Journal of Nonlinear Mathematical Physics

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1499 articles
Research Article

The Classification of Almost Affine (Hyperbolic) Lie Superalgebras

Danil Chapovalov, Maxim Chapovalov, Alexei Lebedev, Dimitry Leites
Pages: 103 - 161
We say that an indecomposable Cartan matrix A with entries in the ground field is almost affine if the Lie (super)algebra determined by it is not finite dimensional or affine (Kac–Moody) but the Lie sub(super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting...
Research Article

The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra

Anahita Eslami Rad, Jean-Pierre Magnot, Enrique G. Reyes
Pages: 103 - 120
Mulase solved the Cauchy problem of the Kadomtsev-Petviashvili (KP) hierarchy in an algebraic category in “Solvability of the super KP equation and a generalization of the Birkhoff decomposition” (Inventiones Mathematicae, 1988), making use of a delicate factorization of an infinite-dimensional group...
Research Article

On a spectral analysis of scattering data for the Camassa-Holm equation

Chueh-Hsin Chang, Tony Wen-Hann Sheu
Pages: 102 - 116
Physical details of the Camassa–Holm (CH) equation that are difficult to obtain in space-time simulation can be explored by solving the Lax pair equations within the direct and inverse scattering analysis context. In this spectral analysis of the completely integrable CH equation we focus solely on the...
Research Article

EPR-B correlations: non-locality or geometry?

A.F. Kracklauer
Pages: 104 - 109
A photoelectron-by-photoelectron classical simulation of EPR-B correlations is dscribed. It is shown that this model can be made compatible with Bell's renowned "no-go" theorem by restricting the source to that which produces only what is known as paired photons.
Research Article

Superposition formulas for integrable vector evolutionary equations on a Sphere

M.Ju. Balakhnev
Pages: 104 - 116
The superposition formulas for solutions of integrable vector evolutionary equations on a sphere are constructed by means of auto-B ?acklund transformation. The equations under consideration were obtained earlier by Sokolov and Meshkov in the frame of the symmetry approach.
Research Article

Solitary Traveling Water Waves of Moderate Amplitude

Anna Geyer
Pages: 104 - 115
We prove the existence of solitary traveling wave solutions for an equation describing the evolution of the free surface for waves of moderate amplitude in the shallow water regime. This nonlinear third-order partial differential equation arises as an approximation of the Euler equations, modeling the...
Research Article

Isochronous Oscillators

F. Calogero, F. Leyvraz
Pages: 103 - 110
We exhibit the solution of the initial-value problem for the system of 2N + 2 oscillators characterized by the Hamiltonian H(p^0,pˇ0,p^_,pˇ_,q^0,qˇ0,q^_,qˇ_)=12[p^02−pˇ02+Ω2(q^02−qˇ02)]        +q^0−Ωqˇ02b∑n=1N[p^n2−pˇn2+ωn2(q^n2−qˇn2)]+p^0−Ωq^0b∑n=1N[−p^npˇn+ωn2q^nqˇn] where N is an arbitrary positive...
Research Article

A Nonlinearly Dispersive Fifth Order Integrable Equation and its Hierarchy

Ashok Das, Ziemowit Popowicz
Pages: 105 - 117
In this paper, we study the properties of a nonlinearly dispersive integrable system of fifth order and its associated hierarchy. We describe a Lax representation for such a system which leads to two infinite series of conserved charges and two hierarchies of equations that share the same conserved charges....
Research Article

On the Exact Solutions of the Nonlinear Wave and (omega)4-Model Equations

A.H. Kara, A.H. Bokhari, F.D. Zaman
Pages: 105 - 111
The nonlinear wave equation with variable long wave velocity and the Gordon-type equations (in particular, the omega-model equation) display a range of symmetry generators, inter alia, translations, Lorentz rotations and scaling - all of which are related to conservation laws. We do a study of the symmetries...
Research Article

Hyperelliptic Addition Law

Victor Buchstaber, Dmitry Leykin
Pages: 106 - 123
Given a family of genus g algebraic curves, with the equation f(x, y, ) = 0, we cosider two fiber-bundles U and X over the space of parameters . A fiber of U is the Jacobi variety of the curve. U is equipped with the natural groupoid structure that induces the canonical addition on a fiber. A fiber of...
Research Article

Adjoint Symmetry Constraints Leading to Binary Nonlinearization

Wen-Xiu Ma, Ruguang Zhou
Pages: 106 - 126
Adjoint symmetry constraints are presented to manipulate binary nonlinearization, and shown to be a slight weaker condition than symmetry constraints in the case of Hamiltonian systems. Applications to the multicomponent AKNS system of nonlinear Schrödinger equations and the multi-wave interaction equations,...
Research Article

Painlevé Analysis and Singular Manifold Method for a (2 + 1) Dimensional Non-Linear Schrödinger Equation

P.G. Estévez, G.A. Hernáez
Pages: 106 - 111
The real version of a (2 + 1) dimensional integrable generalization of the nonlinear Schrödinger equation is studied from the point of view of Painlevé analysis. In this way we find the Lax pair, Darboux transformations and Hirota's functions as well as solitonic and dromionic solutions from an iterative...
Research Article

Reflectionless Analytic Difference Operators I. Algebraic Framework

S.N.M. Ruijsenaars
Pages: 106 - 138
We introduce and study a class of analytic difference operators admitting reflectionless eigenfunctions. Our construction of the class is patterned after the Inverse Scattering Transform for the reflectionless self-adjoint Schrödinger and Jacobi operators corrsponding to KdV and Toda lattice solitons.
Research Article

Isochronous Dynamical System and Diophantine Relations I

F. Calogero, S. Iona
Pages: 105 - 116
We identify a solvable dynamical system — interpretable to some extent as a many-body problem — and point out that — for an appropriate assignment of its parameters — it is entirely isochronous, namely all its nonsingular solutions are completely periodic (i.e., periodic in all degrees of freedom) with...
Research Article

On the Lax pairs of the continuous and discrete sixth Painlevé equations

Robert Conte, Runliang Lin, Micheline Musette
Pages: 107 - 118
Among the recently found discretizations of the sixth Painlevé equation P6, only the one of Jimbo and Sakai admits a discrete Lax pair, which does establish its integrabiity. However, a subtle restriction in this Lax pair prevents the possibility to generalize it in order to find the other missing Lax...
Research Article

On a Two-Parameter Extension of the Lattice KdV System Associated with an Elliptic Curve

Frank W. Nijhoff, Sian E. Puttock
Pages: 107 - 123
A general structure is developed from which a system of integrable partial difference equations is derived generalising the lattice KdV equation. The construction is based on an infinite matrix scheme with as key ingredient a (formal) elliptic Cauchy kernel. The consistency and integrability of the lattice...
Research Article

New Spherically Symmetric Solutions of Nonlinear Schrödinger Equations

Roman M. Cherniha
Pages: 107 - 113
New soliton-like spherically symmetric solutions for nonlinear generalizations of the Schrödiner equation are constructed. A new nonlinear projective invariant Schrödiner equation is suggested and formulae of multiplication of its solutions are found.
Research Article

Invariant Solutions of Nonlinear Diffusion Equations with Maximal Symmetry Algebra

V. A. Galaktionov, S. R. Svirshchevskii
Pages: 107 - 121
Nonlinear n-dimensional second-order diffusion equations admitting maximal Lie algebras of point symmetries are considered. Examples of invariant solutions, as well as of solutions on invariant subspaces for some nonlinear operators, are constructed for arbitrary n. A complete description of all possible...
Research Article

Twisted Symmetries of Differential Equations

Giuseppe Gaeta
Pages: 107 - 136
We review the basic ideas lying at the foundation of the recently developed theory of twisted symmetries of differential equations, and some of its developments.
Research Article

On the Discretization of the Coupled Integrable Dispersionless Equations

Luc Vinet, Guo-Fu Yu
Pages: 106 - 125
We study the integrable discretization of the coupled integrable dispersionless equations. Two semi-discrete version and one full-discrete version of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented.
Research Article

The Group-Theoretical Analysis of Nonlinear Optimal Control Problems with Hamiltonian Formalism

Gülden Gün Polat, Teoman Özer
Pages: 106 - 129
In this study, we pay attention to novel explicit closed-form solutions of optimal control problems in economic growth models described by Hamiltonian formalism by utilizing mathematical approaches based on the theory of Lie groups. For this analysis, the Hamiltonian functions, which are used to define...
Research Article

Meromorphic and formal first integrals for the Lorenz system

Kaiyin Huang, Shaoyun Shi, Wenlei Li
Pages: 106 - 121
The Lorenz system x˙=σ(y−x), y˙=rx−y−xz, z˙=−βz+xy, is completely integrable with two functional independent first integrals when σ = 0 and β, r arbitrary. In this paper, we study the integrability of the Lorenz system when σ, β, r take the remaining values. For the case of σβ ≠ 0, we consider the...
Research Article

Rational solutions to Q3δ in the Adler-Bobenko-Suris list and degenerations

Song-lin Zhao, Da-jun Zhang
Pages: 107 - 132
We derive rational solutions in Casoratian form for the Nijhoff-Quispel-Capel (NQC) equation by using the lattice potential Korteweg-de Vries (lpKdV) equation and two Miura transformations between the lpKdV and the lattice potential modified KdV (lpmKdV) and the NQC equation. This allows us to present...
Research Article

Bilinearization and soliton solutions of N=1 supersymmetric coupled dispersionless integrable system

Arifa Mirza, Mahmood ul Hassan
Pages: 107 - 115
An N=1 supersymmetric generalization of coupled dispersionless (SUSY-CD) integrable system has been proposed by writing its superfield Lax representation. It has been shown that under a suitable variable transformation, the SUSY-CD integrable system is equivalent to N=1 supersymmetric sine-Gordon equation....
Research Article

On Hermitian and Skew-Hermitian Matrix Algebras over Octonions

Arezoo Zohrabi, Pasha Zusmanovich
Pages: 108 - 122
We prove simplicity of algebras in the title, and compute their δ-derivations and symmetric associative forms.
Research Article

Ermakov-Modulated Nonlinear Schrödinger Models. Integrable Reduction

Colin Rogers, Giuseppe Saccomandi, Luigi Vergori
Pages: 108 - 126
Nonlinear Schrödinger equations with spatial modulation associated with integrable Hamiltonian systems of Ermakov-Ray-Reid type are introduced. An algorithmic procedure is presented which exploits invariants of motion to construct exact wave packet representations with potential applications in a wide...
Research Article

Bosonic Realizations of the Colour Heisenberg Lie Algebra

Gunnar Sigurdsson, Sergei D. Silvestrov
Pages: 110 - 128
We describe realizations of the colour analogue of the Heisenberg Lie algebra by power series in non-commuting indeterminates satisfying Heisenberg's canonical commutation relations of quantum mechanics. The obtained formulas are used to construct new operator representations of the colour Heisenberg...
Research Article

Nonlinear Wave Equation in Special Coordinates

Alexander Shermenev
Pages: 110 - 115
Some classical types of nonlinear periodic wave motion are studied in special coodinates. In the case of cylinder coordinates, the usual perturbation techniques leads to the overdetermined systems of linear algebraic equations for unknown coefficients whose compatibility is key step of the investigation....
Research Article

On CP 1 and CP 2 Maps and Weierstrass Representations for Surfaces Immersed into Multi-Dimensional Euclidean Spaces

A.M. Grundland, W.J. Zakrzewski
Pages: 110 - 135
An extension of the classic Enneper­Weierstrass representation for conformally prametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional CP1 and CP2 sigma models which allow the study of the constant mean curvature (CMC) surfaces immersed into Euclidean 3- and 8-dimensional...
Research Article

Jacobi's Last Multiplier and the Complete Symmetry Group of the Euler­Poinsot System

M.C. Nucci, P.G.L. Leach
Pages: 110 - 121
The symmetry approach to the determination of Jacobi's last multiplier is inverted to provide a source of additional symmetries for the Euler­Poinsot system. These addtional symmetries are nonlocal. They provide the symmetries for the representation of the complete symmetry group of the system.
Research Article

Point classification of second order ODEs and its application to Painlevé equations

Vera V. Kartak
Pages: 110 - 129
The first part of this work is a review of the point classification of second order ODEs done by Ruslan Sharipov. His works were published in 1997–1998 in the Electronic Archive at LANL. The second part is an application of this classification to Painlevé equations. In particular, it allows us to solve...
Research Article

Geometric Realization of the Two-Point Velocity Correlation Tensor for Isotropic Turbulence

Vladimir N. Grebenev, Martin Oberlack
Pages: 109 - 120
A new geometric view of homogeneous isotropic turbulence is contemplated employing the two-point velocity correlation tensor of the velocity fluctuations. We show that this correlation tensor generates a family of pseudo-Riemannian metrics. This enables us to specify the geometry of a singled out Eulerian...
Research Article

The Transition of 2-Dimensional Solitons to 1-Dimensional Ones on Hexagonal Lattices

Betti Hartmann, Wojtek J. Zarkzewski
Pages: 111 - 116
We study solitons arising in a system describing the interaction of a two-dimensional discrete hexagonal lattice with an additional electron field (or, in general, an exciton field). We assume that this interaction is electron-phonon-like. In our previous paper [4] we have studied the existence of two-dimensional...
Research Article

Symmetry in Nonlinear Mechanics: Averaging and Normalization Procedures, New Problems and Algorithms

Alexey K. Lopatin
Pages: 111 - 129
The idea of introducing coordinate transformations to simplify the analytic expression of a general problem is a powerful one. Symmetry and differential equations have been close partners since the time of the founding masters, namely, Sophus Lie (1842­1899), and his disciples. To this days, symmetry...
Research Article

Analysis of Timing Jitter for Ultrashort Soliton Communication Systems Using Perturbation Methods

Margarida Facão, Mário Ferreira
Pages: 112 - 117
We analyse timing jitter of ultrashort soliton systems taking into account the major higher order effects, namely, intrapulse Raman scattering and third order dispersion and using adiabatic perturbation theory. We obtain an expression for the soliton arrival time variance that depends on the quintic...
Research Article

On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators

Sofiane Bouarroudj
Pages: 112 - 127
Let Vect(R) be the Lie algebra of smooth vector fields on R. The space of symbols Pol(T R) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(R)-module that becomes trivial once the action is restricted to sl(2) Vect(R). The deformations of Pol(T R), which...
Research Article

Complete Invariant Characterization of Scalar Linear (1+1) Parabolic Equations

Fazal M. Mahomed
Pages: 112 - 123
We obtain a complete invariant characterization of scalar linear (1+1) parabolic equations under equivalence transformations for all the four canonical forms. Firstly semi-invariants under changes of independent and dependent variables and the construction of the relevant transformations that relate...
Research Article

Remarks on the Waterbag Model of Dispersionless Toda Hierarchy

Jen-Hsu Chang
Pages: 112 - 123
We construct the free energy associated with the waterbag model of dToda. Also, relations for conserved densities are investigated.
Research Article

Solvable Systems of Isochronous, Multi-Periodic or Asymptotically Isochronous Nonlinear Oscillators

F. Calogero, F. Leyvraz
Pages: 111 - 120
A simple technique is identified to manufacture solvable nonlinear dynamical systems, and in particular three classes whose generic solutions are, respectively, isochronous, multi-periodic, or asymptotically isochronous.
Research Article

Replicator Dynamics and Mathematical Description of Multi-Agent Interaction in Complex Systems

Vasyl V. Gafiychuk, Anatoliy K. Prykarpatsky
Pages: 113 - 122
We consider the general properties of the replicator dynamical system from the stanpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. A Lyaponuv function for the investigation of the evolution of the system has been proposed. The...
Short Communication

Conditional Invariance and Exact Solutions of a Nonlinear System

Pavlo Myronyk, Natalia Bubenchikova
Pages: 114 - 116
The Lie and Q-conditional invariance of one nonlinear system of PDEs of the thirdorder is searched. The ansatze have been built which reduce the PDEs system to ODEs. The classes of exact solutions of the given system are obtained. The relation between the Korteweg-de Vries equation and Harry-Dym equation...
Research Article

(p, q)-Analog of Two-Dimensional Conformal Field Theory. The Ward Identities and Correlation Functions

I.M. Burban
Pages: 114 - 119
A (p, q)-analog of two-dimensional conformally invariant field theory based on the quantum algebra Upq(su(1, 1)) is proposed. The representation of the algebra Upq(su(1, 1)) on the space of quasi-primary fields is given. The (p, q)-deformed Ward identities of conformal field theory are defined. The two-...
Review Article

On a Class of Linearizable Monge-Ampère Equations

D.J. Arrigo, J.M. Hill
Pages: 115 - 119
Monge-Ampère equations of the form, uxxuyy - u2 xy = F(u, ux, uy) arise in many areas of fluid and solid mechanics. Here it is shown that in the special case F = u4 yf(u, ux/uy), where f denotes an arbitrary function, the Monge-Ampère equation can be linearized by using a sequence of Ampère, point, Legendre...
Research Article

Limit Spectra of Random Gram Matrices

V.I. Serdobolskii
Pages: 116 - 121
Solutions to basic non-linear limit spectral equation for matrices RT R of increasing dmension are investigated, where R are rectangular random matrices with independent normal entries. The analytical properties of limiting normed trace for the resolvent of RT R are investigated, boundaries of limit...
Research Article

Nonisospectral Flows on Semi-infinite Jacobi Matrices

Yurij Berezansky, Michael Shmoish
Pages: 116 - 146
Research Article

Geometric Differences between the Burgers and the Camassa-Holm Equations

Boris Kolev
Pages: 116 - 132
The Burgers equation and the Camassa-Holm equations can both be recast as the Euler equation for a right-invariant metric on the diffeomorphism group of the circle, the L 2-metric for Burgers and the H 1-metric for Camassa-Holm. Their geometric behaviors are however very different. We present a survey...
Research Article

Solitary Waves in Open Channels with Abrupt Turns and Branching Points

André Nachbin, Vanessa da Silva Simões
Pages: 116 - 136
The dynamics of solitary waves is studied in intricate domains such as open channels with sharp-bends and branching points. Of particular interest, the wave characteristics at sharp-bends is rationalized by using the Jacobian of the Schwarz–Christoffel transformation. It is observed that it acts in a...
Research Article

Gauge Transformation and Reciprocal Link for (2+1)-Dimensional Integrable Field Systems

Błażej M. Szablikowski
Pages: 117 - 128
Appropriate restrictions of Lax operators which allows to construction of (2+1dimensional integrable field systems, coming from centrally extended algebra of pseuddifferential operators, are reviewed. The gauge transformation and the reciprocal link between three classes of Lax hierarchies are established.
Research Article

Contact Transformations in Classical Mechanics

Yurij Yaremko
Pages: 117 - 123
Transformations of coordinates of points in an infinite-dimensional graded vector space, the so-called contact transformations, are examined. An infinite jet prolongation of the extended configuration space of N spinless particles is the subspace of this vector space. The dynamical equivalence among...
Research Article

An adapted group-dilation anisotropic multifractal formalism for functions

Anouar Ben Mabrouk
Pages: 117 - 139
Anisotropic phenomena can be observed almost everywhere in nature. This makes them important sub jects for theoretical and experimental studies. In this work, we focus on the study of anisotropic quasi-self-similar signals. It holds that the classical multifractal formalism in all its formulations does...
Research Article

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

Tao Xu, Changjing Liu, Fenghua Qi, Chunxia Li, Dexin Meng
Pages: 116 - 141
In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new...
Research Article

q,k-Generalized Gamma and Beta Functions

Rafael Díaz, Carolina Teruel
Pages: 118 - 134
We introduce the q,k-generalized Pochhammer symbol. We construct q,k and Bq,k, the q,k-generalized gamma and beta functions, and show that they satisfy properties that generalize those satisfied by the classical gamma and beta functions. Moreover, we provide integral representations for q,k and Bq,k.
Research Article

Bihamiltonian Geometry and Separation of Variables for Toda Lattices

Gregorio Falqui, Franco Magri, Marco Pedroni
Pages: 118 - 127
We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton­Jacobi method in the so-called Darboux­Nijenhuis coordinates.
Research Article

A Note on Traveling Wave Solutions to the Two Component Camassa–Holm Equation

Keivan Mohajer
Pages: 117 - 125
In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa–Holm equation are distributional traveling wave solutions to the Camassa–Holm equation provided that the set u-1(c), where c is the speed of the wave, is of measure zero. In...
Research Article

Mean-Field Limit of a Microscopic Individual-Based Model Describing Collective Motions

Carlo Bianca, Christian Dogbe
Pages: 117 - 143
This paper is mainly concerned with a mean-field limit and long time behavior of stochastic microscopic interacting particles systems. Specifically we prove that a class of ODE modeling collective interactions in animals or pedestrians converges in the mean-field limit to the solution of a non-local...
Research Article

Polynomial Growth for Birational Mappings from Four-State Spin Edge Models

J-M Maillard
Pages: 119 - 132
We classify all four-state spin edge models according to their behavior under a specific group of birational symmetry transformations generated from the so-called inversion relations. This analysis uses the measure of complexity of the action of birational symetries of these lattice models, and aims...
Research Article

Manin-Olshansky Triples for Lie Superalgebras

Dimitry Leites, Alexander Shapovalov
Pages: 120 - 125
Following V. Drinfeld and G. Olshansky, we construct Manin triples (g, a, a ) such that g is different from Drinfeld's doubles of a for several series of Lie superalgebras a which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization...
Research Article

Explode-Decay Dromions in the Non-Isospectral Davey-Stewartson I (DSI) Equation

R. Radha, S. Vijayalakshmi, M. Lakshmanan
Pages: 120 - 126
In this letter, we report the existence of a novel type of explode-decay dromions, which are exponentially localized coherent structures whose amplitude varies with time, through Hirota method for a nonisospectral Davey-Stewartson equation I discussed recently by Jiang. Using suitable transformations,...
Research Article

A Nonlinear Transformation of the Dispersive Long Wave Equations in (2+1) Dimensions and its Applications

Mingliang Wang, Yubin Zhou, Zhibin Li
Pages: 120 - 125
A nonlinear transformation of the dispersive long wave equations in (2+1) dimensions is derived by using the homogeneous balance method. With the aid of the transformation given here, exact solutions of the equations are obtained.
Research Article

Nonlinear Schrödinger Equations and the Separation Property

Gerald A. Goldin, George Svetlichny
Pages: 120 - 132
Research Article

Focusing mKdV Breather Solutions with Nonvanishing Boundary Condition by the Inverse Scattering Method

Miguel A. Alejo
Pages: 119 - 135
Using the Inverse Scattering Method with a nonvanishing boundary condition, we obtain an explicit breather solution with nonzero vacuum parameter b of the focusing modified Korteweg–de Vries (mKdV) equation. Moreover, taking the limiting case of zero frequency, we obtain a generalization of the double...
Research Article

Variational symmetries and pluri-Lagrangian systems in classical mechanics

Matteo Petrera, Yuri B. Suris
Pages: 121 - 145
We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with m commuting variational...
Research Article

On characteristic integrals of Toda field theories

Zhaohu Nie
Pages: 120 - 131
Characteristic integrals of Toda field theories associated to general simple Lie algebras are constructed using systematic techniques, and complete mathematical proofs are provided. Plenty of examples illustrating the results are presented in explicit forms.
Research Article

The Classification of the Bifurcations Emerging in the case of an Integrable Hamiltonian System with Two Degrees of Freedom when an Isoenergetic Surface is Non-Compact

Galina Goujvina
Pages: 122 - 129
On a symplectical manifold M4 consider a Hamiltonian system with two degrees of freedom, integrable with the help of an additional integral f. According to the welknown Liouville theorem, non-singular level surfaces of the integrals H and f can be represented as unions of tori, cylinders and planes....
Research Article

Should PT Symmetric Quantum Mechanics Be Interpreted as Nonlinear?

Miloslav Znojil
Pages: 122 - 133
The Feshbach-type reduction of the Hilbert space to the physically most relevant "model" subspace is suggested as a means of a formal unification of the standard quantum mechanics with its recently proposed PT symmetric modification. The resuting "effective" Hamiltonians Heff (E) are always Hermitian,...
Research Article

Bernoulli Numbers and Solitons — Revisited

Grzegorz Rządkowski
Pages: 121 - 126
In the present paper we propose a new proof of the Grosset–Veselov formula connecting one-soliton solution of the Korteweg–de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.
Research Article

The Inhomogeneous Invariance Quantum Group of Q-Deformed Boson Algebra with Continuous Parameters

Azmi Ali Altintas, Metin Arik, Ali Serdar Arikan
Pages: 121 - 125
We present a q-deformed boson algebra using continuous momentum parameters and investigate its inhomogeneous invariance quantum group.
Research Article

Competing Species: Integrability and Stability

P.G.L. Leach, J. Miritzis
Pages: 123 - 133
We examine the classical model of two competing species for integrability in terms of analytic functions by means of the Painlevé analysis. We find that the governing equations are integrable for certain values of the essential parameters of the system. We find that, for all integrable cases with the...
Research Article

Reductions for Some Ordinary Differential Equations Through Nonlocal Symmetries

M. L. Gandarias, M. S. Bruzón
Pages: 123 - 133
In [19] we derive nonlocal symmetries for ordinary differential equations by embedding the given equation in an auxiliary system. Since the nonlocal symmetries of the ODE's are local symmetries of the associated auxiliary system this result provides an algorithmic method to derive this kind of nonlocal...
Research Article

Conservation Laws of The Generalized Riemann Equations

Binfang Gao, Kai Tian, Q. P. Liu, Lujuan Feng
Pages: 122 - 135
Two special classes of conserved densities involving arbitrary smooth functions are explicitly formulated for the generalized Riemann equation at arbitrary N. The particular case when N = 2 covers most of the known conserved densities of the equation, and the result is also extended to the famous Gurevich-Zybin,...
Research Article

Goat cheese for breakfast in Istanbul or Why are certain nonlinear PDEs both widely applicable and integrable? Reminiscences of Francesco Calogero

Robin Bullough
Pages: 124 - 137
It is shown how in the early days of soliton theory 1976-the early 1980's Francesco Calogero maintained a considerable influence on the field and on the work of the athor Robin Bullough in particular. A vehicle to this end was the essentially annual sequence of international conferences Francesco organised...
Research Article

Continuity Equation in Nonlinear Quantum Mechanics and the Galilei Relativity Principle

Wilhelm Fushchych, Vyacheslav Boyko
Pages: 124 - 128
Classes of the nonlinear Schrödinger-type equations compatible with the Galilei relativity principle are described. Solutions of these equations satisfy the continuity equation.
Research Article

Conditional Linearizability Criteria for Third Order Ordinary Differential Equations

Fazal M. Mahomed, Asghar Qadir
Pages: 124 - 133
Using geometric methods for linearizing systems of second order cubically non-linear in the first derivatives ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for conditional linearizability via point transformation with...
Research Article

Approximation of Solitons in the Discrete NLS Equation

Jesus Cuevas, Guillaume James, Panayotis G. Kevrekidis, Boris A. Malomed, Bernardo Sanchez-Rey
Pages: 124 - 136
We study four different approximations for finding the profile of discrete solitons in the one- dimensional Discrete Nonlinear Schrödinger (DNLS) Equation. Three of them are discrete approximations (namely, a variational approach, an approximation to homoclinic orbits and a Green-function approach),...
Research Article

Ideals Generated by Traces or by Supertraces in the Symplectic Reflection Algebra H1,V(I2(2m + 1)) II

I.A. Batalin, S.E. Konstein, I.V. Tyutin
Pages: 123 - 133
The algebra 𝒣≔H1,ν(I2(2m+1)) of observables of the Calogero model based on the root system I2(2m + 1) has an m-dimensional space of traces and an (m + 1)-dimensional space of supertraces. In the preceding paper we found all values of the parameter ν for which either the space of traces contains...
Research Article

Mixed Perturbative Expansion: the Validity of a Model for the Cascading

Hervé Leblond
Pages: 126 - 143
A new type of perturbative expansion is built in order to give a rigorous derivation and to clarify the range of validity of some commonly used model equations. This model describes the evolution of the modulation of two short and localized pulses, fundamental and second harmonic, propagating together...
Research Article

First Integrals Generated by Pseudosymmetries in Nambu-Poisson Mechanics

Mircea Crâşmăreanu
Pages: 126 - 135
Some types of first integrals for Hamiltonian Nambu-Poisson vector fields are obtained by using the notions of pseudosymmetries. In this theory, the homogeneous Hamiltnian vector fields play a special role and we point out this fact. The differential system which describe the SU(2)-monopoles is given...
Research Article

Bilinearization of Coupled Nonlinear Schrödinger Type Equations: Integrabilty and Solitons

K. Porsezian
Pages: 126 - 131
Considering the coupled envelope equations in nonlinear couplers, the question of integrability is attempted. It is explicitly shown that Hirota's bilinear method is one of the simple and alternative techniques to Painlevé analysis to obtain the integrability conditions of the coupled nonlinear Schrödinger...
Research Article

Bilinear Forms of Integrable Lattices Related to Toda and Lotka-Volterra Lattices

Ken-ichi Maruno, Wen-Xiu Ma
Pages: 127 - 139
Hirota's bilinear technique is applied to some integrable lattice systems related to the Bäcklund transformations of the 2DToda, Lotka-Volterra and relativistic LotkVolterra lattice systems, which include the modified Lotka-Volterra lattice system, the modified relativistic Lotka-Volterra lattice system,...
Research Article

Semiclassical Solutions of the Nonlinear Schrödinger Equation

A.V. Shapovalov, A.Yu. Trifonov
Pages: 127 - 138
A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schrödinger equation (NLSE) with an external field. These solutions are considered as multidimensional solitary waves. The center of mass of such a solution is shown to move along with the bicharacteristics...
Research Article

Bi-Hamiltonian Structure of a Three-Component Camassa-Holm Type Equation

Nianhua Li, Q.P. Liu
Pages: 126 - 134
A recently proposed three-component Camassa-Holm equation is considered. It is shown that this system is a bi-Hamiltonian system.
Research Article

Asymptotic Solutions of the Whitham Equations

T. Grava
Pages: 128 - 132
We extend a previous result, namely we show that the solution of the Whitham equtions is asymptotically self-similar for generic monotone polynomial initial data with smooth perturbation.
Research Article

Solitons in a double pendulums chain model, and DNA roto-torsional dynamics

Mariano Cadoni, Roberto De Leo, Giuseppe Gaeta
Pages: 128 - 146
It was first suggested by Englander et al to model the nonlinear dynamics of DNA relevant to the transcription process in terms of a chain of coupled pendulums. In a related paper [4] we argued for the advantages of an extension of this approach based on considering a chain of double pendulums with certain...
Letter to Editor

On the Propagation of Binary Signals in a Two-Dimensional Nonlinear Lattice with Nearest-Neighbor Interactions

Javier Ruiz-Ramírez, Jorge Eduardo Macías-Díaz
Pages: 127 - 136
In this work, we use a computational technique with multiple properties of consistency in order to approximate solutions of a bounded β-Fermi–Pasta–Ulam lattice in two space dimensions subject to harmonic driving in two adjacent boundaries. The processes of nonlinear supratransmission and infratransmission...
Research Article

On Edge Waves in Stratified Water Along a Sloping Beach

Raphael Stuhlmeier
Pages: 127 - 137
Building on previous investigations, we show that Gerstner's famous deep water wave and the related edge wave propagating along a sloping beach, found within the context of water of constant density, can both be adapted to provide explicit free surface flows in incompressible fluids with arbitrary...
Research Article

Solutions of the Extended Kadomtsev–Petviashvili–Boussinesq Equation by the Hirota Direct Method

Asli Pekcan
Pages: 127 - 139
We show that we can apply the Hirota direct method to some non-integrable equations. For this purpose, we consider the extended Kadomtsev–Petviashvili–Boussinesq (eKPBo) equation with M variable which is (uxxx−6uux)x+a11uxx+2∑k=2Ma1kuxxk+∑i,j=2Maijuxixj=0, where aij = aji are constants and xi = (x,...
Research Article

The Elliptic Sinh-Gordon Equation in the Quarter Plane

Guenbo Hwang
Pages: 127 - 140
We study the elliptic sinh-Gordon equation formulated in the quarter plane by using the so-called Fokas method, which is a significant extension of the inverse scattering transform for the boundary value problems. The method is based on the simultaneous spectral analysis for both parts of the Lax pair...
Research Article

Classical quasi-trigonometric r-matrices of Cremmer-Gervais type and their quantization

Julia Y.-Magnusson, Maxim Samsonov, Alexander Stolin
Pages: 129 - 136
We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang-Baxter equation. The method is based on so-called affinization of certain seaweed algebras and their quantum analogues.
Research Article

Properties of the Dominant Behaviour of Quadratic Systems

A. Maharaj, P.G.L Leach
Pages: 129 - 144
We study the dominant terms of systems of Lotka-Volterra-type which arise in the the mathematical modelling of the evolution of many divers natural systems from the viewpoint of both symmetry and singularity analyses. The connections between an increase in the amount of symmetry possessed by the system...
Short Communication

Reduction and Exact Solutions of the Monge-Ampere Equation

Oleg Leibov
Pages: 129 - 131
On the basis of a subgroup structure of the Poincaré group P(1, 3) the ansatzes which reduce the Monge­Ampere equation to differential equations with fewer independent variables have been constructed. The corresponding symmetry reduction has been done. By means of the solutions of the reduced equations...
Research Article

Geodetic Systems on Linear and Affine Groups. Classics and Quantization.

Jan J. Slawianowski
Pages: 130 - 137
Described are classical and quantized systems on linear and affine groups. Unlike the traditional models applied in astrophysics, nuclear physics, molecular vibrations and elasticity, our models are not only kinematically ruled by the affine group, but also their kinetic energies are affinely invariant....
Research Article

Perturbed Lie Symmetry and Systems of Non-Linear Diffusion Equations

R.J. Wiltshire
Pages: 130 - 138
The method of one parameter, point symmetric, approximate Lie group invariants is applied to the problem of determining solutions of systems of pure one-dimensional, diffusion equations. The equations are taken to be non-linear in the dependent variables but otherwise homogeneous. Moreover, the matrix...
Research Article

General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory

Hironobu Kimura, Damiran Tseveenamijil
Pages: 130 - 152
Isomonodromic deformation of linear differential equations on ℙ1 with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GLN+1(ℂ) which appeared in the theory of general...
Research Article

Inverse Scattering Transform and Solitons for Square Matrix Nonlinear Schrödinger Equations with Mixed Sign Reductions and Nonzero Boundary Conditions

Alyssa K. Ortiz, Barbara Prinari
Pages: 130 - 161
The inverse scattering transform (IST) with nonzero boundary conditions at infinity is developed for a class of 2 × 2 matrix nonlinear Schrödinger-type systems whose reductions include two equations that model certain hyperfine spin F = 1 spinor Bose-Einstein condensates, and two novel equations that...
Research Article

Is the Classical Bukhvostov-Lipatov Model Integrable? A Painlevé Analysis

Marco Ameduri, Costas J. Efthimiou
Pages: 132 - 139
In this work we apply the Weiss, Tabor and Carnevale integrability criterion (Painlevé analysis) to the classical version of the two dimensional Bukhvostov-Lipatov model. We are led to the conclusion that the model is not integrable classically, except at a trivial point where the theory can be described...
Research Article

Symmetry Properties and Exact Solutions of the Fokker-Planck Equation

Valery Stohny
Pages: 132 - 136
Symmetry properties of some Fokker-Planck equations are studied. In the one-dimensional case, when symmetry groups turn out to be six-parameter ones, this allows to find changes of variables to reduce such Fokker-Planck equations to the one-dimensional heat equation. The symmetry and the family of exact...
Research Article

On a Certain Fractional q-Difference and its Eigen Function

Atsushi Nagai
Pages: 133 - 142
A fractional q-difference operator is presented and its properties are investigated. Epecially, it is shown that this operator possesses an eigen function, which is regarded as a q-discrete analogue of the Mittag-Leffler function. An integrable nonlinear mapping with fractional q-difference is also presented.
Research Article

KAM Theorem for the Nonlinear Schrödinger Equation

Benoît Grébert, Thomas Kappeler
Pages: 133 - 138
We prove the persistence of finite dimensional invariant tori associated with the dfocusing nonlinear Schrödinger equation under small Hamiltonian perturbations. The invariant tori are not necessarily small.
Research Article

Representations of -algebras and dynamical systems

Vasyl' L. Ostrovs'kyĭ, Yurii S. Samoilenko
Pages: 133 - 150
Consider a -algebra A generated by self-adjoint elements a1, . . . , an (aj = a j , j = 1, . . . , n) and the relations Pk(a1, . . . , an) = 0 (k = 1, . . . , m). (1) Here Pk(·) are polynomials in the non-commuting variables a1, . . . , an over C such that P k (·) = Pk(·). In other words, A is a quotient...
Research Article

Water Waves near a Shoreline in a Flow with Vorticity: Two Classical Examples

Robin Stanley Johnson
Pages: 133 - 156
The equations that describe the classical problem of water waves-inviscid, no surface tension and constant pressure at the surface - are non-dimensionalised and scaled appropriately, and the two examples: traditional gravity waves and edge waves, are introduced. In addition each type of wave is allowed...