# Journal of Nonlinear Mathematical Physics

1499 articles

**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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## On the Coupled Dispersionless-type Equations and the Short Pulse-type Equations

Juan Hu, Jia-Liang Ji, Guo-Fu Yu

Pages: 14 - 26

In this paper, we study the correspondence between the Coupled Dispersionless (CD)-type equations and the Short Pulse (SP)-type equations. From the real and complex modified CD equations, we construct the real and complex Modified Short Pulse (mSP) equations geometrically and algebraically. From the...

**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...

**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**

## 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...

**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**

## 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**

## 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**

## 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**

## 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 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**

## 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**

## 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**

## 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**

## 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

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**

## 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**

## 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**

## 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...

**Research Article**

## Differential Equations for the Components of the Canonical Chua's Circuit and its General Properties

Anatolij Samoilenko, Leon Chua

Pages: 25 - 40

**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...

**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...

**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**

## 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**

## 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**

## 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**

## 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**

## The (*N* + 1)-Dimensional Burgers Equation: A Bilinear Extension, Vortex, Fusion and Rational Solutions

Hongli An, Engui Fan, Manwai Yuen

Pages: 27 - 37

In this paper, by introducing a fractional transformation, we obtain a bilinear formulation for the (N + 1)-dimensional Burgers equation. Such a formulation constitutes a bilinear extension to the (N + 1)-dimensional Cole-Hopf transformation between the (N + 1)-dimensional Burgers system and generalized...

**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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 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**

## 2D reductions of the equation *u*_{yy} = *u*_{tx} + *u*_{y}u_{xx} − *u*_{x}u_{xy} and their nonlocal symmetries

_{yy}

_{tx}

_{y}u

_{xx}

_{x}u

_{xy}

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**

## 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**

## SO(2) and Hamilton-Dirac mechanics

Cestmir Burdik, Eugen Paal, Juri Virkepu

Pages: 37 - 43

Canonical formalism for plane rotations is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide...

**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...

**Research Article**

## Fractal and Chaotic Solutions of the Discrete Nonlinear Schrödinger Equation in Classical and Quantum Systems

H.S. Dhillon, F.V. Kusmartsev, K.E. Kürten

Pages: 38 - 49

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the 4 type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or...

**Research Article**

## Inverse Scattering Transformation for the Fokas–Lenells Equation with Nonzero Boundary Conditions

Yi Zhao, Engui Fan

Pages: 38 - 52

In this article, we focus on the inverse scattering transformation for the Fokas–Lenells (FL) equation with nonzero boundary conditions via the Riemann–Hilbert (RH) approach. Based on the Lax pair of the FL equation, the analyticity, symmetry and asymptotic behavior of the Jost solutions and scattering...

**Research Article**

## High-Frequency Asymptotics for the Helmholtz Equation in a Half-Plane

Min-Hai Huang

Pages: 38 - 47

Base on the integral representations of the solution being derived via Fokas' transform method, the high-frequency asymptotics for the solution of the Helmholtz equation, in a half-plane and subject to the Neumann condition is discussed. For the case of piecewise constant boundary data, full asymptotic...

**Research Article**

## On Asymptotic Methods in the Theory of Differential Equations of Mathematical Physics

Mykola Shkil

Pages: 40 - 50

In 1807 Fourier suggested a original method of solving partial differential equations. The method is known to lead to ordinary differential equations containing some arbitrary parameter...

**Research Article**

## Symmetries, Lagrangian Formalism and Integration of Second Order Ordinary Difference Equations

Vladimir Dorodnitsyn, Roman Kozlov, Pavel Winternitz

Pages: 41 - 56

An integration technique for difference schemes possessing Lie point symmetries is proposed. The method consists of determining an invariant Lagrangian and using a discrete version of Noether's theorem to obtain first integrals. This lowers the order of the invariant difference scheme.

**Research Article**

## Progressive Internal Gravity Waves With Bounded Upper Surface Climbing a Triangular Obstacle

Mina B. Abd-El-Malek, Malak N. Makar

Pages: 41 - 53

In this paper we discuss a theoretical model for the interfacial profiles of progressive non-linear waves which result from introducing a triangular obstacle, of finite height, attached to the bottom below the flow of a stratified, ideal, two layer fluid, bounded from above by a rigid boundary. The derived...

**Research Article**

## Symmetry Properties of the Approximations of Multidimensional Generalized Van Der Pol Equations

Norbert Euler, Marianna Euler

Pages: 41 - 59

**Research Article**

## Poisson brackets, Novikov-Leibniz structures and integrable Riemann hydrodynamic systems

Orest D. Artemovych, Denis Blackmore, Anatolij K. Prykarpatski

Pages: 41 - 72

A general differential-algebraic approach is devised for constructing multi-component Hamiltonian operators as differentiations on suitably constructed loop Lie algebras. The related Novikov-Leibniz algebraic structures are presented and a new non-associative “Riemann” algebra is constructed, which is...

**Research Article**

## On Weak Convergence of Locally Periodic Functions

Dag Lukkassen, Peter Wall

Pages: 42 - 57

We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boandary value problems are also considered.

**Research Article**

## Sp(2) Quantization of Solitonic Theories

V. Calian

Pages: 42 - 47

The gauge-field theoretical formulation of solitonic theories is quantized by using an extended version of the BRST Sp(2) symmetric formalism. The proposed method is based on a modified triplectic geometry which allows us to incorporate the linear and/or nonlinear global symmetries of the model and to...

**Research Article**

## Singular Scattering Matrices

David Atkinson

Pages: 43 - 49

A nonlinear integrodifferential equation is solved by the methods of S-matrix theory. The technique is shown to be applicable to situations in which the effective potential is singular.