Journal of Nonlinear Mathematical Physics

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1493 articles
Corrigendum

Corrigendum

Some remarks on materials with memory: heat conduction and viscoelasticity

Sandra Carillo
Pages: i - iii
Research Article

Preface

Adrian Constantin
Pages: v - v
Research Article

Preface

P. Basarab-Horwath, M. Euler, N. Euler, P. G. L. Leach
Pages: v - v
Editorial

Preface

S. Twareque Ali, Piotr Kielanowski, Anatol Odzijewicz, Martin Schlichenmaier
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 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.
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 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...
Research Article

Compatible Poisson Structures and bi-Hamiltonian Systems Related to Low-dimensional Lie Algebras

Gh. Haghighatdoost, S. Abdolhadi-Zangakani, J. Abedi-Fardad
In this work, we give a method to construct compatible Poisson structures on Lie groups by means of structure constants of their Lie algebras and some vector field. In this way we calculate some compatible Poisson structures on low-dimensional Lie groups. Then, using a theorem by Magri and Morosi, we...
Research Article

Symmetries of Kolmogorov Backward Equation

Roman Kozlov
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

Exact Solutions of a Nonlinear Diffusion Equation with Absorption and Production

Robert Conte
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 Ramsey Dynamical Model and Closed-Form Solutions

Gülden Gün Polat, Teoman Özer
This study focuses on the analysis of Ramsey dynamical model with current Hamiltonian defining an optimal control problem in a neoclassical growth model by utilizing Lie group theory. Lie point symmetries of coupled nonlinear first-order ordinary differential equations corresponding to first-order conditions...
Research Article

Orbits and Lagrangian Symmetries on the Phase Space

Javier Pérez Álvare
In this article, given a regular Lagrangian system L on the phase space TM of the configuration manifold M and a 1-parameter group G of transformations of M whose lifting to TM preserve the canonical symplectic dynamics associated to L, we determine conditions so that its infinitesimal generator produces...
Editorial

Foreword

A.H. Kara, D.P. Mason
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.
Editorial

Foreword

Eugen Paal, Sergei Silvestrov
Pages: 0 - 0
Research Article

Affine Ricci Solitons of Three-Dimensional Lorentzian Lie Groups

Yong Wang
In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.
Research Article

On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations

K. Sethukumarasamy, P. Vijayaraju, P. Prakash
In this article, we explain how to extend the Lie symmetry analysis method for n-coupled system of fractional ordinary differential equations in the sense of Riemann-Liouville fractional derivative. Also, we systematically investigated how to derive Lie point symmetries of scalar and coupled fractional...
Research Article

The Orthogonal and Symplectic Schur Functions, Vertex Operators and Integrable Hierarchies

Linjie Shi, Na Wang, Minru Chen
In this paper, we first construct an integrable system whose solutions include the orthogonal Schur functions and the symplectic Schur functions. We find that the orthogonal Schur functions and the symplectic Schur functions can be obtained by one kind of Boson-Fermion correspondence which is slightly...
Research Article

Lie Symmetries of the Canonical Connection: Codimension One Abelian Nilradical Case

Hassan Almusawa, Ryad Ghanam, Gerard Thompson
This paper studies the canonical symmetric connection ∇ associated to any Lie group G. The salient properties of ∇ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to ∇ in the special case where the Lie algebra...
Research Article

A New Case of Separability in a Quartic Hénon-Heiles System

Nicola Sottocornola
There are four quartic integrable Hénon-Heiles systems. Only one of them has been separated in the generic form while the other three have been solved only for particular values of the constants. We consider two of them, related by a canonical transformation, and we give their separation coordinates...
Research Article

Statistical de Rham Hodge Operators and the Kastler-Kalau-Walze Type Theorem for Manifolds With Boundary

Sining Wei, Yong Wang
In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.
Research Article

A Local Equivariant Index Theorem for Sub-Signature Operators

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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...
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 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...
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 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

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

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...
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 suq(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

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∞(������) 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∞(������)...
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

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

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

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...
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

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

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

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

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

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...
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...
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

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

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

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

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

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

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

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.