# Journal of Nonlinear Mathematical Physics

1499 articles

**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...

**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**

## 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...

**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...

**Research Article**

## Transformation of Auto-Bäcklund Type For Hyperbolic Generalization of Burgers Equation

Ekaterina V. Kutafina

Pages: 411 - 420

We consider the hyperbolic generalization of Burgers equation with polynomial source term. The transformation of auto-Bäcklund type was found. Application of the results is shown in the examples, where kink and bi-kink solutions are obtained from the pair of two stationary ones.

**Short Communication**

## Galilei Invariance of the FokkerPlanck Equation with Nonlinearity

Vadym Cherkasenko

Pages: 416 - 417

Consider equation

**Research Article**

## On Asymptotic Nonlocal Symmetry of Nonlinear Schrödinger Equations

W.W. Zachary, V.M. Shtelen

Pages: 417 - 437

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schrödinger equation, discovered by Fushchych and Segeda in 1977, can...

**Research Article**

## On Exact Solutions of the Lorentz-Maxwell Equations

Igor Revenko

Pages: 417 - 420

New exact solutions are obtained for the systems of classical electrodynamics equations.

**Research Article**

## Symmetry Reductions of Second Heavenly Equation and 2+1-Dimensional Hamiltonian Integrable Systems

D. Yazici, M. B. Sheftel

Pages: 417 - 425

Second heavenly equation of Pleba~nski, presented in a two-component form, is known to be a 3 +1 dimensional multi-Hamiltonian integrable system. We show that one symmetry reduction of this equation yields a two component 2+1Âdimensional multi-Hamiltonian integrable system. For this system, we present...

**Research Article**

## Proper-Time Formulation of Classical Electrodynamics

Tepper L. Gill, W.W. Zachary

Pages: 418 - 425

We show that Maxwell's equations have a generalization associated with the propertime of the source and a new invariance group which leaves this variable fixed for all observers. We show that the second postulate (of Einstein) depends on the anthropocentric view that the only clock to use is the proper-clock...

**Research Article**

## Symmetry Classification of the OneDimensional Second Order Equation of a Hydrodynamic Type

Vyacheslav Boyko

Pages: 418 - 424

The paper contains a symmetry classification of the onedimensional second order equation of a hydrodynamical type L(Lu) + Lu = F(u), where L t + ux. Some classes of exact solutions of this equation are given.

**Research Article**

## On the global dynamics of a three-dimensional forced-damped differential system

Jaume Llibre, Y. Paulina Martínez, Claudia Valls

Pages: 414 - 428

In this paper by using the Poincaré compactification of ℝ3 we make a global analysis of the model x′ = −ax + y + yz, y′ = x − ay + bxz, z′ = cz − bxy. In particular we give the complete description of its dynamics on the infinity sphere. For a + c = 0 or b = 1 this system has invariants. For these values...

**Research Article**

## Semidiscrete Integrable Nonlinear Systems Generated by the New Fourth Order Spectral Operator: Systems of Obverse Type

Oleksiy O. Vakhnenko

Pages: 415 - 425

In the framework of zero-curvature representation we have proposed three distinct versions of semidiscrete integrable nonlinear systems arising due to a proper multifield augment of integrable nonlinear Schrödinger system with the background-controlled intersite resonant couplings. The specification...

**Letter to Editor**

## Peakons Arising as Particle Paths Beneath Small-Amplitude Water Waves in Constant Vorticity Flows

Delia Ionescu-Kruse

Pages: 415 - 422

We present a new kind of particle path in constant vorticity water of finite depth, within the framework of small-amplitude waves.

**Research Article**

## Vectorial Regularization and Temporal Means in Keplerian Motion

Daniel Condurache, Vladimir Martinusi

Pages: 420 - 440

We study the well-known Kepler’s problem by introducing a new vectorial regularization. This helps deduce Kepler’s equations by a simple and unified method. Some integral temporal means are also obtained by means of this regularization. The vectorial eccentricity plays a fundamental part in this approach.

**Research Article**

## Symmetry constraint of MKdV equations by binary nonlinearization

Wenxiu Ma

Pages: 420 - 433

A symmetry constraint for the MKdV integrable hierarchy is presented by binary nonlinearization. The spatial and temporal parts of the Lax pairs and adjoint Lax pairs of MKdV equations are all constrained as finite-dimensional Liouville integrable Hamiltonian systems, whose integrals of motion are explicitly...

**Research Article**

## On Some Generalized Symmetric Integral Operators of Buschman-Erdelyi's Type

N. Virchenko

Pages: 421 - 425

Some new symmetric integral operators with kernels involving the generalized Legendre's function of the first kind Pm,n k (z) are introduced. Some their applications are given.

**Research Article**

## On the Spectral Problem Associated with the Camassa-Holm Equation

Christer Bennewitz

Pages: 422 - 432

We give a basic uniqueness theorem in the inverse spectral theory for a Sturm-Liouville equation with a weight which is not of one sign. It is shown that the theorem may be applied to the spectral problem associated with the Camassa-Holm integrable system which models shallow water waves.

**Research Article**

## Gauge Theory Approach Towards an Explicit Solution of the (Classical) Elliptic Calogero-Moser System

Edwin Langmann

Pages: 423 - 439

We discuss the relation of the trigonometric Calogero-Moser (CM) system to YanMills gauge theories and its generalization to the elliptic case. This yields a liearization of the time evolution of the elliptic CM system and suggests two promising strategies for finding a fully explicit solution of this...

**Research Article**

## A Note on the Appearance of Self-Dual Yang-Mills Fields in Integrable Hierarchies

L. Fehér, A. Gábor

Pages: 423 - 432

A family of mappings from the solution spaces of certain generalized Drinfeld-Sokolov hierarchies to the self-dual Yang-Mills system on R2,2 is described. This provides an extension of the well-known relationship between self-dual connections and integrable hierarchies of AKNS and Drinfeld-Sokolov type.

**Research Article**

## Hk Metrics on the Diffeomorphism Group of the Circle

Adrian Constantin, B. Kolev

Pages: 424 - 430

Each Hk inner product, k N, endows the diffeomorphism group of the circle with a Riemannian structure. For k 1 the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of geodesics holds.

**Research Article**

## Products in the category of 2n-manifolds

Andrew Bruce, Norbert Poncin

Pages: 420 - 453

We prove that the category of 2n-manifolds has all finite products. Further, we show that a 2n-manifold (resp., a 2n-morphism) can be reconstructed from its algebra of global 2n-functions (resp., from its algebra morphism between global 2n-function algebras). These results are of importance...

**Research Article**

## Coadjoint Orbits of Certain Motion Groups and Their Coherent States

Majdi Ben Halima

Pages: 420 - 430

Let ℍn, n ≥ 1, be the (2n+1)-dimensional Heisenberg group and let K be a closed connected subgroup of the unitary group U(n) acting on ℍn by automorphisms. Using the moment map, we provide in this paper a dequantization procedure for all generic admissible coadjoint orbits of the semidirect product G...

**Research Article**

## Nonstandard Separability on the Minkowski Plane

Giuseppe Pucacco, Kjell Rosquist

Pages: 421 - 430

We present examples of nonstandard separation of the natural Hamilton–Jacobi equation on the Minkowski plane 𝕄2. By “nonstandard” we refer to the cases in which the form of the metric, when expressed in separating coordinates, does not have the usual Liouville structure. There are two possibilities:...

**Research Article**

## The Incompressible NavierStokes for the Nonlinear Discrete Velocity Models

A. Bellouquid

Pages: 426 - 445

We establish the incompressible NavierStokes limit for the discrete velocity model of the Boltzmann equation in any dimension of the physical space, for densities which rmain in a suitable small neighborhood of the global Maxwellian. Appropriately scaled families solutions of discrete Boltzmann equation...

**Research Article**

## On New Representations of Galilei Groups

W.I. Fushchych, R.Z. Zhdanov

Pages: 426 - 435

We have constructed new realizations of the Galilei group and its natural extensions by Lie vector fields. These realizations together with the ones obtained by Fushchych & Cherniha (Ukr. Math. J., 1989, 41, N 10, 1161; N 12, 1456) and Rideau & Winternitz (J. Math. Phys., 1993, 34, 558) give a complete...

**Research Article**

## Multiparameter Deformations of the Algebra gln in Terms of Anyonic Oscillators

A.M. Gavrilik, N.Z. Iorgov

Pages: 426 - 431

Generators of multiparameter deformations Uq;s1,s2,...,sn-1 (gln) of the universal enveloping algebra U(gln) are realized bilinearly by means of an appropriately generalized form of anyonic oscillators (AOs). This modification takes into account the parameters s1, ..., sn-1 and yields usual AOs when...

**Research Article**

## A Model of Heart Rate Kinetics in Response to Exercise

James Robert Stirling, Maria Zakynthinaki, Ignacio Refoyo, Javier Sampedro

Pages: 426 - 436

We present a mathematical model, in the form of two coupled ordinary differential equations, for the heart rate kinetics in response to exercise. Our heart rate model is an adaptation of the model of oxygen uptake kinetics of Stirling et al. [21]; a physiological justification for this adaptation, as...

**Letter to Editor**

## Some Examples of Algebraic Geodesics on Quadrics. II

A. M. Perelomov

Pages: 423 - 428

In this note we give new examples of algebraic geodesics on some two-dimensional quadrics, namely, on ellipsoids, one-sheet hyperboloids, and hyperbolic paraboloids. It appears that in all considered cases, such geodesics are rational space curves.

**Research Article**

## Orbital stability of the smooth solitary wave with nonzero asymptotic value for the mCH equation

Chaohong Pan, Lijing Zheng

Pages: 423 - 438

This paper is concerned with orbital stability of the smooth solitary wave with nonzero asymptotic value for the mCH equation
ut−uxxt+2kux+au2ux=2uxuxx+uuxxx.
Under the parametric conditions a > 0 and
k<18a
, an interesting phenomenon is discovered, that is, for the stability there...

**Research Article**

## Bubbles, Clusters and Denaturation in Genomic DNA: Modeling, Parametrization, Efficient Computation

Nikos Theodorakopoulos

Pages: 429 - 447

The paper uses mesoscopic, nonlinear lattice dynamics based (Peyrard–Bishop–Dauxois, PBD) modeling to describe thermal properties of DNA below and near the denaturation temperature. Computationally efficient notation is introduced for the relevant statistical mechanics. Computed melting profiles of long...

**Research Article**

## Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Gazanfer Unal, Ismail Iyigunler, C Masood Khalique

Pages: 430 - 442

Necessary and sufficient conditions for the linearization of one-dimensional nonau- tonomous jump-diffusion stochastic differential equations are given. Stochastic inte- grating factor is introduced to solve the linear jump-diffusion stochastic differential equations. Closed form solutions to certain...

**Research Article**

## Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

Alessandro Michelangeli, Alessandro Olgiati

Pages: 426 - 464

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons...

**Research Article**

## Multiple Hamiltonian Structures and Lax Pairs for BogoyavlenskyVolterra Systems

Stelios P. Kouzaris

Pages: 431 - 450

Results on the Volterra model which is associated to the simple Lie algebra of type An are extended to the BogoyavlenskyVolterra systems of type Bn, Cn and Dn. In paticular we find Lax pairs, Hamiltonian and Casimir functions and multi-Hamiltonian structures. Moreover, we investigate recursion operators,...

**Research Article**

## Linearization of a Second-Order Stochastic Ordinary Differential Equation

Sergey V. Meleshko, Eckart Schulz

Pages: 427 - 441

Necessary and sufficient conditions which allow a second-order stochastic ordinary differential equation to be transformed to linear form are presented. The transformation can be chosen in a way so that all but one of the coefficients in the stochastic integral part vanish. The linearization criteria...

**Research Article**

## A Three State Hard-Core Model on a Cayley Tree

James Martin, Utkir Rozikov, Yuri Suhov

Pages: 432 - 448

We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state (x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson...

**Short Communication**

## On Symmetry Reduction of Nonlinear Generalization of the Heat Equation

Valentyn Marchenko

Pages: 432 - 434

Reductions and classes of new exact solutions are constructed for a class of Galileiinvariant heat equations.

**Research Article**

## Spectral Zeta Functions of a 1D Schrödinger Problem

Joe Watkins

Pages: 428 - 444

We study the spectral zeta functions associated to the radial Schrödinger problem with potential V(x) = x2M + αxM−1 + (λ2 − 1/4)/x2. After directly computing some of the zeta functions, we use the quantum Wronskian equation to give sum rules between them, allowing for instances where the explicit form...

**Research Article**

## Magnetic curves in Sasakian manifolds

Simona Luiza Druţă-Romaniuc, Jun-ichi Inoguchi, Marian Ioan Munteanu, Ana Irina Nistor

Pages: 428 - 447

In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a...

**Research Article**

## Real Forms of the Complex Twisted N=2 Supersymmetric Toda Chain Hierarchy in Real N=1 and Twisted N=2 Superspaces

O. Lechtenfeld, A. Sorin

Pages: 433 - 444

Three nonequivalent real forms of the complex twisted N=2 supersymmetric Toda chain hierarchy (solv-int/9907021) in real N=1 superspace are presented. It is demostrated that they possess a global twisted N=2 supersymmetry. We discuss a new superfield basis in which the supersymmetry transformations are...

**Research Article**

## Noncommutative Hypergeometric and Basic Hypergeometric Equations

Alessandro Conflitti, Michael J. Schlosser

Pages: 429 - 443

Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100(14) (2003) 8138–8141] considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give...

**Research Article**

## Quaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions

John Cobb, Alex Kasman, Albert Serna, Monique Sparkman

Pages: 429 - 452

Quaternion-valued solutions to the non-commutative KdV equation are produced using determinants. The solutions produced in this way are (breather) soliton solutions, rational solutions, spatially periodic solutions and hybrids of these three basic types. A complete characterization of the parameters...

**Research Article**

## Integrability on generalized *q*-Toda equation and hierarchy

Anni Meng, Chuanzhong Li, Shuo Huang

Pages: 429 - 441

In this paper, we construct a new integrable equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the generalized q-Toda equation and a whole integrable generalized q-Toda hierarchy are also constructed....

**Research Article**

## Symmetry Analysis of the Multidimensional Polywave Equation

Olena Roman

Pages: 435 - 440

We present symmetry classification of the polywave equation 2l u = F(u). It is established that the equation in question is invariant under the conformal group C(1, n) iff F(u) = eu , n + 1 = 2l or F(u) = u(n+1+2l)/(n+1-2l) , n + 1 = 2l. Symmetry reduction for the biwave equation 22 u = uk is carried...

**Research Article**

## An Old Method of Jacobi to Find Lagrangians

M. C. Nucci, P. G. L. Leach

Pages: 431 - 441

In a recent paper by Ibragimov a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables one to derive (many) Lagrangians of any second-order differential...

**Research Article**

## On near and the nearest correlation matrix

Pasha Zusmanovich

Pages: 431 - 439

We present an elementary heuristic reasoning based on Arnold's theory of versal deformations in support of a straightforward algorithm for finding a correlation matrix near a given symmetric one.

**Research Article**

## Non-Lie and Discrete Symmetries of the Dirac Equation

Jiri Niederle, Anatolii Nikitin

Pages: 436 - 444

New algebras of symmetries of the Dirac equation are presented, which are formed by linear and antilinear firstorder differential operators. These symmetries are applied to decouple the Dirac equation for a charged particle interacting with an external field.

**Research Article**

## Combination of Inverse Spectral Transform Method and Method of Characteristics: Deformed Pohlmeyer Equation

A.I. Zenchuk

Pages: 437 - 448

We apply a version of the dressing method to a system of four-dimensional nonlinear Partial Differential Equations (PDEs), which contains both Pohlmeyer equation (i.e. nonlinear PDE integrable by the Inverse Spectral Transform Method) and nonlinear matrix PDE integrable by the method of characteristics...

**Research Article**

## Finite and infinite systems of nonlinearly-coupled ordinary differential equations, the solutions of which feature remarkable Diophantine findings

Francesco Calogero

Pages: 433 - 441

We use previous results concerning the time evolution of the zeros xn(t) of time-dependent polynomials pN(z; t) or entire functions F(z; t) of the complex variable z, in order to identify lots of nonlinearly-coupled, finite or infinite, systems of Ordinary Differential Equations the solutions of which...

**Research Article**

## Resonance Broadening Theory of Farley-Buneman Turbulence in the Auroral E-Region

A.M. Hamza

Pages: 438 - 461

The conventional theory of resonance broadening for a two-species plasma in a magnetic field is revised, and applied to an ionospheric turbulence case. The assumptions made in the conventional theory of resonance broadening have, in the past, led to replacing the frequency by + ik2 D in the resonant...

**Research Article**

## Multiscale Analysis of Discrete Nonlinear Evolution Equations: The Reduction of the dNLS

Decio Levi, Rafael Hernandez Heredero

Pages: 440 - 448

In this paper we consider multiple lattices and functions defined on them. We itroduce some slow varying conditions and define a multiscale analysis on the lattice, i.e. a way to express the variation of a function in one lattice in terms of an asymtotic expansion with respect to the other. We apply...

**Research Article**

## Wave Breaking in a Class of Nonlocal Dispersive Wave Equations

Hailiang Liu

Pages: 441 - 466

The Korteweg de Vries (KdV) equation is well known as an approximation model for small amplitude and long waves in different physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. In this work we consider a class of nonlocal dispersive wave equations which also...

**Research Article**

## On Reduction of the Euler Equations by Means of Two-Dimensional Algebras

Halyna Popovych

Pages: 441 - 446

A complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the Euler equations is constructed. Using some of them, the Euler equations are reduced to systems of partial differential equations in two independent variables which are integrated in quadratures.

**Research Article**

## Integrability analysis of the Emden-Fowler equation

K.S. Govinder, P.G.L. Leach

Pages: 443 - 461

The Emden-Fowler equation of index n is studied utilising the techniques of Lie and Painlev Ìe analysis. For general n information about the integrability of this equation is obtained. The link between these two types of analyses is explored. The special cases of n = âˆ’3, 2 are also examined. As a...

**Research Article**

## Periodic solutions of symmetric Kepler perturbations and applications

Angelo Alberti, Claudio Vidal

Pages: 439 - 465

We investigate the existence of several families of symmetric periodic solutions as continuation of circular orbits of the Kepler problem for certain symmetric differentiable perturbations using an appropriate set of Poincaré-Delaunay coordinates which are essential in our approach. More precisely, we...

**Research Article**

## Magnetic fields in 2D and 3D sphere

Jose L. Cabrerizo

Pages: 440 - 450

In this note we study the Landau–Hall problem in the 2D and 3D unit sphere, that is, the motion of a charged particle in the presence of a static magnetic field. The magnetic flow is completely determined for any Riemannian surface of constant Gauss curvature, in particular, the unit 2D sphere. For the...

**Research Article**

## Symmetry, Singularities and Integrability in Complex Dynamics I: The Reduction Problem

P.G.L. Leach, S. Cotsakis, G.P. Flessas

Pages: 445 - 479

Quadratic systems generated using Yang-Baxter equations are integrable in a sense, but we display a deterioration in the possession of the Painlevé property as the number of equations in each `integrable system' increases. Certain intermediate systems are constructed and also tested for the Painlevé...

**Research Article**

## On Subalgebras of the Conformal Algebra AC(2,2)

A.F. Barannyk

Pages: 445 - 454

Subalgebras of the Lie algebra AC(2, 2) of the group C(2, 2), which is the group of conformal transformations of the pseudo-Euclidean space R2,2, are studied. All subalgebras of the algebra AC(2, 2) are splitted into three classes, each of those is characterized by the isotropic rank 0, 1, or 3. We present...

**Research Article**

## On a One-Phase Stefan Problem in Nonlinear Conduction

S. De Lillo, M.C. Salvatori

Pages: 446 - 454

A one phase Stefan problem in nonlinear conduction is considered. The problem is shown to admit a unique solution for small times. An exact solution is obtained which is a travelling front moving with constant speed.

**Research Article**

## A Finite Dimensional Analog of the Krein Formula

I.A. Shereshevskii

Pages: 446 - 457

I offer a simple and useful formula for the resolvent of a small rank perturbation of large matrices. I discuss applications of this formula, in particular, to analytical and numerical solving of difference boundary value problems. I present examples connected with such problems for the difference Laplacian...

**Research Article**

## Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions

T. I. Valchev, A. B. Yanovski

Pages: 442 - 461

This paper is a continuation of our previous work in which we studied a sl(3, ℂ) Zakharov-Shabat type auxiliary linear problem with reductions of Mikhailov type and the corresponding integrable hierarchy of nonlinear evolution equations. Now, we shall demonstrate how one can construct special solutions...

**Research Article**

## A multi-symplectic numerical integrator for the two-component Camassa–Holm equation

David Cohen, Takayasu Matsuo, Xavier Raynaud

Pages: 442 - 453

A new multi-symplectic formulation of the two-component Camassa-Holm equation (2CH) is presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. A multi-symplectic discretisation based on this new formulation is exemplified by means of...

**Research Article**

## Symmetry Reduction for a System of Nonlinear Evolution Equations

Lyudmila Barannyk

Pages: 447 - 452

In this paper we obtain the maximal Lie symmetry algebra of a system of PDEs. We reduce this system to a system of ODEs, using some rank three subalgebras of the finite-dimensional part of the symmetry algebra. The corresponding invariant solutions of the PDEs are obtained.

**Research Article**

## Toda–Heisenberg Chain: Interacting *σ*-Fields in Two Dimensions

G. M. Pritula, V. E. Vekslerchik

Pages: 443 - 459

We study a (2 ± 1)-dimensional system that can be viewed as an infinite number of O(3) σ-fields coupled by a nearest-neighbour Heisenberg-like interaction. We reduce the field equations of this model to an integrable system that is closely related to the two-dimensional relativistic Toda chain and the...

**Research Article**

## Euler–Lagrange Equations for Functionals Defined On Fréchet Manifolds

José Antonio Vallejo

Pages: 443 - 454

We prove a version of the variational Euler–Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.

**Research Article**

## What a Classical r-Matrix Really Is

Boris A. Kupershmidt

Pages: 448 - 488

The notion of classical r-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, where the standard definitions are shown to be deficient, is proposed, the notion of an O-operator. This notion has all the natural properties one would expect form it, but lacks...

**Research Article**

## Implicit Solutions to Some Lorentz Invariant Nonlinear Equations Revisited

David B. Fairlie

Pages: 449 - 456

An implicit solution to the vanishing of the so-called Universal Field Equation, or Bordered Hessian, which dates at least as far back as 1935 [1] is revived, and derived from a much later form of the solution. A linear ansatz for an implicit solution of second order partial differential equations, previously...

**Research Article**

## Rigorous Results in the Scaling Theory of Irreversible Aggregation Kinetics

François Leyvraz

Pages: 449 - 465

The kinetic equations describing irreversible aggregation and the scaling approach dveloped to describe them in the limit of large times and large sizes are tersely reviewed. Next, a system is considered in which aggregates can only react with aggregates of their own size. The existence of a scaling...

**Research Article**

## Biomechanics of DNA: Rotational Oscillations of Bases

L. V. Yakushevich

Pages: 449 - 461

In the present paper we investigate the rotational oscillations of the nitrous bases that form a central base pair in a short DNA fragment consisting of three base pairs. For this purpose we use a simple mechanical model of the DNA fragment where the bases are imitated by pendulums, and the interactions...

**Research Article**

## A New (*γ*_{n}, *σ*_{k})-KP Hierarchy and Generalized Dressing Method

_{n}

_{k}

Yuqin Yao, Yehui Huang, Yunbo Zeng

Pages: 445 - 457

A new (γn, σk)-KP hierarchy (KPH) with two new time series γn and σk, which consists of γn-flow, σk-flow and mixed γn and σk evolution equations of eigenfunctions, is proposed. Two reductions and constrained flows of (γn, σk)-KPH are studied. The dressing method is generalized to the (γn, σk)-KPH and...

**Research Article**

## Explicit Soliton Asymptotics for the Nonlinear Schrödinger Equation on the Half-Line

K. Kalimeris

Pages: 445 - 452

There exists a particular class of boundary value problems for integrable nonlinear evolution equations formulated on the half-line, called linearizable. For this class of boundary value problems, the Fokas method yields a formalism for the solution of the associated initial-boundary value problem, which...

**Research Article**

## Dirac Reduction Revisited

Krzysztof Marciniak, Maciej Blaszak

Pages: 451 - 463

The procedure of Dirac reduction of Poisson operators on submanifolds is discussed within a particularly useful special realization of the general Marsden-Ratiu redution procedure. The Dirac classification of constraints on `first-class' constraints and `second-class' constraints is reexamined.

**Research Article**

## Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

Spyridon Kamvissis, Dmitry Shepelsky, Lech Zielinski

Pages: 448 - 473

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(−2iβx) (or asymptotically periodic, u(x, 0) =α exp(−2iβx)→0 as x→∞), and a Robin boundary condition at x = 0:...

**Research Article**

## Application of Differential Forms to Construction of Nonlocal Symmetries

S.I. Agafonov

Pages: 453 - 457

Differential forms are used for construction of nonlocal symmetries of partial differential equations with conservation laws. Every conservation law allows to introduce a nonlocal variable corresponding to a conserved quantity. A prolongation technique is suggested for action of symmetry operators on...

**Research Article**

## Cohomology of Groups of Diffeomorphisms Related to the Modules of Differential Operators on a Smooth Manifold

Sofiane Bouarroudj

Pages: 455 - 463

Let M be a manifold and T M be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of M with values in the space of linear differential operators acting on C (T M). When M is the n-dimensional sphere, Sn , we use this

**Research Article**

## The Finite-Dimensional Moser Type Reduction of Modified Boussinesq and Super-Korteweg-de Vries Hamiltonian Systems via the Gradient-Holonomic Algorithm and Dual Moment Maps. Part I

A.K. Prykarpatsky, O.E. Hentosh, D.L. Blackmore

Pages: 455 - 469

The Moser type reductions of modified Boussinessq and super-Korteweg-de Vries equations upon the finite-dimensional invariant subspaces of solutions are considered. For the Hamiltonian and Liouville integrable finite-dimensional dynamical systems concerned with the invariant subspaces, the Lax representations...

**Research Article**

## Quantizing preserving Noether symmetries

M.C. Nucci

Pages: 451 - 463

A procedure which obviates the constraint imposed by the conflict between consistent quantization and the invariance of the Hamiltonian description under nonlinear canonical transformation is proposed. This new quantization scheme preserves the Noether point symmetries of the underlying Lagrangian in...

**Research Article**

## Seiberg-Witten-like Equations on 7-Manifolds

Nedim Degirmenci, Nülifer Özdemir

Pages: 457 - 461

The Seiberg-Witten equations are of great importance in the study of topology of smooth four-dimensional manifolds. In this work, we propose similar equations for 7-dimensional compact manifolds with G2-structure.

**Research Article**

## Bi-Hamiltonian Representation, Symmetries and Integrals of Mixed Heavenly and Husain Systems

M. B. Sheftel, D. Yazici

Pages: 453 - 484

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here...

**Research Article**

## Hierarchies of *q*-discrete Painlevé equations

Huda Alrashdi, Nalini Joshi, Dinh Thi Tran

Pages: 453 - 477

In this paper, we construct a new hierarchy based on the third q-discrete Painlevé equation (qPIII) and also study the hierarchy of the second q-discrete Painlevé equation (qPII). Both hierarchies are derived by using reductions of the general lattice modified Korteweg-de Vries equation. Our results...

**Research Article**

## Asymptotic Approximation of Hyperbolic Weakly Nonlinear Systems

A. Krylovas, R. Ciegis

Pages: 458 - 470

An averaging method for getting uniformly valid asymptotic approximations of the solution of hyperbolic systems of equations is presented. The averaged system of equtions disintegrates into independent equations for non-resonance systems. We consider the resonance conditions for some classes of solutions....

**Research Article**

## Fundamental Solutions of the Axial Symmetric Goursat Problem

A.A. Borghardt, D.Ya. Karpenko, N.Yu. Nosenko

Pages: 458 - 463

Fundamental solutions (FS) with a given boundary condition on the characteristics of relativistic problems with axial symmetry are considered. This is so-called the Goursat problem (GP) or zero plane formalism in Dirac's terminology, or modification of the proper time method in the Fock-Nambu-Schwinger...

**Research Article**

## Integration of the stochastic logistic equation via symmetry analysis

Giuseppe Gaeta

Pages: 454 - 467

We apply the recently developed theory of symmetry of stochastic differential equations to a stochastic version of the logistic equation, obtaining an explicit integration, i.e. an explicit formula for the process in terms of any single realization of the driving Wiener process.

**Research Article**

## Nonlocal symmetry constraints and exact interaction solutions of the (2+1) dimensional modified generalized long dispersive wave equation

Junchao Chen, Yong Chen

Pages: 454 - 472

In this paper, nonlocal symmetry of the (2+1) dimensional modified generalized long dispersive wave system and its applications are investigated. The nonlocal symmetry related to the eigenfunctions in Lax pairs is derived, and infinitely many nonlocal symmetries are obtained. By introducing three potentials,...

**Research Article**

## Orbital Linearization in the Quadratic Lotka–Volterra Systems Around Singular Points Via Lie Symmetries

Jaume Giné, Susanna Maza

Pages: 455 - 464

In this paper, we consider linearizability and orbital linearizability properties of the Lotka–Volterra system in the neighborhood of a singular point with eigenvalues 1 and -q. In this paper we give the explicit smooth near-identity change of variables that linearizes or orbital linearizes such Lotka–Volterra...

**Research Article**

## Periodic Traveling Water Waves with Isobaric Streamlines

Henrik Kalisch

Pages: 461 - 471

It is shown that in water of finite depth, there are no periodic traveling waves with the property that the pressure in the underlying fluid flow is constant along streamlines. In the case of infinite depth, there is only one such solution, which is due to Gerstner.

**Research Article**

## On the Integrability of a Class of Nonlinear Dispersive Wave Equations

Ivanov ROSSEN

Pages: 462 - 468

We investigate the integrability of a class of 1+1 dimensional models describing nolinear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.

**Research Article**

## Mode-Coupling and Nonlinear Landau Damping Effects in Auroral Farley-Buneman Turbulence

A.M. Hamza

Pages: 462 - 470

The fundamental problem of Farley-Buneman turbulence in the auroral E-region has been discussed and debated extensively in the past two decades. In the present paper we intend to clarify the different steps that the auroral E-region plasma has to undergo before reaching a steady state. The mode-coupling...

**Research Article**

## Partial integrability of the anharmonic oscillator

Robert Conte

Pages: 462 - 473

We consider the anharmonic oscillator with an arbitrary-degree anharmonicity, a damping term and a forcing term, all coefficients being time-dependent: u â€²â€² + g1 (x)u â€² + g2 (x)u + g3 (x)u n + g4 (x) = 0, n real. Its physical applications range from the atomic Thomas-Fermi model to Emden gas dynamics...

**Research Article**

## On the Application of a Generalized Version of the Dressing Method to the Integration of Variable Coefficient *N*-Coupled Nonlinear Schrödinger Equation

Ting Su, Huihui Dai, Xian Guo Geng

Pages: 458 - 476

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are...

**Research Article**

## Derivation of asymptotical formulas for resolution of systems of differential equations with symmetrical matrices

M.I. Shkil, P.F. Samusenko

Pages: 463 - 467

Asymptotic formulae for resolution of L-diagonal systems of ordinary differential equations with symmetrical matrices are derived.

**Research Article**

## Helicoidal Peyrard–Bishop Model of DNA Dynamics*

Slobodan Zdravković

Pages: 463 - 484

A nonlinear model describing DNA dynamics, called helicoidal Peyrard–Bishop model, is described. It is shown that the model can explain a local opening of a DNA helix during transcription. An impact of friction forces is also studied. It is pointed out that a role of viscosity is crucial for DNA-RNA...

**Research Article**

## On Vortex Solutions and Links between the Weierstrass System and the Complex Sine-Gordon Equations

P. Bracken, P.P. Goldstein, A.M. Grundland

Pages: 464 - 486

The connection between the complex Sine and Sinh-Gordon equations associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlevé Test and analyse the possibility of deriving the Bäcklund transformation...

**Research Article**

## The LandauGinzberg Theory for the Two-Dimensional Bose Gas

Harry L. Morrison, Achilles D. Speliotopoulos

Pages: 464 - 474

Using results from sheaf theory combined with the phenomenological theory of the two-dimensional superfluid, the precipitation of quantum vortices is shown to be the genesis of a macroscopic order parameter for a phase transition in two dimensions.

**Research Article**

## Quaternionic Integrability

Giuseppe Gaeta

Pages: 461 - 474

Standard (Arnold–Liouville) integrable systems are intimately related to complex rotations. One can define a generalization of these, sharing many of their properties, where complex rotations are replaced by quaternionic ones, and more generally by the action of a Clifford group. Such a generalization...

**Research Article**

## Some Water Wave Equations and Integrability

Yishen Li

Pages: 466 - 481

A theory of bidirectional solitons on water is developed by using the classical Boussnesq equation. Moreover, analytical multi-solitons of Camassa-Holm equation are presented.

**Research Article**

## Hopf magnetic curves in the anti-de Sitter space 13

Giovanni Calvaruso, Marian Ioan Munteanu

Pages: 462 - 484

We consider the anti-de Sitter space 13 and the hyperbolic Hopf fibration h:13(1)→2(1/2). Using their description in terms of paraquaternions, we study the magnetic curves of the hyperbolic Hopf vector field. A complete classification is obtained for light-like magnetic curves, showing in particular...

**Research Article**

## G2-Calogero-Moser Lax operators from reduction

Andreas Fring, Nenad Manojlović

Pages: 467 - 478

We construct a Lax operator for the G2-Calogero-Moser model by means of a double reduction procedure. In the first reduction step we reduce the A6-model to a Bmodel with the help of an embedding of the B3-root system into the A6-root system together with the specification of certain coupling constants....

**Research Article**

## On the ten classes of scale-invariant nonlinear wave equations for vector fields

P.V. Marko

Pages: 468 - 473

We describe all systems of three equations of the form 2uj = Fj(u1, u2, u3), j = 1, 3 invariant under the extended Poincaré group. As a result, we have obtained ten classes of ~P(1, 3)-invariant nonlinear partial differential equations for real vector fields.