Journal of Nonlinear Mathematical Physics

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1493 articles
Research Article

Superalgebras for the 3D Harmonic Oscillator and Morse Quantum Potentials

H. Fakhri
Pages: 361 - 375
In addition to obtaining supersymmetric structure related to the partner Hamiltonans, we get another supersymmetric structure via factorization method for both the 3D harmonic oscillator and Morse quantum potentials. These two supersymmetries induce also an additional supersymmetric structure involving...
Research Article

Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax

Jarmo Hietarinta
Pages: 358 - 389
We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of “multidimensional consistency” (MDC): it should be...
Research Article

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

J.M. Escobar, J. Núñez, P. Pérez-Fernández
Pages: 358 - 374
In this paper, we deal with contractions of Lie algebras. We use two invariant functions of Lie algebras as a tool, named ψ and φ function, respectively, which have a great application in Physics due to their remarkable properties. We focus the study of these functions in the frame of the filiform Lie...
Research Article

Symmetry Preserving Discretization of SL(2,R) Invariant Equations

Anne Bourlioux, Raphaël Rebelo, Pavel Winternitz
Pages: 362 - 372
Nonlinear ODEs invariant under the group SL(2,R) are solved numerically. We show that solution methods incorporating the Lie point symmetries provide better results than standard methods.
Research Article

Reductions of Integrable Lattices

B. Grammaticos, A. Ramani, J. Satsuma, R. Willox
Pages: 363 - 371
We present a novel method for the reduction of integrable two-dimensional discrete systems to one-dimensional mappings. The procedure allows for the derivation of nonautonomous systems, which are typically discrete (difference or q) Painlevé equtions, or of autonomous ones. In the latter case we produce...
Research Article

Dark Equations

B.A. Kupershmidt
Pages: 363 - 445
Observing the Universe, astronomers have concluded that the motion of stars can not be accounted for unless one assumes that most of the mass in the Universe is carried on by a "dark matter", so far impervious to all attempts at being detected. There is now a similar concept of "dark energy". I shall...
Research Article

The Cohomology of the Variational Bicomplex Invariant under the Symmetry Algebra of the Potential Kadomtsev-Petviashvili Equation

Juha Pohjanpelto
Pages: 364 - 376
The variational bicomplex of forms invariant under the symmetry algebra of the potential Kadomtsev-Petviashvili equation is described and the cohomology of the associated Euler-Lagrange complex is computed. The results are applied to a characterization problem of the Kadomtsev-Petviashvili equation by...
Research Article

Differential Constraints Compatible with Linearized Equations

Ahmet Satir
Pages: 364 - 370
Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.
Research Article

A generalized Schrödinger equation via a complex Lagrangian of electrodynamics

Nicoleta Aldea, Gheorghe Munteanu
Pages: 361 - 373
In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation. By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics,...
Research Article

Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries

A. Ghose Choudhury, Partha Guha, Barun Khanra
Pages: 365 - 382
In this paper we compute first integrals of nonlinear ordinary differential equations using the extended Prelle-Singer method, as formulated by Chandrasekar et al in J. Math. Phys. 47 (2), 023508, (2006). We find a new first integral for the Painlev´e-Gambier XXII equation. We also derive the first integrals...
Research Article

Poisson Homology of r-Matrix Type Orbits I: Example of Computation

Alexei Kotov
Pages: 365 - 383
In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang­Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e...
Research Article

Feynman-Jackson integrals

Rafael Díaz, Eddy Pariguan
Pages: 365 - 376
We introduce perturbative Feynman integrals in the context of q-calculus generalizing the Gaussian q-integrals introduced by Diaz and Teruel. We provide analytic as well as combinatorial interpretations for the Feynman-Jackson integrals.
Research Article

Lie Algebras and Superalgebras Defined by a Finite Number of Relations: Computer Analysis

V.P. Gerdt, V.V. Kornyak
Pages: 367 - 373
The presentation of Lie (super)algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. It is very important, for instance, for investigation of the particular Lie (super)algebras arising in different (super)symmetric...
Research Article

Exact Solutions of the Modified Gross–Pitaevskii Equation in “Smart“ Periodic Potentials in the Presence of External Source

Thokala Soloman Raju, Prasanta K. Panigrahi
Pages: 367 - 376
We report wide class of exact solutions of the modified Gross–Pitaevskii equation (GPE) in “smart” Jacobian elliptic potentials, in the presence of external source. Solitonlike solutions, singular solutions, and periodic solutions are found using a recently developed fractional transform in which all...
Research Article

Solving Simultaneously Dirac and Ricatti Equations

J. Casahorrán
Pages: 371 - 382
We analyse the behaviour of the Dirac equation in d = 1 + 1 with Lorentz scalar potential. As the system is known to provide a physical realization of supersymmetric quantum mechanics, we take advantage of the factorization method in order to enlarge the restricted class of solvable problems. To be precise,...
Research Article

Factorisation of recursion operators of some Lagrangian systems

Dmitry K. Demskoi
Pages: 368 - 378
We observe that recursion operator of an S-integrable hyperbolic equation that degenerates into a Liouvile-type equation admits a particular factorisation. This observation simplifies the construction of such operators. We use it to find a new quasi-local recursion operator for a triplet of scalar fields....
Research Article

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

Chuanzhong Li, Tao Song
Pages: 368 - 382
The noncommutative Toda hierarchy is studied with the help of Moyal deformation by a reduction on the non-commutative two dimensional Toda hierarchy. Further we generalize the noncommutative Toda hierarchy to the extended noncommutative Toda hierarchy. To survey on its integrability, we construct the...
Research Article

Symmetries of the Relativistic Two-Particle Model with Scalar-Vector Interaction

Askold Duviryak
Pages: 372 - 378
A relativistic two-particle model with superposition of time-asymmetric scalar and vector interactions is proposed and its symmetries are considered. It is shown that first integrals of motion satisfy nonlinear Poisson-bracket relations which include the Poincaré algebra and one of the algebras so(1,3),...
Research Article

Link Invariants and Lie Superalgebras

P. Grozman, D. Leites
Pages: 372 - 379
Berger and Stassen reviewed skein relations for link invariants coming from the simple Lie algebras g. They related the invariants with decomposition of the tensor square of the g-module V of minimal dimension into irreducible components. (If V V , one should also consider the decompositions of V V and...
Research Article

New Quasi-Exactly Solvable Difference Equation

Ryu Sasaki
Pages: 373 - 384
Exact solvability of two typical examples of the discrete quantum mechanics, i.e. the dynamics of the Meixner-Pollaczek and the continuous Hahn polynomials with full parameters, is newly demonstrated both at the Schr¨odinger and Heisenberg picture levels. A new quasiexactly solvable difference equation...
Research Article

Galilean-invariant Nonlinear PDEs and their Exact Solutions

Roman M. Cherniha
Pages: 374 - 383
All systems of (n+1)-dimensional quasilinear evolutional second- order equations invariant under the chain of algebras AG(1.n) AG1(1.n) AG2(1.n) are described. The obtained results are illustrated by examples of nonlinear Schrödinger equations.
Research Article

µ -symmetry and µ -conservation law for the extended mKdV equation

Kh. Goodarzi, M. Nadjafikhah
Pages: 371 - 381
In this paper, we obtain µ -symmetry and µ -conservation law of the extended mKdV equation. The extended mKdV equation dose not admit a variational problem since it is of odd order. First we obtain µ -conservation law of the extended mKdV equation in potential form because it admits a variational problem,...
Research Article

A Whittaker-Shannon-Kotelnikov sampling theorem related to the Askey-Wilson functions

Fethi Bouzeffour
Pages: 375 - 388
A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Askey-Wilson functions is proved. Applications to finite continuous Askey-Wilson transform are given.
Research Article

Infinite-Dimensional and Colored Supermanifolds

Vladimir Molotkov
Pages: 375 - 446
Here the theory of finite-dimensional supermanifolds is generalized in two directions. First, we introduce infinite-dimensional supermanifolds “locally isomorphic” to arbitrary Banach (or, more generally, locally convex) superspaces. This is achieved by considering supermanifolds as functors (equipped...
Research Article

Normal Forms for Coupled Takens-Bogdanov Systems

David Mumo Malonza
Pages: 376 - 398
The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module. In this paper Groebner basis methods are used to determine a Groebner basis...
Research Article

Geometry of the Recursion Operators for the GMV System

A. B. Yanovski, G. Vilasi
Pages: 373 - 390
We consider the Recursion Operator approach to the soliton equations related to a auxiliary linear system introduced recently by Gerdjikov, Mikhailov and Valchev (GMV system) and their interpretation as dual of Nijenhuis tensors on the manifold of potentials.
Research Article

Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂1|1 and Deformations of the Superspace of Symbols

Imed Basdouri, Mabrouk Ben Ammar, Nizar Ben Fraj, Maha Boujelbene, Kaouthar Kamoun
Pages: 373 - 409
Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra 𝒦(1) of contact vector fields on the (1, 1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute...
Research Article

On Shtelen's Solution of the Free Linear Schrödinger Equation

W.W. Zachary
Pages: 377 - 382
The solution of the three-dimensional free Schrödinger equation due to W.M. Shtelen based on the invariance of this equation under the Lorentz Lie algebra so(1,3) of nonlocal transformations is considered. Various properties of this solution are examined, including its extension to n 3 spatial dimensions...
Research Article

Nonlocal Symmetries and the Complete Symmetry Group of 1 + 1 Evolution Equations

S.M. Myeni, P.G.L. Leach
Pages: 377 - 392
The complete symmetry group of a 1 + 1 linear evolution equation has been demon- strated to be represented by the six-dimensional Lie algebra of point symmetries sl(2, R)?s W , where W is the three-dimensional Heisenberg-Weyl algebra. The infinite number of solution symmetries does not play a role in...
Letter to Editor

On some connections between the Gompertz function and special numbers

Grzegorz Rządkowski, Wojciech Rządkowski, Paweł Wójcicki
Pages: 374 - 380
In the present paper we show that the Gompertz function, the Fisher–Tippett and the Gumbel probability distributions are related to both Stirling numbers of the second kind and Bernoulli numbers. Especially we prove for the Gumbel probability density function an analog of the Grosset–Veselov formula...
Research Article

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

Hui Mao, Q.P. Liu, Lingling Xue
Pages: 375 - 386
In this paper, we construct a Darboux transformation and the related Bäcklund transformation for the super-symmetric Sawada-Kotera (SSK) equation. The associated nonlinear superposition formula is also worked out. We demonstrate that these are natural extensions of the similar results of the Sawada-Kotera...
Research Article

Relativistic Two-Body Problem: Existence and Uniqueness of Two-Sided Solutions to Functional-Differential Equations of Motion

V.I. Zhdanov
Pages: 379 - 384
We study a class of explicitly Poincare-invariant equations of motion (EMs) of two point bodies with a finite speed of propagation of interactions (combination of retarded and advanced ones) that may be considered as functional-differential equations or differential equations with deviating argument...
Research Article

Theory of Economic Equilibrium

Nikolai Gonchar
Pages: 380 - 400
New concepts of economics such as an average demand matrix of society, strategy of a firm and consumer behaviour, and others are introduced. We give sufficient conditions for technological mapping under which there exist both the Walras equlibrium state and optimal Walras equilibrium one. We obtain the...
Research Article

A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)

Darryl D. Holm, Andrew N.W. Hone
Pages: 380 - 394
We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x). When b = 2 and g is the peakon kernel (i.e. g(x) = exp(-|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained...
Research Article

A Large Time Asymptotics for Transparent Potentials for the Novikov–Veselov Equation at Positive Energy

A. V. Kazeykina, R. G. Novikov
Pages: 377 - 400
In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov–Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized...
Research Article

The Transfer Integral Operator Method in the Study of DNA Unzipping and Bubble Formation

Z. Rapti, K. Ø. Rasmussen, A. R. Bishop
Pages: 381 - 396
In this work we reexamine the unzipping and bubble formation of DNA in the context of the Peyrard–Bishop–Dauxois DNA model using the transfer integral operator method. After a brief overview of the method, we use it to calculate the probabilities that consecutive base-pairs are stretched beyond a threshold...
Research Article

The Structure of Gelfand-Levitan-Marchenko Type Equations for Delsarte Transmutation Operators of Linear Multi-Dimensional Differential Operators and Operator Pencils. Part 2

Jolanta Golenia, Anatolij K. Prykarpatsky, Yarema A. Prykarpatsky
Pages: 381 - 408
The differential-geometric and topological structure of Delsarte transmutation opertors their associated Gelfand-Levitan-Marchenko type equations are studied making use of the de Rham-Hodge-Skrypnik differential complex. The relationships with spetral theory and special Berezansky type congruence properties...
Research Article

The Influence of Quantum Field Fluctuations on Chaotic Dynamics of Yang­Mills System

V.I. Kuvshinov, A.V. Kuzmin
Pages: 382 - 388
On example of the model field system we demonstrate that quantum fluctuations of non-abelian gauge fields leading to radiative corrections to Higgs potential and spontaneous symmetry breaking can generate order region in phase space of inherently chaotic classical field system. We demonstrate on the...
Research Article

Primary Branch Solutions of First Order Autonomous Scalar Partial Differential Equations via Lie Symmetry Approach

S. Y. Lou, Ruo Xia Yao
Pages: 379 - 392
A primary branch solution (PBS) is defined as a solution with m independent n − 1 dimensional arbitrary functions for an m order n dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be determined by using Lie symmetry group approach companying with the introduction...
Research Article

Integrability of Certain Deformed Nonlinear Partial Differential Equations

R. Sahadevan, L. Nalinidevi
Pages: 379 - 396
A systematic investigation of certain higher order or deformed soliton equations with (1 + 1) dimensions, from the point of complete integrability, is presented. Following the procedure of Ablowitz, Kaup, Newell and Segur (AKNS) we find that the deformed version of Nonlinear Schrodinger equation, Hirota...
Research Article

Symmetry analysis for Whitham-Broer-Kaup equations

Zhiyong Zhang, Xuelin Yong, Yufu Chen
Pages: 383 - 397
We investigate a further group analysis of Whitham-Broer-Kaup(for short WBK) equations. An optimal system of one-dimensional subalgebras is derived and used to construct reduced equations and similarity solutions. Moreover, a special case of WBK equations is linearized and some new solutions are obtained....
Research Article

Lie Superalgebra Valued Self-Dual Yang-Mills Fields and Symmetry Reduction

M. Legaré
Pages: 383 - 391
Self-dual Yang-Mills fields with values in a Lie superalgebra on the four-dimensional Euclidean space and pseudo-Euclidean space of signature (2,2) can be reduced by subgroups of the corresponding conformal group to integrable systems with anticommuting degrees of freedom. Examples of reductions are...
Research Article

The Nonabelian Liouville-Arnold Integrability by Quadratures Problem: a Symplectic Approach

Anatoliy K. Prykarpatsky
Pages: 384 - 410
A symplectic theory approach is devised for solving the problem of algebraic-analytical construction of integral submanifold imbeddings for integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on canonically symplectic phase spaces.
Research Article

Two-Parameter Deformation of the Oscillator Algebra and (p, q)­Analog of Two-Dimensional Conformal Field Theory

Ivan Burban
Pages: 384 - 391
The two-parameter deformation of canonical commutation relations is discussed. The self-adjointness property of the (p, q)-deformed position Q and momentum P operators is investigated. The (p, q)-analog of two-dimensional conformal field theory based on the (p, q)-deformation of the su(1, 1) subalgebra...
Research Article

Products of distributions and collision of a δ-wave with a δ′-wave in a turbulent model

C.O.R. Sarrico, A. Paiva
Pages: 381 - 394
We study the possibility of collision of a δ-wave with a stationary δ′-wave in a model ruled by equation f (t)ut+[u2−β(x−γ(t))u]x = 0, where f, β and γ are given real functions and u = u(x, t) is the state variable. We adopt a solution concept which is a consistent extension of the classical solution...
Research Article

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

M.B. Sheftel, A.A. Malykh
Pages: 385 - 395
We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of three- and twodimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions...
Short Communication

Symmetry of a Two-Particle Equation for Parastates

S.P. Onufriichuk, O.I. Prylypko
Pages: 385 - 387
We study hidden symmetry of a two-particle system of equations for parastates. Invariance operators are described for various potentials. It is a well-known fact that the systems of partial differential equations have a hidden symmetry, which can not be observed in the classical approach of Lie [1].
Research Article

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

Dan-dan Xu, Da-jun Zhang, Song-lin Zhao
Pages: 382 - 406
The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation KM + MK = r sT we introduce a scalar function S(i, j) = sT Kj (I + M)−1Kir which is defined as same as in discrete case. S(i, j) satisfy some recurrence...
Research Article

New extension of dispersionless Harry Dym hierarchy

Hongxia Wu, Jingxin Liu, Yunbo Zeng
Pages: 383 - 398
The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new...
Research Article

The Toy Top, an Integrable System of Rigid Body Dynamics

Boris A. Springborn
Pages: 387 - 410
A toy top is defined as a rotationally symmetric body moving in a constant gravittional field while one point on the symmetry axis is constrained to stay in a horizontal plane. It is an integrable system similar to the Lagrange top. Euler-Poisson equtions are derived. Following Felix Klein, the special...
Research Article

A two-point boundary value problem on a Lorentz manifold arising in A. Poltorak's concept of reference frame

Yuri E. Gliklikh, Peter S. Zykov
Pages: 388 - 397
In A. Poltorak's concept, the reference frame in General Relativity is a certain manifold equipped with a connection. The question under consideration here is whether it is possible to join two events in the space-time by a time-like geodesic if they are joined by a geodesic of the reference frame connection...
Short Communication

On Symmetry of the Generalized Breit Equation

S.P. Onufriichuk, O.I. Prylypko
Pages: 388 - 390
In this paper we find the complete set of symmetry operators for the two-particle Breit equation in the class of first-order differential operators with matrix coefficients. A new integral of motion is obtained.
Research Article

The Scattering Approach for the Camassa—­Holm equation

Jonatan Lenells
Pages: 389 - 393
We present an approach proving the integrability of the Camassa­—Holm equation for initial data of small amplitude.
Research Article

Integrability Conditions for Complex Homogeneous Kukles Systems

Jaume Giné, Claudia Valls
Pages: 387 - 398
In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the form ẋ = x + Pn(x, y), ẏ = −y, where Pn(x,y) is a homogeneous polynomial of degree n, called the complex homogeneous Kukles systems of degree n. We characterize all the homogeneous...
Research Article

Nonlinear Maxwell Equations

G.A. Kotelnikov
Pages: 391 - 395
The infinite series of Lorentz and Poincaré-invariant nonlinear versions of the Maxwell equations are suggested. Some properties of these equations are considered.
Research Article

Conformally Invariant Ansätze for the Maxwell Field

Victor Lahno
Pages: 392 - 400
A general procedure for construction of conformally invariant Ansätze for the Maxwell field is suggested. Ansätze invariant with respect to inequivalent three-parameter subgroups of the conformal group are constructed.
Research Article

Similarity Reductions of the Zabolotskaya-Khokhlov Equation with a Dissipative Term

Masayoshi Tajiri
Pages: 392 - 397
Similarity reductions of the Zabolotskaya-Khokhlov equation with a dissipative term to one-dimensional partial differential equations including Burgers' equation are investigated by means of Lie's method of infinitesimal transformation. Some similarity solutions of the Z-K equation are obtained.
Research Article

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

A.M. Grundland, B. Huard
Pages: 393 - 419
In this paper we employ a “direct method” to construct rank-k solutions, express- ible in Riemann invariants, to hyperbolic system of first order quasilinear differential equations in many dimensions. The most important feature of our approach is the analysis of group invariance properties of these solutions...
Research Article

Two-component generalizations of the Novikov equation

Hongmin Li
Pages: 390 - 403
Some two-component generalizations of the Novikov equation, except the Geng-Xue equation, are presented, as well as their Lax pairs and bi-Hamiltonian structures. Furthermore, we study the Hamiltonians of the Geng-Xue equation and discuss the homogeneous and local properties of them.
Research Article

On Einstein Equations on Manifolds and Supermanifolds

D. Leites, E. Poletaeva, V. Serganova
Pages: 394 - 425
The Einstein equations (EE) are certain conditions on the Riemann tensor on the real Minkowski space M. In the twistor picture, after complexification and compactifcation M becomes the Grassmannian Gr4 2 of 2-dimensional subspaces in the 4-dimesional complex one. Here we answer for which of the classical...
Research Article

Laplace Invariants for General Hyperbolic Systems

Chris Athorne, Halis Yilmaz
Pages: 391 - 410
We consider the generalization of Laplace invariants to linear differential systems of arbitrary rank and dimension. We discuss completeness of certain subsets of invariants.
Research Article

The Eigenvectors of the Heisenberg Hamiltonian with Elliptic Form of the Exchange Spin Interaction

V.I. Inozemtsev
Pages: 395 - 403
The eigenvectors of the Hamiltonian HN of N-site quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvetors via the solutions to the system of highly transcendental...
Research Article

The Point of Maximum Curvature as a Marker for Physiological Time Series

James Robert Stirling, Maria Zakynthinaki
Pages: 396 - 406
We present a geometric analysis of the model of Stirling et al. [14]. In particular we analyze the curvature of a heart rate time series in response to a step like increment in the exercise intensity. We present solutions for the point of maximum curvature which can be used as a marker of physiological...
Research Article

On Infinitesimal Symmetries of the Self-Dual Yang-Mills Equations

T.A. Ivanova
Pages: 396 - 404
Infinite-dimensional algebra of all infinitesimal transformations of solutions of the self-dual Yang-Mills equations is described. It contains as subalgebras the infinitedimensional algebras of hidden symmetries related to gauge and conformal transformations.
Research Article

*-Representations of the Quantum Algebra Uq(sl(3))

L.B. Turovskaya
Pages: 396 - 401
Studied in this paper are real forms of the quantum algebra Uq(sl(3)). Integrable operator representations of *-algebras are defined. Irreducible representations are classified up to a unitary equivalence.
Research Article

Proper rational and analytic first integrals for asymmetric 3-dimensional Lotka-Volterra systems

Jaume Llibre, Clàudia Valls
Pages: 393 - 404
We go beyond in the study of the integrability of the classical model of competition between three species studied by May and Leonard [19], by considering a more realistic asymmetric model. Our results show that there are no global analytic first integrals and we provide all proper rational first integrals...
Research Article

Integrable Boundary Conditions for the Hirota-Miwa Equation and Lie Algebras

Ismagil Habibullin, Aigul Khakimova
Pages: 393 - 413
Systems of discrete equations on a quadrilateral graph related to the series DN(2) of the affine Lie algebras are studied. The systems are derived from the Hirota-Miwa equation by imposing boundary conditions compatible with the integrability property. The Lax pairs for the systems are presented. It...
Research Article

Bayesian Approach to the Determination of the Kinetic Parameters of DNA Hairpins Under Tension

Marco Ribezzi-Crivellari, Mario Wagner, Felix Ritort
Pages: 397 - 410
In this paper we propose a Bayesian scheme for the determination of the unfolding and refolding kinetic rates of DNA hairpins under tension. This method is based on the hypothesis that the unfolding-refolding dynamics is well described by a Markov Chain. The results from the Bayesian method are contrasted...
Research Article

Darboux integrability of a generalized Friedmann-Robertson-Walker Hamiltonian system

Jaume Llibre, Claudia Valls
Pages: 394 - 406
We study the Darboux first integrals of a generalized Friedmann-Robertson-Walker Hamiltonian system.
Research Article

Euler-Poincaré Formalism of (Two Component) Degasperis-Procesi and Holm-Staley type Systems

Partha Guha
Pages: 398 - 429
In this paper we propose an Euler-Poincaré formalism of the Degasperis and Procesi (DP) equation. This is a second member of a one-parameter family of partial dif- ferential equations, known as b-field equations. This one-parameter family of pdes includes the integrable Camassa-Holm equation as a first...
Research Article

Doubly periodic waves of a discrete nonlinear Schrodinger system with saturable nonlinearity

Robert Conte, K. W. Chow
Pages: 398 - 409
A system of two discrete nonlinear Schr¨odinger equations of the Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly periodic wave, whose long wave limit is also derived. As a by-product, several new solutions of the elliptic type are provided for NLS-type discrete and continuous...
Research Article

The Bäcklund and the Galilei Invariant Transformations Constructed by Similarity Variables for Soliton Equations

Shunji Kawamoto
Pages: 398 - 404
The Painlevé-test has been applied to checking the integrability of nonlinear PDEs, since similarity solutions of many soliton equations satisfy the Painlevé equation. As is well known, such similarity solutions can be obtained by the infinitesimal transformation, that is, the classical similarity analysis,...
Research Article

Algebro-geometric Constructions of Quasi Periodic Flows of the Discrete Self-dual Network Hierarchy and Applications

Dong Gong, Xianguo Geng
Pages: 395 - 427
In this paper we obtain the discrete integrable self-dual network hierarchy associated with a discrete spectral problem. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the discrete self-dual network hierarchy are straightened using the Abel-Jacobi coordinates....
Research Article

Sundman Symmetries of Nonlinear Second-Order and Third-Order Ordinary Differential Equations

Norbert Euler, Marianna Euler
Pages: 399 - 421
We investigate the Sundman symmetries of second-order and third-order nonlinear odinary differential equations. These symmetries, which are in general nonlocal tranformations, arise from generalised Sundman transformations of autonomous equations. We show that these transformations and symmetries can...
Research Article

Two New Solvable Dynamical Systems of Goldfish Type

F. Calogero
Pages: 397 - 414
Two new solvable dynamical systems of goldfish type are identified, as well as their isochronous variants. The equilibrium configurations of these isochronous variants are simply related to the zeros of appropriate Laguerre and Jacobi polynomials.
Research Article

On Radial Schrödinger Equations in Curved Spaces and Their Spectra Through Nonlinear Constraints

J. Beckers, N. Debergh
Pages: 401 - 408
First, we determine the radial Schrödinger equation in D-dimensional curved spaces when central problems are considered. Second, we develop the so-called factorization method on the basis of supersymmetric arguments for solving such radial equations when D = 1, 2, 3-harmonic oscillator and D = 3-hydrogen...
Research Article

Eigenvectors of the recursion operator and a symmetry structure for the coupled KdV hierarchy

Sen-Yue Lou
Pages: 401 - 413
It is shown that eigenvectors of the recursion operator L with the eigenvalue i and the inverse of the recursion operator Li L-i for the coupled KdV hierarchy (CKdVH) can be obtained in terms of squared eigenfunctions of the associated linear problem. The symmetry structure and corresponding infinite...
Research Article

Three-Generation Distler-Kachru Models

Yu.I. Samoilenko, Yu.M. Malyuta, N.N. Aksenov
Pages: 402 - 408
Research Article

Explicit solutions for a nonlinear model on the honeycomb and triangular lattices

V.E. Vekslerchik
Pages: 399 - 422
We study a simple nonlinear model defined on the honeycomb and triangular lattices. We propose a bilin-earization scheme for the field equations and demonstrate that the resulting system is closely related to the well-studied integrable models, such as the Hirota bilinear difference equation and the Ablowitz-Ladik...
Research Article

The description of reflection coefficients of the scattering problems for finding solutions of the Korteweg–de Vries equations

Pham Loi Vu
Pages: 399 - 432
The results of inverse scattering problem associated with the initial-boundary value problem (IBVP) for the Korteweg–de Vries (KdV) equation with dominant surface tension are formulated. The necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients of the...
Research Article

Canonically Transformed Detectors Applied to the Classical Inverse Scattering Problem

C. Jung, T.H. Seligman, J.M. Torres
Pages: 404 - 411
The concept of measurement in classical scattering is interpreted as an overlap of a particle packet with some area in phase space that describes the detector. Considering that usually we record the passage of particles at some point in space, a common detector is described e.g. for one-dimensional systems...
Research Article

Non-Lie Symmetries and Supersymmetries

Anatolii Nikitin
Pages: 405 - 415
Appeared more than one century ago, the classical Lie approach serves as a powerful tool in investigations of symmetries of partial differential equations. In the last three decades there appear essential generalizations of this approach. They are the modern version of the Lie-Bäcklund symmetries [1],...
Research Article

How to Find Discrete Contact Symmetries

Peter E. Hydon
Pages: 405 - 416
This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism...
Research Article

Semidiscrete Integrable Nonlinear Systems Generated by the New Fourth-Order Spectral Operator: Local Conservation Laws

Oleksiy O. Vakhnenko
Pages: 401 - 414
Starting with the semidiscrete integrable nonlinear Schrödinger system on a zigzag-runged ladder lattice we have presented the generalization and an essentially off-diagonal enlargement of its spectral operator which in the framework of zero-curvature equation allows to generate at least two new types...
Research Article

On the Origin of Fractional Shapiro Steps in Systems of Josephson Junctions with Few Degrees of Freedom

A. Valizadeh, M.R. Kolahchi, J.P. Straley
Pages: 407 - 416
We investigate the origin of fractional Shapiro steps in arrays consisting of a few overdamped Josephson junctions. We show that when the symmetry reduces the equations to that of a single junction equation, only integer steps appear. Otherwise, fractional steps will appear when the evolution equations...
Research Article

Bilinear Identities and Squared Eigenfunction Symmetries of the BCr-KP Hierarchy

Lumin Geng, Huizhan Chen, Na Li, Jipeng Cheng
Pages: 404 - 419
The BCr-KP hierarchy is an important sub hierarchy of the KP hierarchy, which includes the BKP and CKP hierarchies as the special cases. Some properties of the BCr-KP hierarchy and its constrained case are investigated in this paper, including bilinear identities and squared eigenfunction symmetries....
Research Article

On Classification of Subalgebras of the Poincaré Algebra

Leonid F. Barannyk
Pages: 409 - 417
The substantiation of the algorithm for classifying subalgebras of the Poincaré algebra AP(1, n) up to P(1, n)-conjugacy is completed
Research Article

Nonlocal Symmetry of Nonlinear Wave Equations

V.A. Tychynin
Pages: 409 - 413
A class of nonlinear wave equations is considered. Symmetry of these equations is extended using nonlocal transformations.
Research Article

Functional Representation of the Volterra Hierarchy

V.E Vekslerchik
Pages: 409 - 431
In this paper I study the functional representation of the Volterra hierarchy (VH). Using the Miwa's shifts I rewrite the infinite set of Volterra equations as one functional equation. This result is used to derive a formal solution of the associated linear problem, a generating function for the conservation...
Research Article

Ideals generated by traces or by supertraces in the symplectic reflection algebra H1,ν (I2(2m + 1))

S.E. Konstein, I.V. Tyutin
Pages: 405 - 425
For each complex number ν, an associative symplectic reflection algebra ℋ := H1,ν (I2(2m + 1)), based on the group generated by root system I2(2m + 1), has an m-dimensional space of traces and an (m + 1)-dimensional space of supertraces. A (super)trace sp is said to be degenerate if the corresponding...
Research Article

Asymptotically isochronous systems

Francesco Calogero, David Gomez-Ullate
Pages: 410 - 426
Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the same fixed period. We focus on two such mechanisms, emphasizing...
Research Article

Phenomenological Modeling of DNA Overstretching

Ray W. Ogden, Giuseppe Saccomandi, Ivonne Sgura
Pages: 411 - 427
A phenomenological model based on the three-dimensional theory of nonlinear elasticity is developed to describe the phenomenon of overstretching in the force-extension curve for double-stranded DNA (dsDNA). By using the concept of a material with multiple reference configurations a single formula is...
Research Article

r-Matrices for Relativistic Deformations of Integrable Systems

Yuri B. Suris
Pages: 411 - 447
We include the relativistic lattice KP hierarchy, introduced by Gibbons and Kupershmidt, into the r-matrix framework. An r-matrix account of the nonrelativistic lattice KP hierarchy is also provided for the reader's convenience. All relativistic constructions are regular one-parameter perturbations of...
Research Article

Dromion Perturbation for the Davey-Stewartson-1 Equations

O.M. Kiselev
Pages: 411 - 422
The perturbation of the dromion of the Davey-Stewartson-1 equation is studied over the large time.
Research Article

Dark Equations and Their Light Integrability

Denis Blackmore, Anatolij K. Prykarpatski
Pages: 407 - 428
A relatively new approach to analyzing integrability, based upon differential-algebraic and symplectic techniques, is applied to some “dark equations ”of the type introduced by Boris Kupershmidt. These dark equations have unusual properties and are not particularly well-understood. In particular, dark...
Research Article

Rogue wave management in an inhomogeneous Nonlinear Fibre with higher order effects

J.S. He, Y.S. Tao, K. Porsezian, A.S. Fokas
Pages: 407 - 419
We consider an inhomogeneous Hirota equation with variable dispersion and nonlinearity. We introduce a novel transformation which maps this equation to a constant coefficient Hirota equation. By employing this transformation we construct the rogue wave solution of the inhomogeneous Hirota equation. Furthermore,...
Research Article

Reflection Symmetries of q-Bernoulli Polynomials

Boris A. Kupershmidt
Pages: 412 - 422
A large part of the theory of classical Bernoulli polynomials Bn(x)'s follows from their reflection symmetry around x = 1/2: Bn(1 - x) = (-1)n Bn(x). This symmetry not only survives quantization but has two equivalent forms, classical and quantum, depending upon whether one reflects around 1/2 the classical...
Research Article

q-Deformed Dressing Operators And Modified Integrable Hierarchies

I. Mukhopadhaya, A. Roy Chowdhury
Pages: 414 - 419
A q-deformation of the dressing operator introduced by Sato is suggested. It is shown that it produces q-deformation of known integrable heirarchies, with the infinite number of conservation laws. A modification introduced by Kupershmidt when incorporated leads to both modified and deformed integrable...
Short Communication

Conditional and Lie Symmetry of Nonlinear Wave Equation

Victor Repeta
Pages: 414 - 416
Group classification of the nonlinear wave equation is carried out and the conditional invariance of the wave equation with the nonlinearity F(u) = u is found.
Research Article

On the Nonlocal Symmetries of the μ-Camassa–Holm Equation

Ognyan Christov
Pages: 411 - 427
The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm and the Hunter–Saxton equations. This equation admits quadratic pseudo-potentials which allow us to compute some first-order nonlocal symmetries. The found symmetries preserve...