Journal of Nonlinear Mathematical Physics

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902 articles
Research Article

µ-Holomorphic Projective Connections and Conformal Covariance

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

Correctors for Some Nonlinear Monotone Operators

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

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

A Note on q-Bernoulli Numbers and Polynomials

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

Noetherian first integrals

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

Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

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

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

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...
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

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

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

Proper-Time Relativistic Dynamics and the Fushchych-Shtelen Transformation

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

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

Existence of Periodic Solutions of a Type of Nonlinear Impulsive Delay Differential Equations with a Small Parameter

Jehad O Alzabut
Pages: 13 - 21
The Banach fixed point theorem is used to prove the existence of a unique( w) periodic solution of a new type of nonlinear impulsive delay differential equation with a small parameter.
Research Article

Reduction of Order for Systems of Ordinary Differential Equations

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...
Research Article

Hierarchies of Difference Equations and Bäcklund Transformations

Peter A CLARKSON, Andrew N W HONE, Nalini JOSHI
Pages: 13 - 26
In this paper we present a method for deriving infinite sequences of difference equations containing well known discrete Painlevé equations by using the Bäcklund transformtions for the equations in the second Painlevé equation hierarchy.
Research Article

Particle Trajectories in Linearized Irrotational Shallow Water Flows

Delia Ionesco-Kruse
Pages: 13 - 27
We investigate the particle trajectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending...
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

On the Fourth-Order Accurate Compact ADI Scheme for Solving the Unsteady Nonlinear Coupled Burgers' Equations

Pages: 13 - 34
The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical...
Research Article

Symmetry of the Schrödinger Equation with Variable Potential

Pages: 13 - 22
We study symmetry properties of the Schrödinger equation with the potential as a new dependent variable, i.e., the transformations which do not change the form of the class of equations. We also consider systems of the Schrödinger equations with certain conditions on the potential. In addition we investigate...
Research Article

Nonlinear Models in Quantum Optics through Quantum Algebras

Pages: 13 - 17
The suq(2) algebra is shown to provide a natural dynamical algebra for some nonlnear models in Quantum Optics. Applications to the computation of eigenvalues and eigenvectors for the Hamiltonian describing second harmonics generation are proposed.
Research Article

A Truncation for Obtaining all the First Degree Birational Transformations of the Painlevé Transcendents

Robert CONTE, Micheline MUSETTE
Pages: 14 - 28
A birational transformation is one which leaves invariant an ordinary differential eqution, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation of Okamoto for the mater Painlevé equation P6. Then we improve...
Research Article

Links Between Different Analytic Descriptions of Constant Mean Curvature Surfaces

Pages: 14 - 21
Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system
Research Article

A novel approach to the theory of Padé approximants

Christopher ATHORNE
Pages: 15 - 27
By associating polynomials and power series expansions with sln(C) modules we dscribe the theory of Padé approximants in terms of tensor products of representations and interpret their recurrence relations algebraically. The treatment links with the theory of Hirota derivatives and discrete integrable...
Research Article

Generalised Symmetries and the Ermakov-Lewis Invariant

Pages: 15 - 26
Generalised symmetries and point symmetries coincide for systems of first-order odinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the sytem of first-order ordinary differential equations...
Research Article

q-Euler numbers and polynomials associated with p-adic q-integrals

Taekyun KIM
Pages: 15 - 27
The main purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials. In particular, by using the q-Volkenborn integration on Zp, we construct p-adic q-Euler numbers and polynomials of higher order. We also define new generating functions of multiple...
Review Article

A Heat Transfer with a Source: the Complete Set of Invariant Difference Schemes

Pages: 16 - 50
In this letter we present the set of invariant difference equations and meshes which preserve the Lie group symmetries of the equation ut = (K(u)ux)x +Q(u). All special cases of K(u) and Q(u) that extend the symmetry group admitted by the differential equation are considered. This paper completes the...
Research Article

Two-Photon Algebra and Integrable Hamiltonian Systems

Pages: 18 - 22
The two-photon algebra h6 is used to define an infinite class of N-particle Hamiltonian systems having (N -2) additional constants of the motion in involution. By constrution, all these systems are h6-coalgebra invariant. As a straightforward application, a new family of (quasi)integrable N-dimensional...
Research Article

Deforming the Lie Superalgebra of Contact Vector Fields on S1|1 Inside the Lie Superalgebra of Superpseudodifferential Operators on S1|1

Pages: 19 - 33
We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie supealgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced...
Research Article

Non-coordinate case of graded differential algebra with ternary differential

Pages: 21 - 26
In this article, we generalize a construction of graded q-differential algebra with ternary differential satisfying the property d3 = 0 and q-Leibniz rule on the non-coordinate case, that is on the case where the differentials of generators of underlying algebra do not coincide with generators of bimodule...
Research Article

The Heun Equation and the Calogero-Moser-Sutherland System III: The Finite-Gap Property and the Monodromy

Pages: 21 - 46
A new approach to the finite-gap property for the Heun equation is constructed. The relationship between the finite-dimensional invariant space and the spectral curve is clarified. The monodromies are calculated and are expressed as hyperelliptic integrals. Applications to the spectral problem for the...
Research Article

On Integrability of Differential Constraints Arising from the Singularity Analysis

Pages: 21 - 25
Integrability of differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordnary differential equations are obtained in this way, which are integrable by quadrtures in spite of very complicated branching of their...
Research Article

Singularity Analysis and Integrability of a Simplified Multistrain Model for the Transmission of Tuberculosis and Dengue Fever

MC Nucci, P.G.L. Leach
Pages: 22 - 34
We apply singularity analysis to a caricature of the simplified multistrain model of Castillo-Chavez and Feng (J Math Biol 35 (1997) 629–656) for the transmission of tuberculosis and the coupled two-stream vector-based model of Feng and Velasco- Hern ?andez (J Math Biol 35 (1997) 523–544) to identify...
Research Article

On the Calculation of Finite-Gap Solutions of the KdV Equation

Pages: 22 - 33
A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these solutions in the form of rational functions of the elliptic Weierstrass...
Research Article

Boundary Algebra and Exact Solvability of the Asymmetric Exclusion Process

Boyka Aneva
Pages: 22 - 33
We consider a lattice driven diffusive system withUq(su(2)) invariance in the bulk. Within the matrix product states approach the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra. Boundary processes amount to the appearance of parameter dependent...
Research Article

Peakon-Antipeakon Interaction

Pages: 23 - 27
Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa­ Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon.
Research Article

Stochastic Cohomology of the Frame Bundle of the Loop Space

Pages: 23 - 40
We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we impose...
Research Article

Lie Symmetries, Infinite-Dimensional Lie Algebras and Similarity Reductions of Certain (2+1)-Dimensional Nonlinear Evolution Equations

Pages: 24 - 39
The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik ­ Novikov ­ Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation,...
Research Article

Four Dimensional Lie Symmetry Algebras and Fourth Order Ordinary Differential Equations

Pages: 24 - 35
Realizations of four dimensional Lie algebras as vector fields in the plane are explcitly constructed. Fourth order ordinary differential equations which admit such Lie symmetry algebras are derived. The route to their integration is described.
Research Article

Complex Lie Symmetries for Variational Problems

Sajid Ali, Fazal M Mahomed, Asghar Qadir
Pages: 25 - 35
We present the use of complex Lie symmetries in variational problems by defining a complex Lagrangian and considering its Euler-Lagrange ordinary differential equation. This Lagrangian results in two real “Lagrangians” for the corresponding system of partial differential equations, which satisfy Euler-Lagrange...
Research Article

Hierarchy of Chaotic Maps with an Invariant Measure and their Compositions

Pages: 26 - 41
We give a hierarchy of many-parameter families of maps of the interval [0, 1] with an invariant measure and using the measure, we calculate Kolmogorov­Sinai entropy of these maps analytically. In contrary to the usual one-dimensional maps these maps do not possess period doubling or period-n-tupling...
Research Article

On the matrix 3 × 3 exact solvable models of the type G2

Pages: 27 - 36
We study the exact solvable 3 × 3 matrix model of the type G2. We apply the construction similar to that one, which give the 2 × 2 matrix model. But in the studied case the construction does not give symmetric matrix potential. We conceive that the exact solvable 3 × 3 matrix potential model of the type...
Research Article

A Hopf C-algebra associated with an action of SUq(1,1) on a two-parameter quantum deformation of the unit disc

Pages: 27 - 45
We define a Hopf C -algebra associated with an action of the quantum group SUq(1, 1) on a two-parameter quantum deformation of the unit disc, which has a left comodule structure over this Hopf C -algebra. Mathematics Subject Classification (1991). 81C05.
Research Article

A New Discrete Hénon-Heiles System

Alan K COMMON, Andrew N W HONE, Micheline MUSETTE
Pages: 27 - 40
By considering the Darboux transformation for the third order Lax operator of the Sawada-Kotera hierarchy, we obtain a discrete third order linear equation as well as a discrete analogue of the Gambier 5 equation. As an application of this result, we consider the stationary reduction of the fifth order...
Short Communication

Uniqueness of Steady Symmetric Deep-Water Waves with Vorticity

Pages: 27 - 30
Given a steady and symmetric deep-water wave we prove that the surface profile and the vorticity distribution determine the wave motion completely throughout the fluid.
Research Article

Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation

Shuyin Wu, Zhaoyang Yin
Pages: 28 - 49
In the paper, several problems on the periodic Degasperis-Procesi equation with weak dissipation are investigated. At first, the local well-posedness of the equation is established by Kato’s theorem and a precise blow-up scenario of the solutions is given. Then, several critera guaranteeing the blow-up...
Research Article

Response Functions of Spiral Wave Solutions of the Complex Ginzburg­Landau Equation

Pages: 28 - 34
Dynamics of spiral waves in perturbed two-dimensional autowave media can be dscribed asymptotically in terms of Aristotelean dynamics. We apply this general thory to the spiral waves in the Complex Ginzburg­Landau equation (CGLE). The RFs are found numerically. In this work, we study the dependence of...