and study their discrete dynamics. We show that application to the orbit analysis over these fields of the Hasse-Weil bound for the number of points on an algebraic curve gives a strong indication of the existence of an integral for...

**Research Article**

## On Integrable Discretization of the Inhomogeneous Volterra Lattice

V.V. Konotop

Pages: 166 - 171

An integrable discretization of an inhomogeneous Volterra lattice is introduced. Some aspects of dynamics of a soliton affected by a random force are discussed.

**Research Article**

## Lump Solutions for PDE's: Algorithmic Construction and Classification

P.G. Estévez, J. Prada

Pages: 166 - 175

In this paper we apply truncated Painleve expansions to the Lax pair of a PDE to derive gauge Backlund transformations of this equation. It allows us to construct an algorithmic method to derive solutions by starting from the simplest one. Actually, we use this method to obtain an infinite set of lump...

**Research Article**

## A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

Agata Bezubik, Agata Dbrowska, Aleksander Strasburger

Pages: 167 - 173

This article summarizes a new, direct approach to the determination of the expansion into spherical harmonics of the exponential ei(x|y) with x, y Rd . It is elementary in the sense that it is based on direct computations with the canonical decomposition of homogeneous polynomials into harmonic components...

**Research Article**

## On the discretization of Laine equations

Kostyantyn Zheltukhin, Natalya Zheltukhina

Pages: 166 - 177

We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such an equation does not exist. It is also shown that all constructed semi-discrete...

**Research Article**

## Geometry of rank 2 distributions with nonzero Wilczynski invariants*

Boris Doubrov, Igor Zelenko

Pages: 166 - 187

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ℝ5 with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric...

**Research Article**

## Point Symmetries of Controlled Systems and Their Applications

Victor I. Lehenkyj

Pages: 168 - 172

A problem of finding point symmetries of controlled systems is discussed, basic theorems and algorithms are formulated. The application to some problems of flight dynamics is suggested.

**Research Article**

## Lagrangians for Dissipative Nonlinear Oscillators: The Method of Jacobi Last Multiplier

M. C. Nucci, K. M. Tamizhmani

Pages: 167 - 178

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several equations, including a class of equations recently studied...

**Research Article**

## Initial-boundary value problem for the two-component nonlinear Schrödinger equation on the half-line

Jian Xu

Pages: 167 - 189

We present a 3×3 Riemann-Hilbert problem formalism for the initial-boundary value problem of the two-component nonlinear Schrödinger (2-NLS) equation on the half-line. And we also get the Dirichlet to Neuemann map through analysising the global relation in this paper.

**Research Article**

## The Relation between a 2D Lotka-Volterra equation and a 2D Toda Lattice

Claire R. Gilson, Jonathan J.C. Nimmo

Pages: 169 - 179

It is shown that the 2-discrete dimensional Lotka-Volterra lattice, the two dmensional Toda lattice equation and the recent 2-discrete dimensional Toda lattice equation of Santini et al can be obtained from a 2-discrete 2-continuous dimensional Lotka-Volterra lattice.

**Research Article**

## The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains

Maxim Chapovalov, Dimitry Leites, Rafael Stekolshchik

Pages: 169 - 215

The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite...

**Research Article**

## Separabilty in HamiltonJacobi Sense in Two Degrees of Freedom and the Appell Hypergeometric Functions

Vladimir Dragovi

Pages: 170 - 177

A huge family of separable potential perturbations of integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid is given through the Appell hypegeometric functions F4 of two variables, leading to an interesting connection between two classical theories: separability in HamiltonJacobi...

**Research Article**

## Quantized W-algebra of sl(2, 1) and Quantum Parafermions of Uq(^sl(2))

Jintai Ding, Boris Feigin

Pages: 170 - 183

In this paper, we establish the connection between the quantized W-algebra of sl(2, 1) and quantum parafermions of Uq(^sl(2)) that a shifted product of the two quantum parafermions of Uq(^sl(2)) generates the quantized W-algebra of sl(2, 1).

**Research Article**

## Generalization of the Equivalence Transformations

S.V. Meleshko

Pages: 170 - 174

This report is devoted to generalization of the equivalence transformations. Let a system of differential equations be given. Almost all systems of differential equations have arbitrary elements: arbitrary functions or arbitrary constants.

**Research Article**

## Commutativity of Pfaffianization and Bäcklund Transformations: The Leznov Lattice

Chun-Xia Li, Jun-Xiao Zhao, Xing-Biao Hu

Pages: 169 - 178

In this paper, we first obtain Wronskian solutions to the Bäcklund transformation of the Leznov lattice and then derive the coupled system for the Bäcklund transformation through Pfaffianization. It is shown the coupled system is nothing but the Bäcklund transformation for the coupled Leznov lattice...

**Research Article**

## Liouvillian integrability of a general Rayleigh-Duffing oscillator

Jaume Giné, Claudia Valls

Pages: 169 - 187

We give a complete description of the Darboux and Liouville integrability of a general Rayleigh-Duffing oscillator through the characterization of its polynomial first integrals, Darboux polynomials and exponential factors.

**Research Article**

## On bi-hamiltonian structure of some integrable systems

A.V. Tsiganov

Pages: 171 - 185

We classify quadratic Poisson structures on so (4) and e(3), which have the same foliations by symplectic leaves as canonical Lie-Poisson tensors. The separated variables for some of the corresponding bi-integrable systems are constructed.

**Research Article**

## Fredholm Property of Operators from 2D String Field Theory

Hai-Long Her

Pages: 170 - 184

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ-instanton equation. Solutions of ζ-instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining...

**Research Article**

## No periodic orbits for the type A Bianchi's systems

Claudio A. Buzzi, Jaume Llibre

Pages: 170 - 179

It is known that the 6 models of Bianchi class A have no periodic solutions. In this article we provide a new, direct, unified and easier proof of this result.

**Research Article**

## Some Fourth-Order Ordinary Differential Equations which Pass the Painlevé Test

Nicolai A. Kudryashov

Pages: 172 - 177

An approach to the Painlevé analysis of fourth-order ordinary differential equations is presented. Some fourth-order ordinary differential equations which pass the Painlevé test are found.

**Research Article**

## On non-Lie ansatzes and new exact solutions of the classical Yang-Mills equations

Renat Zhdanov, Wilhelm Fushchych

Pages: 172 - 181

We ssuggest an effective method for reducing Yang-Mills equations to systems of ordinary differential equations. With the use of this method, we construct wide families of new exact solutions of the Yang-Mills equations. Analysis of the solutions obtained shows that they correspond to conditional symmetry...

**Research Article**

## Exact Solutions of a Nonlinear Diffusion Equation with Absorption and Production

Robert Conte

Pages: 171 - 181

We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy in the framework of a turbulence model with a single equation.

**Research Article**

## On generalized Lax equations of the Lax triple of the BKP and CKP hierarchies

Xiao-Li Wang, Jian-Qin Mei, Min-Li Li, Zhao-wen Yan

Pages: 171 - 182

Based on the Lax triple (Bm, Bn, L) of the BKP and CKP hierarchies, we derive the nonlinear evolution equations from the generalized Lax equation. The solutions of some evolution equations are presented, such as soliton and rational solutions.

**Research Article**

## Some Remarks on Materials with Memory: Heat Conduction and Viscoelasticity

Sandra Carillo

Pages: 173 - 178

Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alterntive to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear...

**Research Article**

## Integrable Systems and Metrics of Constant Curvature

Maxim Pavlov

Pages: 173 - 191

In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with metrics of constant curvature. Kaup-Boussinesq system has three local Hamiltonian structures and one nonlocal...

**Short Communication**

## Mathematical Simulation of Heat Transfer in Relaxing Media

V.M. Bulavatsky, I.I. Yuryk

Pages: 173 - 174

We find a numerically-analytical solution of a boundary problem for the third-order partial differential equation, which describes the mass and heat transfer in active media.

**Research Article**

## Wigner Quantization on the Circle and R+

G. Chadzitaskos, J. Tolar

Pages: 174 - 178

We construct a deformation quantization for two cases of configuration spaces: the multiplicative group of positive real numbers R+ and the circle S1 . In these cases we define the momenta using the Fourier transform. Using the identification of symbols of quantum observables -- real functions on the...

**Research Article**

## A note on Bernoulli polynomials and solitons

Khristo N. Boyadzhiev

Pages: 174 - 178

The dependence on time of the moments of the one-soliton KdV solutions is given by Bernoulli polynomials. Namely, we prove the formula R x n sech 2 (x âˆ’ t) dx = 2 Ï€ n (âˆ’i) n Bn ( 1 2 + t Ï€ i) , expressing the moments of the one-soliton function sech 2 (xâˆ’t) in terms of the Bernoulli polynomials...

**Research Article**

## On Finite-Gap Elliptic Solutions of the KdV Equation

A.M. Korostil'

Pages: 175 - 179

We present a simple and general method for calculation of finitegap elliptic solutions of the KdV equation as an isospectral deformation of Schrödinger potential based on their representation by rational functions of the elliptic Weierstrass functions.

**Research Article**

## Symmetry Group of Vlasov-Maxwell Equations in Plasma Theory

V.F. Kovalev, S.V. Krivenko, V.V. Pustovalov

Pages: 175 - 180

**Research Article**

## Harmonic Maps Between Noncompact Manifolds

Anestis Fotiadis

Pages: 176 - 184

We describe the problem of finding a harmonic map between noncompact manifold. Given some sufficient conditions on the domain, the target and the initial map, we prove the existence of a harmonic map that deforms the given map.

**Research Article**

## On Lie Group Classification of a Scalar Stochastic Differential Equation

Roman Kozlov

Pages: 177 - 187

Lie point symmetry group classification of a scalar stochastic differential equation (SDE) with one-dimensional Brownian motion is presented. First we prove that the admitted symmetry group is at most three-dimensional. Then the classification is carried out with the help of Lie algebra realizations...

**Research Article**

## Subgroup Type Coordinates and the Separationof Variables in Hamilton-Jacobi and Schrödinger Equations

E.G. Kalnins, Z. Thomova, P. Winternitz

Pages: 178 - 208

Separable coordinate systems are introduced in complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically) in terms of subgroup chains. Finally the explicit solutions...

**Research Article**

## Construction of Variable Mass Sine-Gordon and Other Novel Inhomogeneous Quantum Integrable Models

Anjan Kundu

Pages: 178 - 182

The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model,...

**Short Communication**

## A Remark on Rational Isochronous Potentials

Oleg A. Chalykh, Alexander P. Veselov

Pages: 179 - 183

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form U(x) = 1 2 2 x2 or U(x) = 1 8 2 x2 + c2 x-2 .

**Research Article**

## Poisson configuration spaces, von Neumann algebras, and harmonic forms

Alexei Daletskii

Pages: 179 - 184

We give a short review of recent results on L2 -cohomology of infinite configuration spaces equipped with Poisson measures.

**Research Article**

## On sl(2)-equivariant quantizations

S. Bouarroudj, M. Iadh Ayari

Pages: 179 - 187

By computing certain cohomology of Vect(M ) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-pro jective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M ). Contrariwise, for pro jective embeddings sl(2)-equivariant...

**Research Article**

## Symmetry Solutions of a Third-Order Ordinary Differential Equation which Arises from Prandtl Boundary Layer Equations

R. Naz, Fazal M. Mahomed, David P. Mason

Pages: 179 - 191

The similarity solution to Prandtl’s boundary layer equations for two-dimensional and radial flows with vanishing or constant mainstream velocity gives rise to a third-order ordinary differential equation which depends on a parameter ?. For special values of ? the third-order ordinary differential equation...

**Research Article**

## Relating the Bottom Pressure and the Surface Elevation in the Water Wave Problem

B. Deconinck, K. L. Oliveras, V. Vasan

Pages: 179 - 189

An overview is presented of recent progress on the relation between the pressure at the bottom of the flat water bed and the elevation of the free water boundary within the context of the one-dimensional, irrotational water wave problem. We present five different approaches to this problem. All are compared...

**Research Article**

## A Group Classification of a System of Partial Differential Equations Modeling Flow in Collapsible Tubes

M. Molati, F. M. Mahomed, C. Wafo Soh

Pages: 179 - 208

The purpose of this work is to perform group classification of a coupled system of partial differential equations (PDEs) modeling a flow in collapsible tubes. This system of PDEs contains unknown functions of the dependent variables whose forms are specified via the classification with respect to subalgebras...

**Research Article**

## Non-isospectral lattice hierarchies in 2 + 1 dimensions and generalized discrete Painlevé hierarchies

P.R. Gordoa, A. Pickering, Z.N. Zhu

Pages: 180 - 196

In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of discrete equations, which we obtain once again...

**Research Article**

## Soliton Collisions and Ghost Particle Radiation

Hiêú D. Nguyêñ

Pages: 180 - 198

This paper investigates the nature of particle collisions for three-soliton solutions of the Korteweg-de Vries (KdV) equation by describing mathematically the interaction of soliton particles and generation of ghost particle radiation. In particular, it is proven that a collision between any two soliton...

**Research Article**

## Nonlinear Quantum Dynamical Equation for Self-Acting Electron

V.P. Oleinik

Pages: 180 - 189

From the action principle, the quantum dynamical equation is obtained both relativistically and gauge invariant, which is analogous to the Dirac equation and describes behaviour of an arbitrary number of self-acting charged particles. It is noted that solutions of this equation are indicative of the...

**Research Article**

## A New Set of Admitted Transformations for Autonomous Stochastic Ordinary Differential Equations

Sergey V. Meleshko, Eckart Schulz

Pages: 179 - 196

This paper investigates symmetries of autonomous ordinary stochastic differential equations. Change of time includes the stochastic process itself, and is uniquely determined by the transformation of the spatial variable. As a particular feature, the time change by an admitted Lie symmetry group may...

**Letter to Editor**

## A Note on the Gauss Decomposition of the Elliptic Cauchy Matrix

L. Fehér, C. Klimčík, S. Ruijsenaars

Pages: 179 - 182

Explicit formulas for the Gauss decomposition of elliptic Cauchy type matrices are derived in a very simple way. The elliptic Cauchy identity is an immediate corollary.

**Research Article**

## Solutions of the (2 + 1)-Dimensional KP, SK and KK Equations Generated by Gauge Transformations from Nonzero Seeds

Jingsong He, Xiaodong Li

Pages: 179 - 194

By using gauge transformations, we manage to obtain new solutions of (2 + 1)-dimensional Kadomtsev–Petviashvili (KP), Kaup–Kuperschmidt (KK) and Sawada–Kotera (SK) equations from nonzero seeds. For each of the preceding equations, a Galilean type transformation between these solutions u2 and the previously...

**Research Article**

## The Mixed Kuper-Camassa-Holm-Hunter-Saxton Equations

Ling Zhang, Beibei Hu

Pages: 179 - 187

In this paper, a mixed Kuper-CH-HS equation by a Kupershmidt deformation is introduced and its integrable properties are studied. Moreover, that the equation can be viewed as a constraint Hamiltonian flow on the coadjoint orbit of Neveu-Schwarz superalgebra is shown.

**Research Article**

## New Integrable and Linearizable Nonlinear Difference Equations

R. Sahadevan, G. Nagavigneshwari

Pages: 179 - 190

A systematic investigation to derive nonlinear lattice equations governed by partial difference equations (PΔΔE) admitting specific Lax representation is presented. Further it is shown that for a specific value of the parameter the derived nonlinear PΔΔE's can be transformed into a linear PΔΔE's...

**Research Article**

## On Negatons of the Toda Lattice

Cornelia Schiebold

Pages: 181 - 193

Negatons are a solution class with the following characteristic properties: They consist of solitons which are organized in groups. Solitons belonging to the same group are coupled in the sense that they drift apart from each other only logarithmically. The groups themselves rather behave like particles....

**Research Article**

## Reflectionless Analytic Difference Operators III. Hilbert Space Aspects

S.N.M. Ruijsenaars

Pages: 181 - 209

In the previous two parts of this series of papers, we introduced and studied a large class of analytic difference operators admitting reflectionless eigenfunctions, focusing on algebraic and function-theoretic features in the first part, and on connections with solitons in the second one. In this third...

**Research Article**

## Exactly Integrable Systems Connected to Semisimple Algebras of Second Rank A2, B2, C2, G2

A.N. Leznov

Pages: 181 - 197

Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental representations of the corresponding semisimple groups.

**Research Article**

## r-Matrix for the Restricted KdV Flows with the Neumann Constraints

Ruguang Zhou

Pages: 181 - 189

Under the Neumann constraints, each equation of the KdV hierarchy is decomposed into two finite dimensional systems, including the well-known Neumann model. Like in the case of the Bargmann constraint, the explicit Lax representations are deduced from the adjoint representation of the auxiliary spectral...

**Research Article**

## Parasupersymmetric Quantum Mechanics with Arbitrary p and N

Zoya Symenoh

Pages: 181 - 185

The generalization of parasupersymmetric quantum mechanics generated by an arbitrary number of parasupercharges and characterized by an arbitrary order of paraquantization is given. The relations for parasuperpotentials are obtained. It is shown that parasuperpotentials can be explicitly expressed via...

**Research Article**

## Lie symmetries and nonlocally related systems of the continuous and discrete dispersive long waves system by geometric approach

Shou-Fu Tian, Tian-Tian Zhang, Pan-Li Ma, Xing-Yong Zhang

Pages: 180 - 193

By using the extended Harrison and Estabrook's differential forms approach, in this paper, we investigate the Lie symmetries of the continuous and discrete dispersive long waves system, respectively. Based on this method, two closed ideals written in terms of a set of differential forms are constructed...

**Research Article**

## Elliptic even finite-gap potentials and spectra of the Schrödinger operator

A.M. Korostil'

Pages: 182 - 192

A simple method for calculating finite-gap elliptic potentials and corresponding spectra of the one-dimensional Schrödinger operator which is based on a general representation of the potentials in a form of rational functions of the Weierstrass function and trace formulae is proposed. It is shown that...

**Research Article**

## On an integrable differential-difference equation with a source

Gegenhasi, Xing-Biao Hu

Pages: 183 - 192

**Research Article**

## Hidden N = (2|2) Supersymmetry of the N = (1|1) Supersymmetric Toda Lattice Hierarchy

O. Lechtenfeld, A. Sorin

Pages: 183 - 195

An N = (2|2) superfield formulation of the N = (1|1) supersymmetric Toda lattice hierarchy is proposed, and its five real forms are presented.

**Research Article**

## Nonlinear Renormalization Group Flow and Optimization

Sen-Ben Liao

Pages: 183 - 187

Renormalization group flow equations for scalar 4 are generated using smooth smearing functions. Numerical results for the critical exponent in d = 3 are caculated by polynomial truncation of the blocked potential. It is shown that the covergence of with the order of truncation can be improved by fine...

**Research Article**

## A Systematic Method of Finding Linearizing Transformations for Nonlinear Ordinary Differential Equations I: Scalar Case

V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan

Pages: 182 - 202

In this paper we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and...

**Research Article**

## Symmetries of Kolmogorov Backward Equation

Roman Kozlov

Pages: 182 - 193

The note provides the relation between symmetries and first integrals of Itô stochastic differential equations and symmetries of the associated Kolmogorov Backward Equation (KBE). Relation between the symmetries of the KBE and the symmetries of the Kolmogorov forward equation is also given.

**Research Article**

## Extended Prelle-Singer Method and Integrability/Solvability of a Class of Nonlinear nth Order Ordinary Differential Equations

V.K. Chandrasekar, M. Senthilvelan, M. Laksmanan

Pages: 184 - 201

We discuss a method of solving nth order scalar ordinary differential equations by extending the ideas based on the Prelle-Singer (PS) procedure for second order ordnary differential equations. We also introduce a novel way of generating additional integrals of motion from a single integral. We illustrate...

**Research Article**

## Generalized Operator Yang-Baxter Equations, Integrable ODEs and Nonassociative Algebras

I.Z. Golubchik, V.V. Sokolov

Pages: 184 - 197

Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative...

**Research Article**

## The Lorenz System has a Global Repeller at Infinity

Harry Gingold, Daniel Solomon

Pages: 183 - 189

It is well known that the celebrated Lorenz system has an attractor such that every orbit ends inside a certain ellipsoid in forward time. We complement this result by a new phenomenon and by a new interpretation. We show that “infinity” is a global repeller for a set of parameters wider than that usually...

**Research Article**

## The determinant representation of an *N*-fold Darboux transformation for the short pulse equation

Shuzhi Liu, Lihong Wang, Wei Liu, Deqin Qiu, Jingsong He

Pages: 183 - 194

We present an explicit representation of an N-fold Darboux transformation T̃N for the short pulse equation, by the determinants of the eigenfunctions of its Lax pair. In the course of the derivation of T̃N, we show that the quasi-determinant is avoidable, and it is contrast to a recent paper (J. Phys....

**Research Article**

## Canonical Analysis of Symmetry Enhancement with Gauged Grassmannian Model

Sang-Ok Hahn, Phillial Oh, Cheonsoo Park, Sunyoung Shin

Pages: 185 - 190

We study the Hamiltonian structure of the gauge symmetry breaking and enhancment. After giving a general discussion of these phenomena in terms of the constrained phase space, we perform a canonical analysis of the Grassmannian nonlinear sigma model coupled with Chern-Simons term, which contains a free...

**Research Article**

## Nonclassical Potential System Approach for a Nonlinear Diffusion Equation

M.L. Gandarias, M.S. Bruzon

Pages: 185 - 196

In this paper we consider a class of equations that model diffusion. For some of these equations nonclassical potential symmetries are derived by using a modified system approach. These symmetries allow us to increase the number of exact known solutions. These solutions are unobtainable from classical...

**Research Article**

## Second-Order Differential Invariants for Some Extensions of the Poincaré Group and Invariant Equations

Irina Yehorchenko

Pages: 186 - 195

It is well-known that symmetry properties are extremely important for choosing differential equations which can be suitable for description of real physical processes. We present functional bases of second-order differential invariants for various representations of some extensions of the Poincaré group...

**Research Article**

## Solitary waves in helicoidal models of DNA dynamics

Giuseppe Gaeta, Laura Venier

Pages: 186 - 204

We analyze travelling solitary wave solutions in the Barbi-Cocco-Peyrard and in a simplified version of the Cocco-Monasson models of nonlinear DNA dynamics. We identify conditions to be satisfied by parameters for such solutions to exist, and provide first order ODEs whose solutions give the required...

**Research Article**

## The Lie symmetry group of the general Liénard-type equation

Ágota Figula, Gábor Horváth, Tamás Milkovszki, Zoltán Muzsnay

Pages: 185 - 198

We consider the general Liénard-type equation u¨=∑k=0nfku˙k for n ≥ 4. This equation naturally admits the Lie symmetry ∂∂t. We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions f0,..., fn. Moreover, we give an equivalent...

**Research Article**

## Torsional Travelling Waves in DNA

Tomasz Lipniacki

Pages: 188 - 194

The simple model of the non-linear DNA dynamics [4] is pursued in order to study the local untwisting of DNA double helix. It is shown how the advancing RNA polymerase may force the motion of the torsional solitary wave along DNA.

**Research Article**

## A multidimensional superposition principle: numerical simulation and analysis of soliton invariant manifolds I

Alexander A. Alexeyev

Pages: 188 - 229

In the framework of a multidimensional superposition principle involving an analytical approach to nonlinear PDEs, a numerical technique for the analysis of soliton invari- ant manifolds is developed. This experimental methodology is based on the use of computer simulation data of solitonâ€“perturbation...

**Research Article**

## Coisotropic quasi-periodic motions for a constrained system of rigid bodies

Ihor Parasyuk

Pages: 189 - 201

We consider a constrained system of four rigid bodies located in axisymmetric potential and gyroscopic force fields and interacting by means of angular velocities. We describe an integrable case (not in Liouville sence!) when 12-dimensional phase space of the above system is fibered by the coisotropic...

**Research Article**

## Appell Bases on Sequence Spaces

M. Maldonado, J. Prada, M. J. Senosiain

Pages: 189 - 194

We study conditions for a sequence of Appell polynomials to be a basis on a sequence space.

**Research Article**

## Generalized Solvable Structures and First Integrals for ODEs Admitting an 𝔰𝔩(2, ℝ) Symmetry Algebra

Paola Morando, Concepción Muriel, Adrián Ruiz

Pages: 188 - 201

The notion of solvable structure is generalized in order to exploit the presence of an 𝔰𝔩(2, ℝ) algebra of symmetries for a kth-order ordinary differential equation ℰ with k > 3. In this setting, the knowledge of a generalized solvable structure for ℰ allows us to reduce ℰ to a family of second-order...

**Research Article**

## Nonlocal symmetries of Plebański’s second heavenly equation

Aleksandra Lelito, Oleg I. Morozov

Pages: 188 - 197

We study nonlocal symmetries of Plebański’s second heavenly equation in an infinite-dimensional covering associated to a Lax pair with a non-removable spectral parameter. We show that all local symmetries of the equation admit lifts to full-fledged nonlocal symmetries in the infinite-dimensional covering....

**Research Article**

## Non-Abelian Lie algebroids over jet spaces

Arthemy V. Kiselev, Andrey O. Krutov

Pages: 188 - 213

We associate Hamiltonian homological evolutionary vector fields – which are the non-Abelian variational Lie algebroids’ differentials – with Lie algebra-valued zero-curvature representations for partial differential equations.

**Research Article**

## Lie Symmetries, Kac-Moody-Virasoro Algebras and Integrability of Certain (2+1)-Dimensional Nonlinear Evolution Equations

M. Lakshmanan, M. Senthil Velan

Pages: 190 - 211

In this paper we study Lie symmetries, Kac-Moody-Virasoro algebras, similarity reductions and particular solutions of two different recently introduced (2+1)-dimensional nonlinear evolution equations, namely (i) (2+1)-dimensional breaking soliton equation and (ii) (2+1)-dimensional nonlinear Schrödinger...

**Research Article**

## States of a Charged Particle with a Tensor-Like Mass in External Constant Magnetic Field

A.A. Borghardt, D.Ya. Karpenko, D.V. Kashkakha

Pages: 190 - 194

Solutions of the Schrödinger equation for a particle with a tensor-like mass are considered. It is shown that the problem of determination of the coherent states in this case is reduced to integration of the nonlinear system of ordinary differential equations.

**Short Communication**

## On the complexified affine metaplectic representation

Ole Rask

Pages: 191 - 193

We study exponentiability of the infinite polynomials with maximal degree 2 of cration and annihilation operators, which give a Fock Space-representation of the coplexification of the affine symplectic group.

**Research Article**

## Novel *solvable* many-body problems

Oksana Bihun, Francesco Calogero

Pages: 190 - 212

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type (“acceleration equals forces”) which determine the motion of points in the complex plane. These models are solvable, namely their conﬁguration at any time can...