Journal of Nonlinear Mathematical Physics

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1486 articles
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.
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...
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

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...
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

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 E8 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

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

The Two-Component Camassa–Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws

Marianna Euler, Norbert Euler, Thomas Wolf
Pages: 13 - 22
Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa–Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor.43 (2010) 492001 (20pp)]....
Research Article

Global Solvability of a Fragmentation-Coagulation Equation With Growth and Restricted Coagulation

Jacek Banasiak, Suares Clovis Oukouomi Noutchie, Ryszard Rudnicki
Pages: 13 - 26
We consider a fragmentation-coagulation equation with growth, where the nonlinear coagulation term, introduced in O. Arino and R. Rudnicki [2], is designed to model processes in which only a part of particles in the aggregates is capable of coalescence. We introduce various growth models, describing...
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

Nonlinear Models in Quantum Optics through Quantum Algebras

Angel Ballesteros, Sergey Chumakov
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

Symmetry of the Schrödinger Equation with Variable Potential

Wilhelm Fushchych, Zoya Symenoh
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

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...
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

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

Samir F. Radwan
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

Approximate Partial Noether Operators of the Schwarzschild Spacetime

Ibrar Hussain, F. M. Mahomed, Asghar Qadir
Pages: 13 - 25
The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and...
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

E.V. Ferapontov, A.M. Grundland
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

Solutions of Burgers, Reynolds, and Navier–Stokes Equations via Stochastic Perturbations of Inviscid Flows

Yuri E. Gliklikh
Pages: 15 - 29
We show that a certain stochastic perturbation of the flow of perfect incompressible fluid under some special external force on the flat n-dimensional torus yields a solution of Navier–Stokes equation without external force in the tangent space at unit of volume preserving diffeomorphism group. If that...
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...
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

R. Goodall, P.G.L. Leach
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...
Review Article

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

Vladimir Dorodnitsyn, Roman Kozlov
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

A Direct procedure on the Integrability of Nonisospectral and Variable-Coefficient MKdV Equation

Xiaorui Hu, Yong Chen
Pages: 16 - 26
An elementary and systematic method based on binary Bell polynomials is applied to nonisospectral and variable-coefficient MKdV (vcMKdV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite local conservation laws are obtained step by step, without too much clever...
Research Article

A note on minimization of rational surfaces obtained from birational dynamical systems

A. S. Carstea, T. Takenawa
Pages: 17 - 33
In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable...
Research Article

Multicomplex solitons

Julia Cen, Andreas Fring
Pages: 17 - 35
We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and nonlocal Korteweg-de Vries equation and elaborate on how multi-soliton...
Research Article

The compatibility of additional symmetry and gauge transformations for the constrained discrete Kadomtsev-Petviashvili hierarchy

Maohua Li, Jipeng Cheng, Jingsong He
Pages: 17 - 31
In this paper, the compatibility between the gauge transformations and the additional symmetry of the constrained discrete Kadomtsev-Petviashvili hierarchy is given, which preserves the form of the additional symmetry of the cdKP hierarchy, up to shifting of the corresponding additional flows by ordinary...
Research Article

Generalized Wronskian and Grammian Solutions to a Isospectral B-type Kadomtsev-Petviashvili equation

Jianjun Cheng, Zhen Wang, Hongqing Zhang
Pages: 17 - 33
Generally speaking, the BKP hierarchy which only has Pfaffian solutions. In this paper, based on the Grammian and Wronskian derivative formulae, generalized Wronskian and Grammian determinant solutions are obtained for the isospectral BKP equation (the second member on the BKP hierarchy) in the Hirota...
Research Article

Symmetry Constraint of the Differential-difference KP Hierarchy and a Second Discretization of the ZS-AKNS System

Kui Chen, Xiao Deng, Da-jun Zhang
Pages: 18 - 35
In this paper we construct a squared-eigenfunction symmetry of the scalar differential-difference KP hierarchy. Through a constraint of the symmetry, Lax triad of the differential-difference KP hierarchy is reduced to a known discrete spectral problem and a semidiscrete AKNS hierarchy. The discrete spectral...
Research Article

Two-Photon Algebra and Integrable Hamiltonian Systems

Angel Ballesteros, Francisco J. Herranz
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

N. Ben Fraj, S. Omri
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

On the oscillons in the signum-Gordon model

Z. Świerczyński
Pages: 20 - 28
New periodic solutions of signum-Gordon equation are presented. We first find solutions φ0(x, t) defined for (x, t) ∈ ℝ × [0, T ] and satisfying the condition φ0(x, 0) = φ0(x, T ) = 0. Then these solutions are extended to the whole spacetime by using (2.4).
Research Article

On Integrability of Differential Constraints Arising from the Singularity Analysis

S. Yu. Sakovich
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

Non-coordinate case of graded differential algebra with ternary differential

Nadezda Bazunova
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

Kouichi Takemura
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 the dynamics of a class of rational Kolmogorov systems

Rachid Boukoucha, Ahmed Bendjeddou
Pages: 21 - 27
In this paper we are intersted in studying the existence of a First integral and the non-existence of limit cycles of rational Kolmogorov systems of the form {x′=x(P(x,y)+R(x,y)S(x,y)),y′=y(Q(x,y)+R(x,y)S(x,y)), where P (x, y) , Q (x, y) , R (x, y) , S (x, y) are homogeneous polynomials of degree...
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

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

M.C. 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

A.M. Korostil
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

Experimental Study of the Velocity Field in Solitary Water Waves

Hung-Chu Hsu, Yang-Yih Chen, Chu-Yu Lin, Chia-Yan Cheng
Pages: 23 - 33
We describe experiments that have been conducted to investigate the velocity field in a solitary water wave. The horizontal and vertical velocity components were measured. The experimental results show that the horizontal velocity component is monotonically increasing with the distance to the wave crest...
Research Article

Stochastic Cohomology of the Frame Bundle of the Loop Space

R. Léandre
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

Peakon-Antipeakon Interaction

R. Beals, D.H. Sattinger, J. Szmigielski
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

Four Dimensional Lie Symmetry Algebras and Fourth Order Ordinary Differential Equations

T. Cerquetelli, N. Ciccoli, M.C. Nucci
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

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

M. Lakshmanan, M. Senthil Velan
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

The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble

Shulin Lyu, James Griffin, Yang Chen
Pages: 24 - 53
We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function w(x;t)=xαe−x−tx,x∈[0,∞),α>−1,t≥0 , are greater than s. This probability is expressed as the quotient of Dn(s,t) and its value at s = 0, where Dn(s,t) denotes the determinant of the...
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

The Bogomolny Decomposition for Systems of Two Generalized Nonlinear Partial Differential Equations of the Second Order

Ł. Stȩpień, D. Sokalska, K. Sokalski
Pages: 25 - 34
Using a concept of strong necessary conditions we derive the Bogomolny decomposition for systems of two generalized elliptic and parabolic nonlinear partial differential equations (NPDE) of the second order. The generalization means that the equation coefficients depend on the field variables. According...
Research Article

Hierarchy of Chaotic Maps with an Invariant Measure and their Compositions

M.A. Jafarizadeh, S. Behnia
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...
Short Communication

Uniqueness of Steady Symmetric Deep-Water Waves with Vorticity

Mats EHRNSTRÖM
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

Applications of Solvable Structures to the Nonlocal Symmetry-Reduction of Odes

Diego Catalano Ferraioli, Paola Morando
Pages: 27 - 42
An application of solvable structures to the reduction of ODEs with a lack of local symmetries is given. Solvable structures considered here are all defined in a nonlocal extension, or covering space, of a given ODE. Examples of the reduction procedure are provided.
Research Article

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

Yury Chapovsky
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

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

C. Burdik, S. Posta, O. Navratil
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 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...
Research Article

On the Recursion Operator for the Noncommutative Burgers Hierarchy

Sandra Carillo, Cornelia Schiebold
Pages: 27 - 37
The noncommutative Burgers recursion operator is constructed via the Cole–Hopf transformation, and its structural properties are studied. In particular, a direct proof of its hereditary property is given.
Research Article

Spinors on Kahler–Norden Manifolds

Nedim Değirmenci, Şenay Karapazar
Pages: 27 - 34
It is known that the complex spin group Spin(n, ℂ) is the universal covering group of complex orthogonal group SO(n, ℂ). In this work we construct a new kind of spinors on some classes of Kahler–Norden manifolds. The structure group of such a Kahler–Norden manifold is SO(n, ℂ) and has a lifting to Spin(n,...
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

The Initial-Boundary Value Poblem for the Korteweg-de Vries Equation on the Positive Quarter-Plane

Pham Loi Vu
Pages: 28 - 43
The paper deals with a problem of developing an inverse-scattering transform for solving the initial-boundary value problem (IBVP) for the Korteweg-de Vries equation on the positive quarter-plane: pt - 6ppx + pxxx = 0, x 0, t 0, (a) with the given initial and boundary conditions: p(x, 0) = p(x), p(x)...
Research Article

The Intermediate Surface Diffusion Flow on Spheres

Joachim Escher
Pages: 28 - 46
It is shown that solutions to the intermediate surface diffusion flow are real analytic in space and time, provided the initial surface is real diffeomorphic to a Euclidean sphere.
Research Article

Integrable 1D Toda cellular automata

Mariusz Białecki
Pages: 28 - 35
First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation, and next we perform a reduction of dKP to the discrete 1D Toda equation.
Research Article

Symmetries of Separating Nonlinear Schrödinger Equations

George Svetlichny
Pages: 28 - 35
We review here the main properties of symmetries of separating hierarchies of nonlinear Schrödinger equations and discuss the obstruction to symmetry liftings from (n)particles to a higher number. We argue that for particles with internal degrees of freedom, new multiparticle effects must appear at each...
Research Article

Response Functions of Spiral Wave Solutions of the Complex Ginzburg­Landau Equation

I.V. Biktasheva, V.N. Biktashev
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...
Research Article

A new solvable many-body problem of goldfish type

Oksana Bihun, Francesco Calogero
Pages: 28 - 46
A new solvable many-body problem of goldfish type is introduced and the behavior of its solutions is tersely discussed.
Research Article

Nonlinear Self-Adjointness and Conservation Laws for the Hyperbolic Geometric Flow Equation

Kênio A. A. Silva
Pages: 28 - 43
We study the nonlinear self-adjointness of a class of quasilinear 2D second order evolution equations by applying the method of Ibragimov. Which enables one to establish the conservation laws for any differential equation. We first obtain conditions determining the self-adjointness for a sub-class in...
Research Article

Bäcklund Transformations on Coadjoint Orbits of the Loop Algebra ~gl(r)

Yuri Fedorov
Pages: 29 - 46
There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra ~gl(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral curves....
Research Article

On Peakon and Kink-peakon Solutions to a (2 + 1) Dimensional Generalized Camassa-Holm Equation

Eric Tovar, Haicheng Gu, Zhijun Qiao
Pages: 29 - 40
In this paper, we study a (2 + 1)-dimensional generalized Camassa-Holm (2dgCH) equation with both quadratic and cubic nonlinearity. We derive a peaked soliton (peakon) solution, double-peakon solutions, and kink-peakon solutions. In particular, weak kink - peakon solution is the first time to address...
Research Article

Integrability of Lie Systems through Riccati Equations

José F. Cariñena, Javier de Lucas
Pages: 29 - 54
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyze a geometric method to construct integrability conditions for Riccati equations following these approaches. Our procedure provides us with a unified geometrical viewpoint that allows us to analyze...
Research Article

Bilinear Identities on Schur Symmetric Functions

Dimitri Gurevich, Pavel Pyatov, Pavel Saponov
Pages: 31 - 48
A series of bilinear identities on the Schur symmetric functions is obtained with the use of Plücker relations.
Research Article

On Bilinear Invariant Differential Operators Acting on Tensor Fields on the Symplectic Manifold

Pavel Grozman
Pages: 31 - 37
Let M be an n-dimensional manifold, V the space of a representation : GL(n)GL(V ). Locally, let T(V ) be the space of sections of the tensor bundle with fiber V over a sufficiently small open set U M, in other words, T(V ) is the space of tensor fields of type V on M on which the group Diff(M) of diffeomorphisms...
Research Article

Existence of Dual Equations by Means of Strong Necessary Conditions - Analysis of Integrability of Partial Differential Nonlinear Equations

K. Sokalski, T. Wietecha, D. Sokalska
Pages: 31 - 52
A concept of strong necessary conditions for extremum of functional has been aplied for analysis an existence of dual equations for a system of two nonlinear Partial Differential Equations (PDE) in 1+1 dimensions. We consider two types of the dual equations: the Bäcklund transformations and the Bogomolny...
Research Article

Some Group Theoretical Aspects of Nonlinear Quantal Oscillators

K. Andriopoulos, P.G.L. Leach
Pages: 32 - 42
We investigate the algebraic properties of the time-dependent Schrödinger equations of certain nonlinear oscillators introduced by Calogero and Graffi (Calogero F & Graffi S, On the quantisation of a nonlinear Hamiltonian oscillator Physics Letters A 313 (2003) 356-362; Calogero F, On the quantisation...
Research Article

Exact solutions of the Gerdjikov-Ivanov equation using Darboux transformations

Halis Yilmaz
Pages: 32 - 46
We study the Gerdjikov-Ivanov (GI) equation and present a standard Darboux transformation for it. The solution is given in terms of quasideterminants. Further, the parabolic, soliton and breather solutions of the GI equation are given as explicit examples.
Research Article

Exact Travelling Wave Solutions of a Beam Equation

J. C. Camacho, M. S. Bruzón, J. Ramírez, M. L. Gandarias
Pages: 33 - 49
In this paper we make a full analysis of the symmetry reductions of a beam equation by using the classical Lie method of infinitesimals and the nonclassical method. We consider travelling wave reductions depending on the form of an arbitrary function. We have found several new classes of solutions that...
Research Article

Solutions of Adler's Lattice Equation Associated with 2-Cycles of the Bäcklund Transformation

James Atkinson, Frank Nijhoff
Pages: 34 - 42
The Bäcklund transformation (BT) of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Bäcklund parameters) as a 2-cycle of the BT. In...
Research Article

Combinatorics of Matrix Factorizations and Integrable Systems

Anton Dzhamay
Pages: 34 - 47
We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial–geometric way using a diagram of a cube. In this...
Research Article

On Constant Vorticity Flows Beneath Two-Dimensional Surface Solitary Waves

Raphael Stuhlmeier
Pages: 34 - 42
We demonstrate that, for a two-dimensional, steady, solitary wave profile, a flow of constant vorticity beneath the wave must likewise be steady and two-dimensional, and the vorticity will point in the direction orthogonal to that of wave propagation. Constant vorticity is the hallmark of a harmonic...
Research Article

On the Structure of the Bäcklund Transformations for the Relativistic Lattices

Vsevolod E. Adler
Pages: 34 - 56
The Bäcklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the Bäcklund...
Research Article

On Decomposition of the ABS Lattice Equations and Related Bäcklund Transformations

Danda Zhang, Da-jun Zhang
Pages: 34 - 53
The Adler-Bobenko-Suris (ABS) list contains scalar quadrilateral equations which are consistent around the cube, and have D4 symmetry and tetrahedron property. Each equation in the ABS list admits a beautiful decomposition. We revisit these decomposition formulas and by means of them we construct Bäcklund...
Research Article

On geometric quantization of the Dirac magnetic monopole

Graham M. Kemp, Alexander P. Veselov
Pages: 34 - 42
We give a simple derivation of the spectrum of the Dirac magnetic monopole on a unit sphere S2 based on geometric quantization and the Frobenius reciprocity formula. The starting point is the calculation by Novikov and Schmelzer of the canonical symplectic structure on the coadjoint orbits of the isometry...
Research Article

Conservations laws for critical Kohn-Laplace equations on the Heisenberg group

Yuri Dimitrov Bozhkov, Igor Leite Freire
Pages: 35 - 47
Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group H, carried out in a preceding work, we estab- lish the conservation laws for the critical Kohn-Laplace equations via the Noether’s Theorem.
Research Article

Threshold Behavior for Nonlinear Wave Equations

Piotr Bizoń
Pages: 35 - 41
In this brief contribution, which is based on my talk at the conference, I discuss the dynamics of solutions of nonlinear wave equations near the threshold of singularity formation. The heuristic picture of threshold behavior is first presented in a general setting and then illustrated with three examples.
Research Article

Variational Methods for Solving Nonlinear Boundary Problems of Statics of Hyper-Elastic Membranes

V.A. Trotsenko
Pages: 35 - 50
A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics [1]­[6]. In the present paper, using the variational method for solving nonlinear boundary problems of statics of hyper-elastic membranes under the regular...
Research Article

Bäcklund-Transformation-Related Recursion Operators: Application to the Self-Dual Yang–Mills Equation

C. J. Papachristou, B. Kent Harrison
Pages: 35 - 49
By using the self-dual Yang–Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by a non-auto-Bäcklund transformation. We prove the Lie-algebra isomorphism between the symmetries of the SDYM...
Research Article

Geometrization of the Leading Term in Acoustic Gaussian Beams

Matias F. Dahl
Pages: 35 - 45
We study Gaussian beams for the wave equation on a Riemannian manifold. For the transport equation we geometrize the leading term at the center of the Gaussian beam. More precisely, if u(x,t)=eiPθ(x,t)(u0(x,t)+u1(x,t)iP+u2(x,t)(iP)2+⋯) is a Gaussian beam propagating along a geodesic c, then we show...
Research Article

2D reductions of the equation uyy = utx + uyuxxuxuxy and their nonlocal symmetries

P. Holba, I.S. Krasil'shchik, O.I. Morozov, P. Vojčák
Pages: 36 - 47
We consider the 3D equation uyy = utx + uyuxx − uxuxy and its 2D symmetry reductions: (1) uyy = (uy + y) uxx − uxuxy − 2 (which is equivalent to the Gibbons-Tsarev equation) and (2) uyy = (uy + 2x)uxx + (y − ux)uxy − ux. Using the corresponding reductions of the known Lax pair for the 3D equation, we...
Research Article

Symmetries and Differential Forms

A.H. Davison, A.H. Kara
Pages: 36 - 43
The method for writing a differential equation or system of differential equations in terms of differential forms and finding their symmetries was devised by Harrison and Estabrook (1971). A modification to the method is demonstrated on a wave equation with variable speed, and the modified method is...
Research Article

Singularity Analysis and a Function Unifying

C. Géronimi, P.G.L. Leach, M.R. Feix
Pages: 36 - 48
The classical (ARS) algorithm used in the Painlevé test picks up only those functions analytic in the complex plane. We complement it with an iterative algorithm giving the leading order and the next terms in all cases. This algorithm works both for an ascending series (about a singularity at finite...
Research Article

Second Order Dynamic Inclusions

Martin Bohner, Christopher C Tisdell
Pages: 36 - 45
The theory of dynamic inclusions on a time scale is introduced, hence accommodating the special cases of differential inclusions and difference inclusions. Fixed point theory for set-valued upper semicontinuous maps, Green's functions, and upper and lower solutions are used to establish existence results...
Research Article

On the Self-Similar Solutions of Generalized Hydrodynamics Equations and Nonlinear Wave Patterns

V.A. Danylenko, V.A. Vladimirov
Pages: 36 - 43
Solutions of the system of dynamical equations of state and equations of the balance of mass and momentum are studied. The system possesses families of periodic, quasiperiodic and soliton-like invariant solutions. Self-similar solutions of this generalized hydrodynamic system are studied. Various complicated...
Research Article

Realizations of the Witt and Virasoro Algebras and Integrable Equations

Qing Huang, Renat Zhdanov
Pages: 36 - 56
In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order differential operators over the space ℝ3. We prove that none of them admits non-trivial central extension, which...
Research Article

On Pauli graded contractions of sl(3, C)

Miloslav Havlicek, Jiri Patera, Edita Pelantova, Jiri Tolar
Pages: 37 - 42
We consider a special fine grading of sl(3, C), where the grading subspaces are geerated by 3 × 3 generalized Pauli matrices. This fine grading decomposes sl(3, C) into eight one­dimensional subspaces. Our aim is to find all contractions of sl(3, C) which preserve this grading. We have found that the...