# Journal of Nonlinear Mathematical Physics

1499 articles

**Corrigendum**

## Corrigendum

### Some remarks on materials with memory: heat conduction and viscoelasticity

Sandra Carillo

Pages: i - iii

**Research Article**

## Preface to the Special Issue of Jnmp in Memory of F. A. Berezin

Dimitry Leites

Pages: v - viii

**Book Review**

## Book Review by P A Clarkson

P.A. Clarkson

Pages: 0 - 0

Peter E Hydon: Symmetry Methods for Differential Equations: A Beginner's Guide Cambridge Texts in Applied Mathematics, Cambridge University Press, 2000.

**Book Review**

## Book Reviews by F Calogero

F. Calogero

Pages: 0 - 0

Four books published by Birkhäuser are reviewed.

**Book Review**

## Book Reviews by F Calogero

F. Calogero

Pages: 0 - 0

Seven books published by Birkhäuser are reviewed.

**Book Review**

## Book Review by B A Kupershmidt

B.A. Kupershmidt

Pages: 0 - 0

Five books are reviewed, namely Bruce C Berndt: Ramanujan's Notebooks. Part I. (With a foreword by S Chadrasekhar). Springer-Verlag, New York Berlin, 1985. 357 pages. --: Ramanujan's Notebooks. Part II. Springer-Verlag, New York Berlin, 1989. 359 pages. --: Ramanujan's Notebooks. Part III. Springer-Verlag,...

**Book Review**

## Book Reviews by F Calogero

F. Calogero

Pages: 0 - 0

Three books are reviewed, namely Masao Nagasawa: Schroedinger Equations and Diffusion Theory. Birkhaeuser, Basel Boston Berlin, 1993. 332 pages. David Wick (with a mathematical appendix by William Farris): The Infamous Bounary - Seven Decades of Controversy in Quantum Physics. Birkhaeuser. Boston Basel...

**Book Review**

## Book Review by P G L Leach

P.G.L. Leach

Pages: 0 - 0

N H Ibragimov: Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley, New York, 1999, 347 pages.

**Research Article**

## On a graded q-differential algebra

Viktor Abramov

Pages: 1 - 8

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

**Research Article**

## A Note on Surface Profiles for Symmetric Gravity Waves with Vorticity

Mats Ehrnström

Pages: 1 - 8

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

**Research Article**

## Asymptotic behavior of discrete holomorphic maps zc and log(z)

Sergey I. Agafonov

Pages: 1 - 14

It is shown that discrete analogs of zc and log(z), defined via particular "integrable" circle patterns, have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painlevé-II equations, asymptotics of these solutions providing...

**Research Article**

## A Chorin-Type Formula for Solutions to a Closure Model for the von K´arm´anHowarth Equation 1

V.N. Grebenev, M. Oberlack

Pages: 1 - 9

The article is devoted to studying the Millionshtchikov closure model (a particular case of a model introduced by Oberlack [14]) for isotropic turbulence dynamics which appears in the context of the theory of the von K´arm´an-Howarth equation. We write the model in an abstract form that enables us to...

**Research Article**

## On a "Quasi" Integrable Discrete Eckhaus Equation

M.J. Ablowitz, C.D. Ahrens

Pages: 1 - 12

In this paper, a discrete version of the Eckhaus equation is introduced. The discretiztion is obtained by considering a discrete analog of the transformation taking the cotinuous Eckhaus equation to the continuous linear, free Schrödinger equation. The resulting discrete Eckhaus equation is a nonlinear...

**Review Article**

## So. . . what was the question?

Gérard G. Emch

Pages: 1 - 8

An overview of the lectures at the 2002 Bialowiea Workshop is presented. The symbol* after a proper name indicates that a copy of the corresponding contribution to the proceedings was communicated to the author of this summary.

**Research Article**

## On the Quantization of Yet Another Two Nonlinear Harmonic Oscillators

Francesco Calogero

Pages: 1 - 6

In two previous papers the quantization was discussed of three one-degree-of-freedom Hamiltonians featuring a constant c, the value of which does not influence at all the corresponding classical dynamics (which is characterized by isochronous solutions, all of them periodic with period 2: "nonlinear...

**Research Article**

## Comparisons Between Vector and Matrix Padé Approximants

Claude Brezinski

Pages: 1 - 12

In this paper, we compare the degrees and the orders of approximation of vector and matrix Padé approximants for series with matrix coefficients. It is shown that, in this respect, vector Padé approximants have better properties. Then, matrixvector Padé approximants are defined and constructed. Finally,...

**Research Article**

## NavierStokes Equations with Nonhomogeneous Dirichlet Data

Herbert Amann

Pages: 1 - 11

We discuss the solvability of the time-dependent incompressible NavierStokes equtions with nonhomogeneous Dirichlet data in spaces of low regularity.

**Research Article**

## Noncentral Extensions as Anomalies in Classical Dynamical Systems

Jorge E. Solomin, Marcela Zuccalli

Pages: 1 - 9

A two cocycle is associated to any action of a Lie group on a symplectic manifold. This allows to enlarge the concept of anomaly in classical dynamical systems considered by F Toppan in [J. Nonlinear Math. Phys. 8, Nr. 3 (2001), 518533] so as to encompass some extensions of Lie algebras related to noncanonical...

**Research Article**

## Hidden Symmetries, First Integralsvand Reduction of Order of Nonlinear Ordinary Differential Equations

Barbara Abraham-Shrauner

Pages: 1 - 9

The reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples....

**Research Article**

## From Bi-Hamiltonian Geometry to Separation of Variables: Stationary Harry-Dym and the KdV Dressing Chain

Maciej Blaszak

Pages: 1 - 13

Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordnates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.

**Research Article**

## Kovalevski Exponents and Integrability Properties in Class A Homogeneous Cosmological Models

Marek Szydłowski, Marek Biesiada

Pages: 1 - 10

Qualitative approach to homogeneous anisotropic Bianchi class A models in terms of dynamical systems reveals a hierarchy of invariant manifolds. By calculating the Kovalevski Exponents according to Adler - van Moerbecke method we discuss how algebraic integrability property is distributed in this class...

**Research Article**

## Complex Angle Variables for Constrained Integrable Hamiltonian Systems

S. Abenda, Yu Fedorov

Pages: 1 - 4

We propose Dirac formalism for constraint Hamiltonian systems as an useful tool for the algebro-geometrical and dynamical characterizations of a class of integrable systems, the so called hyperelliptically separable systems. As a model example, we apply it to the classical geodesic flow on an ellipsoid.

**Research Article**

## UV Manifold and Integrable Systems in Spaces of Arbitrary Dimension

A.N. Leznov

Pages: 1 - 7

The 2n dimensional manifold with two mutually commutative operators of differetiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.

**Research Article**

## Lax Pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2 + 1)-Dimensional Equation

Song-Ju Yu, Kouichi Toda

Pages: 1 - 13

We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2 + 1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional...

**Research Article**

## Solutions of WDVV Equations in Seiberg-Witten Theory from Root Systems

R. Martini, P.K.H. Gragert

Pages: 1 - 4

We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.

**Research Article**

## On the Moyal Quantized BKP Type Hierarchies

Dolan Chapa Sen, A. Roy Chowdhury

Pages: 1 - 7

Quantization of BKP type equations are done through the Moyal bracket and the formalism of pseudo-differential operators. It is shown that a variant of the dressing operator can also be constructed for such quantized systems.

**Research Article**

## Integrability of the Perturbed KdV Equation for Convecting Fluids: Symmetry Analysis and Solutions

J.M. Cerveró, O. Zurrón

Pages: 1 - 23

As an example of how to deal with nonintegrable systems, the nonlinear partial differential equation which describes the evolution of long surface waves in a convecting fluid ut + (uxxx + 6uux) + 5uux + (uxxx + 6uux)x = 0, is fully analyzed, including symmetries (nonclassical and contact transformatons),...

**Research Article**

## Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves

David Henry

Pages: 1 - 7

We show that within the framework of linear theory the particle paths in a periodic gravity-capillary or pure capillary deep-water wave are not closed.

**Research Article**

## Harmonic maps between quaternionic Kahler manifolds

S. Ianuş, R. Mazzocco, G.E. Vilcu

Pages: 1 - 8

**Research Article**

## Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations

Rossen I. Ivanov

Pages: 1 - 12

The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H 1 and H 1 right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a...

**Research Article**

## Nonlocal Extensions of Similarity Methods

George Bluman

Pages: 1 - 24

Similarity methods include the calculation and use of symmetries and conservation laws for a given partial differential equation (PDE). There exists a variety of software to calculate and use local symmetries and local conservation laws. However, it is often the case that a given PDE admits no useful...

**Research Article**

## Quantum Integrability of the Dynamics on a Group Manifold

V. Aldaya, M. Calixto, J. Guerrero, F.F. Lopez-Ruiz

Pages: 1 - 12

We study the dynamics of a particle moving on the SU(2) group manifold. An exact quantization of this system is accomplished by finding the unitary and irreducible representations of a finite-dimensional Lie subalgebra of the whole Poisson algebra in phase space. In fact, the basic position and momentum...

**Research Article**

## An Invertible Transformation and Some of its Applications

M. Bruschi, F. Calogero, F. Leyvraz, M. Sommacal

Pages: 1 - 31

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained...

**Research Article**

## Bosonization Method for Second Super Quantization

Alexander Dynin

Pages: 1 - 13

A boson-fermion correspondence allows an analytic definition of functional super derivative, in particular, and a bosonic functional calculus, in general, on Bargmann Gelfand triples for the second super quantization. A Feynman integral for the super transformation matrix elements in terms of bosonic...

**Research Article**

## The Shape of Soliton-Like Solutions of a Higher-Order Kdv Equation Describing Water Waves

Kostis Andriopoulos, Tassos Bountis, K. Van Der Weele, Liana Tsigaridi

Pages: 1 - 12

We study the solitary wave solutions of a non-integrable generalized KdV equation proposed by Fokas [A. S. Fokas, Physica D87, 145 (1995)], aiming to describe unidirectional waves in shallow water with greater accuracy than the standard KdV equation. This generalized equation includes higher-order terms...

**Research Article**

## Local and Nonlocal Symmetries in Mathematical Physics

Norbert Euler, Enrique G. Reyes

Pages: 1 - 2

**Research Article**

## Preface to Special Issue on the Geometry of the Painlevé equations

Nalini Joshi, Masatoshi Noumi, Hidetaka Sakai, Claude M. Viallet

Pages: 1 - 2

**Research Article**

## Variational Derivation of the Green–Naghdi Shallow-Water Equations

Delia Ionescu-Kruse

Pages: 1 - 12

We consider the two-dimensional irrotational water-wave problem with a free surface and a flat bottom. In the shallow-water regime and without smallness assumption on the wave amplitude we derive, by a variational approach in the Lagrangian formalism, the Green–Naghdi equations (1.1). The second equation...

**Research Article**

## The AKNS Hierarchy Revisited: A Vertex Operator Approach and its Lie-Algebraic Structure

Denis Blackmore, Anatoliy K. Prykarpatsky

Pages: 1 - 15

A novel approach — based upon vertex operator representation — is devised to study the AKNS hierarchy. It is shown that this method reveals the remarkable properties of the AKNS hierarchy in relatively simple, rather natural and particularly effective ways. In addition, the connection of this vertex...

**Research Article**

## Liouvillian Integrability of a Modified Michaelis-Menten Equation

Claudia Valls

Pages: 1 - 8

In this work we consider the modified Michaelis-Menten equation in biochemistry
x˙=-a(E-y)x+by, y˙=a(E-y)x-(b+r)y, z˙=ry.
It models the enzyme kinetics. We contribute to the understanding of its global dynamics, or more precisely, to the topological structure of its orbits by studying the integrability...

**Letter to Editor**

## On Non-Commutative Integrable Burgers Equations

Metin Gürses, Atalay Karasu, Refik Turhan

Pages: 1 - 6

We construct the recursion operators for the non-commutative Burgers equations using their Lax operators. We investigate the existence of any integrable mixed version of left- and right-handed Burgers equations on higher symmetry grounds.

**Letter to Editor**

## The Periodic *μ*-*b*-Equation and Euler Equations on the Circle

Martin Kohlmann

Pages: 1 - 8

In this paper, we study the μ-variant of the periodic b-equation and show that this equation can be realized as a metric Euler equation on the Lie group Diff∞(𝕊) if and only if b = 2 (for which it becomes the μ-Camassa–Holm equation). In this case, the inertia operator generating the metric on Diff∞(𝕊)...

**Letter to Editor**

## Some Examples of Algebraic Geodesics on Quadrics

A. M. Perelomov

Pages: 1 - 5

In this note we give the conditions for the existence of algebraic geodesics on some two-dimensional quadrics, namely, on hyperbolic paraboloids and elliptic paraboloids. It appears that in some cases, such geodesics are the rational space curves.

**Research Article**

## Nurowski’s Conformal Class of a Maximally Symmetric (2,3,5)-Distribution and its Ricci-flat Representatives

Matthew Randall

Pages: 1 - 13

We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters k = 2 and k = 3 naturally show up in the conformal rescaling that takes a representative metric in Nurowski’s conformal class associated to a maximally symmetric (2,3,5)-distribution...

**Letter to Editor**

## Euler’s triangle and the decomposition of tensor powers of the adjoint 𝔰𝔩(2)-module

Askold M. Perelomov

Pages: 1 - 6

By considering a relation between Euler’s trinomial problem and the problem of decomposing tensor powers of the adjoint 𝔰𝔩(2)-module I derive some new results for both problems, as announced in arXiv:1902.08065.

**Research Article**

## A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Xianguo Geng, Yunyun Zhai, Bo Xue, Jiao Wei

Pages: 1 - 23

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which...

**Research Article**

## A Multidimensional Superposition Principle: Classical Solitons IV

Alexander A. Alexeyev

Pages: 1 - 33

This article continues the series of the works of 1998–2007 years devoted to the Multidimensional Superposition Principle, the concept easily explaining both classical soliton and more complex wave interactions in nonlinear PDEs and allowing one, in particular, to construct the general Superposition...

**Letter to Editor**

*Integrable Hamiltonian N*-body problems of goldfish type featuring *N arbitrary functions*

Francesco Calogero

Pages: 1 - 6

A simple application of a neat formula relating the time evolution of the N zeros of a (monic) time-dependent polynomial of degree N in the complex variable w to the time evolution of its N coefficients allows to identify integrable Hamiltonian N-body problems in the plane featuring N arbitrary functions,...

**Research Article**

## Automorphisms of the q-deformed algebra su_{q}(1, 1) and d-Orthogonal polynomials of q-Meixner type

Ali Zaghouani

Pages: 1 - 20

Starting from an operator given as a product of q-exponential functions in irreducible representations of the positive discrete series of the q-deformed algebra suq(1, 1), we express the associated matrix elements in terms of d-orthogonal polynomials. An algebraic setting allows to establish some properties...

**Research Article**

## New solutions with peakon creation in the Camassa–Holm and Novikov equations

Marcus Kardell

Pages: 1 - 16

In this article we study a new kind of unbounded solutions to the Novikov equation, found via a Lie symmetry analysis. These solutions exhibit peakon creation, i.e., these solutions are smooth up until a certain finite time, at which a peak is created. We show that the functions are still weak solutions...

**Research Article**

## Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian

Sergi Simon

Pages: 1 - 16

This is an example of application of Ziglin-Morales-Ramis algebraic studies in Hamiltonian integrability, more specifically the result by Morales, Ramis and Simó on higher-order variational equations, to the well-known Friedmann-Robertson-Walker cosmological model. A previous paper by the author formalises...

**Research Article**

## Non-linear Schrödinger Equations, Separation and Symmetry

George Svetlichny

Pages: 2 - 26

We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to...

**Research Article**

## Solutions of the buoyancy-drag equation with a time-dependent acceleration

Serge E. Bouquet, Robert Conte, Vincent Kelsch, Fabien Louvet

Pages: 3 - 17

We perform the analytic study of the buoyancy-drag equation with a time-dependent acceleration γ(t) by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of γ(t) define a time-dependent Hamiltonian from which the buoyancy-drag...

**Research Article**

## Bäcklund transformations for certain rational solutions of Painlevé VI

Johan van de Leur, Henrik Aratyn

Pages: 3 - 16

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial...

**Research Article**

## Taming Spatiotemporal Chaos by Impurities in the Parametrically Driven Damped Nonlinear Schrödinger Equation

N.V. Alexeeva, I.V. Barashenkov, G.P. Tsironis

Pages: 5 - 12

Solitons of the parametrically driven, damped nonlinear Schrödinger equation become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the chaos can be suppressed by introducing localized inhomogeneities in the parameters of the equation. The pinning of the...

**Research Article**

## Neumann and Bargmann Systems Associated with an Extension of the Coupled KdV Hierarchy

Zhimin Jiang

Pages: 5 - 12

An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the hierarchy is established by using the trace identity. The isospectral problem is nonlinearized as to be finite-dimensional completely integrable...

**Research Article**

## Application of the theory of separability of second-order differential equations

W. Sarlet, F. Cantrijn, E. Martínez

Pages: 5 - 24

**Research Article**

## Gauge Transformations for a Family of Nonlinear Schrödinger Equations

Gerald A. Goldin

Pages: 6 - 11

An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equations introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled...

**Research Article**

## µ-Holomorphic Projective Connections and Conformal Covariance

Mohamed Kachkachi

Pages: 7 - 12

At the quantum level of a bidimensional conformal model, the conformal symmtry is broken by the diffeomorphism anomaly and the conformal covariance is not maintained. Here we interpret geometrically this conformal covariance as an exact holomorphy condition on a two-dimensional Riemann surface on which...

**Letter to Editor**

## On the Geodesic Flow on the Group of Diffeomorphisms of the Circle with a Fractional Sobolev Right-Invariant Metric

Marcus Wunsch

Pages: 7 - 11

We show that the geodesic flow on the infinite-dimensional group of diffeomorphisms of the circle, endowed with a fractional Sobolev metric at the identity, is described by the generalized Constantin–Lax–Majda equation with parameter a=−12.

**Research Article**

## Bright *N*-Solitons For The Intermediate Nonlinear Schrödinger Equation

Yohei Tutiya

Pages: 7 - 23

A previously unknown bright N-soliton solution for an intermediate nonlinear Schrödinger equation of focusing type is presented. This equation is constructed as a reduction of an integrable system related to a Sato equation of a 2-component KP hierarchy for certain differential-difference dispersion...

**Letter to Editor**

## Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

S.E. Konstein, I.V. Tyutin

Pages: 7 - 11

If G is a finite Coxeter group, then symplectic reflection algebra H := H1,η (G) has Lie algebra 𝔰𝔩2 of inner derivations and can be decomposed under spin: H = H0 ⊕ H1/2 ⊕ H1 ⊕ H3/2 ⊕ ... We show that if the ideals ℐi (i = 1,2) of all the vectors from the kernel of degenerate bilinear forms Bi(x,y)...

**Research Article**

## Miura and auto-Backlund transformations for the *q*-deformed KP and *q*-deformed modified KP hierarchies

Jipeng Cheng

Pages: 7 - 19

The Miura and anti-Miura transformations between the q-deformed KP and the q-deformed modified KP hierarchies are investigated in this paper. Then the auto-Backlund transformations for the q-deformed KP and q-deformed modified KP hierarchies are constructed through the combinations of the Miura and anti-Miura...

**Research Article**

## Correctors for Some Nonlinear Monotone Operators

Johan Byström

Pages: 8 - 30

In this paper we study homogenization of quasi-linear partial differential equations of the form -div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. Here the sequence (h) tends to 0 as h and the map a (x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions....

**Research Article**

## Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation

M.L. Gandarias, M.S. Bruzon

Pages: 8 - 12

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the...

**Research Article**

## An invariant p-adic q-integral associated with q-Euler numbers and polynomials

Ismail Naci Cangül, Veli Kurt, Yilmaz Simsek, Hong Kyung Pak, Seog-Hoon Rim

Pages: 8 - 14

The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using...

**Research Article**

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

V. Abramov, O. Liivapuu

Pages: 9 - 20

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

**Research Article**

## A Note on q-Bernoulli Numbers and Polynomials

A.S. Hegazi, M. Mansour

Pages: 9 - 18

In this paper, we define a new q-analogy of the Bernoulli polynomials and the Bernoulli numbers and we deduced some important relations of them. Also, we dduced a q-analogy of the Euler-Maclaurin formulas. Finally, we present a relation between the q-gamma function and the q-Bernoulli polynomials.

**Research Article**

## Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

Harald Grosse, Raimar Wulkenhaar

Pages: 9 - 20

We first review regularization methods based on matrix geometry which provide an ultraviolet cut-off for scalar fields respecting the symmetries. Sections of bundles over the sphere can be quantized, too. This procedure even allows to regularize supesymmetry without violating it. Recently, this work...

**Research Article**

## Noetherian first integrals

P.G.L. Leach, G.P. Flessas

Pages: 9 - 21

From time to time one finds claims in the literature that first integrals/invariants of Lagrangian systems are nonnoetherian. Such claims diminish the contribution of Noether in the topic of integrability. We provide an explicit demonstration of noethe- rian symmetries associated with the integrals which...

**Research Article**

## Convergence to the Time Average by Stochastic Regularization

Olga Bernardi, Franco Cardin, Massimiliano Guzzo

Pages: 9 - 27

In Ergodic Theory it is natural to consider the pointwise convergence of finite time averages of functions with respect to the flow of dynamical systems. Since the pointwise convergence is too weak for applications to Hamiltonian Perturbation Theory, requiring differentiability, we first introduce regularized...

**Research Article**

## Non-Isospectral 1+1 Hierarchies Arising from a Camassa Holm Hierarchy in 2+1 Dimensions

P. G. Estévez, J. D. Lejarreta, C. Sardón

Pages: 9 - 28

The non-isospectral problem (Lax pair) associated with a hierarchy in 2 + 1 dimensions that generalizes the well known Camassa–Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries...

**Short Communication**

## A Note on the Degasperis-Procesi Equation

Octavian G. Mustafa

Pages: 10 - 14

We prove that smooth solutions of the Degasperis-Procesi equation have infinite proagation speed: they loose instantly the property of having compact support.

**Research Article**

## On the Cauchy Problem for a Nonlinearly Dispersive Wave Equation

Zhaoyang Yin

Pages: 10 - 15

We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite time. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate...

**Research Article**

## The Economy of Complete Symmetry Groups for Linear Higher Dimensional Systems

K. Andriopoulos, P.G.L. Leach

Pages: 10 - 23

The complete symmetry groups of systems of linear second order ordinary differential equations are considered in the context of the simple harmonic oscillator. One finds that in general the representation of the complete symmetry group is not unique and in the particular case of a four-dimensional system...

**Research Article**

## The Matrix KadomtsevPetviashvili Equation as a Source of Integrable Nonlinear Equations

Attilio Maccari

Pages: 11 - 20

A new integrable class of DaveyStewartson type systems of nonlinear partial diffrential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev Petviashvili equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling....

**Research Article**

## The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

Nicolas Burq, Patrick Gérard, Nikolay Tzvetkov

Pages: 12 - 27

We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbirary surfaces and three-dimensional manifolds, generalizing...

**Research Article**

## Proper-Time Relativistic Dynamics and the Fushchych-Shtelen Transformation

Tepper L. Gill, James Lindesay, M.F. Mahmood, W.W. Zachary

Pages: 12 - 27

We report on a new formulation of classical relativistic Hamiltonian mechanics which is based on a proper-time implementation of special relativity using a transformation from observer proper-time, which is not invariant, to system proper-time which is a canonical contact transformation on extended phase-space....

**Research Article**

## Remarks on the mass spectrum of two-dimensional Toda lattice of *E*_{8} type

Askold M. Perelomov

Pages: 12 - 16

A simple procedure for obtaining the mass spectrum of 2-dimensional Toda lattice of E8 type is given.

**Research Article**

## Some Symmetry Classifications of Hyperbolic Vector Evolution Equations

Stephen C. Anco, Thomas Wolf

Pages: 13 - 31

Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several O(N)-invariant classes of hyperbolic equations Utx = f(U, Ut, Ux) for an N-component vector U(t, x) are considered. In each class we find all scalinhomogeneous equations admitting a higher symmetry of least...

**Research Article**

## Reduction of Order for Systems of Ordinary Differential Equations

C. Wafo Soh, F.M. Mahomed

Pages: 13 - 20

The classical reduction of order for scalar ordinary differential equations (ODEs) fails for a system of ODEs. We prove a constructive result for the reduction of order for a system of ODEs that admits a solvable Lie algebra of point symmetries. Applications are given for the case of a system of two...