# Journal of Nonlinear Mathematical Physics

1499 articles

**Research Article**

## Correctors for the Homogenization of Monotone Parabolic Operators

Nils Svanstedt

Pages: 268 - 283

In the homogenization of monotone parabolic partial differential equations with oscilations in both the space and time variables the gradients converges only weakly in Lp . In the present paper we construct a family of correctors, such that, up to a remainder which converges to zero strongly in Lp ,...

**Corrigendum**

## The Lie Algebra sl(2, R) and so-called Kepler-Ermakov Systems

P.G.L. Leach, Ayse Karasu Kalkanli

Pages: 269 - 275

A recent paper by Karasu (Kalkanli) and Yildirim (Journal of Nonlinear Mathematical Physics 9 (2002) 475-482) presented a study of the Kepler-Ermakov system in the context of determining the form of an arbitrary function in the system which was compatible with the presence of the sl(2, R) algebra characteristic...

**Short Communication**

## Remarks on Quantization of Classical r-Matrices

Boris A. Kupershmidt

Pages: 269 - 272

If a classical r-matrix r is skewsymmetric, its quantization R can lose the skewsymmetry property. Even when R is skewsymmetric, it may not be unique.

**Research Article**

## Symmetries of Maxwell-Bloch Equations

Pantelis A. Damianou, Paschalis G. Paschali

Pages: 269 - 277

We study symmetries of the real Maxwell-Bloch equations. We give a Lax pair, biHamiltonian formulations and we find a symplectic realization of the system. We have also constructed a hierarchy of master symmetries which is used to generate nonlinear Poisson brackets. In addition we have calculated the...

**Research Article**

## Rota-Baxter operators on pre-Lie algebras

Xiuxian Li, Dongping Hou, Chengming Bai

Pages: 269 - 289

Rota-Baxter operators or relations were introduced to solve certain analytic and com- binatorial problems and then applied to many fields in mathematics and mathematical physics. In this paper, we commence to study the Rota-Baxter operators of weight zero on pre-Lie algebras. Such operators satisfy (the...

**Research Article**

## Final evolutions of a class of May-Leonard Lotka-Volterra systems

Claudio A. Buzzi, Robson A. T. Santos, Jaume Llibre

Pages: 267 - 278

We study a particular class of Lotka-Volterra 3-dimensional systems called May-Leonard systems, which depend on two real parameters a and b, when a + b = −1. For these values of the parameters we shall describe its global dynamics in the compactification of the non-negative octant of ℝ3 including its...

**Research Article**

## A special class of holomorphic mappings and the Faddeev-Hopf model

Radu Slobodeanu

Pages: 270 - 282

Pseudo horizontally weakly conformal maps [16] extend both holomorphic and (semi)conformal maps into an almost Hermitian manifold. We find critical points for the (generalized) Faddeev-Hopf model [28] in this larger class.

**Research Article**

## Massless Pseudo-scalar Fields and Solution of the Federbush Model

S.E. Korenblit, V.V. Semenov

Pages: 271 - 284

The formal Heisenberg equations of the Federbush model are linearized and then are directly integrated applying the method of dynamical mappings. The fundamental role of two-dimensional free massless pseudo-scalar fields is revealed for this procedure together with their locality condition taken into...

**Research Article**

## Pseudo-Stabilization of Prolonged Group Actions I. The Order Zero Case

Peter Olver

Pages: 271 - 277

It is shown that every point transformation group whose prolonged orbit dimensions pseudo-stabilize at order 0 is equivalent, under a change of variables, to the elementary similarity group consisting of translations and dilatations.

**Research Article**

## Tidal Tensors in the Description of Gravity and Electromagnetism

Nicoleta Voicu

Pages: 269 - 284

In 2008–2009, L. F. O. Costa and C. A. R. Herdeiro proposed a new gravito-electromagnetic analogy, based on tidal tensors. We show that connections on the tangent bundle of the space-time manifold help in finding an advantageous geometrization of their ideas. Moreover, the combination tidal tensors —...

**Research Article**

## Parametric Solution of Certain Nonlinear Differential Equations in Cosmology

Jennie D'Ambroise, Floyd L. Williams

Pages: 269 - 278

We obtain in terms of the Weierstrass elliptic ℘-function, sigma function, and zeta function an explicit parametrized solution of a particular nonlinear, ordinary differential equation. This equation includes, in special cases, equations that occur in the study of both homogeneous and inhomogeneous cosmological...

**Research Article**

## The Investigation into New Equations in (2 + 1) Dimensions

Kouichi Toda, Song-Ju Yu

Pages: 272 - 277

First of all, we show the existence of the Lax pair for the Calogero Korteweg-de Vries(CKdV) equation. Next we modify T operator that is one of the Lax pair for the CKdV equation for the search of the (2 + 1)-dimensional case and propose a new equation in (2 + 1) dimensions. We call it the (2 + 1)-dimensional...

**Research Article**

## Quest for Universal Integrable Models

Partha Guha

Pages: 273 - 293

In this paper we discuss a universal integrable model, given by a sum of two WessZumino-Witten-Novikov (WZWN) actions, corresponding to two different orbits of the coadjoint action of a loop group on its dual, and the Polyakov-Weigmann cocycle describing their interactions. This is an effective action...

**Research Article**

## Traces on the Algebra of Observables of the Rational CalogeroModel Based on the Root System

S.E. Konstein, I.V. Tyutin

Pages: 271 - 294

It is shown that HW (ℛ) (η), the algebra of observables of the rational Calogero model based on the root system ℛ ⊂ ℝN, has Tℛ independent traces, where Tℛ is the number of conjugacy classes of elements without eigenvalue 1 belonging to the Coxeter group W (ℛ) ⊂ End ℝN generated by the root system ℛ.
Simultaneously,...

**Research Article**

## Classical Poisson Structure for a Hierarchy of OneDimensional Particle Systems Separable in Parabolic Coordinates

J.C. Eilbeck, V.Z. Enol'skii, V.B. Kuznetsov, D.V. Leykin

Pages: 275 - 294

We consider a hierarchy of many-particle systems on the line with polynomial potentials separable in parabolic coordinates. The first non-trivial member of this hierarchy is a generalization of an integrable case of the Hénon-Heiles system. We give a Lax representation in terms of 2 × 2 matrices for...

**Research Article**

## Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides

Francesco Calogero, Farrin Payandeh

Pages: 273 - 280

The evolution equations mentioned in the title of this paper read as follows:
x˜n=P(n)(x1,x2), n=1,2,
where ℓ is the “discrete-time” independent variable taking integer values (ℓ = 0, 1, 2, ...), xn ≡ xn(ℓ) are the 2 dependent variables, x˜n≡xn(ℓ+1), and the 2 functions P(n)(x1, x2), n = 1, 2, are...

**Research Article**

## A Family of Linearizations of Autonomous Ordinary Differential Equations with Scalar Nonlinearity

Fethi Belkhouche

Pages: 276 - 288

This paper deals with a method for the linearization of nonlinear autonomous diferential equations with a scalar nonlinearity. The method consists of a family of approximations which are time independent, but depend on the initial state. The family of linearizations can be used to approximate the derivative...

**Research Article**

## A Common Integrable Structure in the Hermitian Matrix Model and Hele-Shaw Flows

L. Martinez Alonso, E. Medina

Pages: 277 - 287

It is proved that the system of string equations of the dispersionless 2-Toda hierarchy which arises in the planar limit of the hermitian matrix model also underlies certain processes in Hele-Shaw flows.

**Research Article**

## Speed selection for coupled wave equations

Mariano Cadoni, Giuseppe Gaeta

Pages: 275 - 297

We discuss models for coupled wave equations describing interacting fields, focusing on the speed of travelling wave solutions. In particular, we propose a general mechanism for selecting and tuning the speed of the corresponding (multi-component) travelling wave solutions under certain physical conditions....

**Research Article**

## Change of the Time for the Toda Lattice

A.V. Tsiganov

Pages: 278 - 282

For the Toda lattice we consider properties of the canonical transformations of the extended phase space, which preserve integrability. At the special values of integrals of motion the integral trajectories, separated variables and the action variables are invariant under change of the time. On the other...

**Research Article**

## On the Equivalence of Matrix Differential Operators to Schrödinger Form

F. Finkel, N. Kamran

Pages: 278 - 286

We prove a generalization to the case of s × s matrix linear differential operators of the classical theorem of E. Cotton giving necessary and sufficient conditions for equivalence of eigenvalue problems for scalar linear differential operators. The conditions for equivalence to a matrix Schrödinger...

**Research Article**

## Symplectic Symmetries of Hamiltonian Systems

Ihor Parasyuk

Pages: 278 - 282

The goal of this paper is to describe some interesting phenomena which occur in Hamiltonian systems with symplectic (locally Hamiltonian) symmetries.

**Research Article**

## Affine Ricci Solitons of Three-Dimensional Lorentzian Lie Groups

Yong Wang

Pages: 277 - 291

In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

**Research Article**

## Irreducible Characters and Clebsch-Gordan Series for the Exceptional Algebra E6: An Approach through the Quantum Calogero-Sutherland Model

J. Fernández-Núñez, W. Garcia-Fuertes, A.M. Perelomov

Pages: 280 - 301

We re-express the quantum Calogero-Sutherland model for the Lie algebra E6 and the particular value of the coupling constant = 1 by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan...

**Research Article**

## On Darboux transformations for the derivative nonlinear Schrödinger equation

Jonathan J.C. Nimmo, Halis Yilmaz

Pages: 278 - 293

We consider Darboux transformations for the derivative nonlinear Schrödinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the...

**Research Article**

## Conservation Laws for Self-Adjoint First-Order Evolution Equation

Igor Leite Freire

Pages: 279 - 290

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting...

**Research Article**

## A generalization of the Landau-Lifschitz equation: breathers and rogue waves

Ruomeng Li, Xianguo Geng, Bo Xue

Pages: 279 - 294

A generalization of the Landau-Lifschitz equation with uniaxial anisotropy is proposed, which can also reduce to the derivative nonlinear Schrödinger equation under an infinitesimal parameter. Based on the gauge transformation between Lax pairs, an N-fold generalized Darboux transformation is constructed...

**Research Article**

## On a q-Difference Painlevé III Equation: II. Rational Solutions

Kenji Kajiwara

Pages: 282 - 303

Rational solutions for a q-difference analogue of the Painlevé III equation are consdered. A Determinant formula of JacobiTrudi type for the solutions is constructed.

**Research Article**

## Painlevé Test and Higher Order Differential Equations

Uğurhan Muğan, Fahd Jrad

Pages: 282 - 310

Starting from the second Painlevé equation, we obtain Painlevé type equations of higher order by using the singular point analysis.

**Research Article**

## On Representation of the PQ Pair Solution at the Singular Point Neighborhood

N.V. Ustinov

Pages: 283 - 288

The compatible expansion in series of solutions of both the equations of PQ pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition...

**Research Article**

## On an inverse scattering algorithm for the Camassa-Holm equation

Octavian G. Mustafa, Donal O'Regan

Pages: 283 - 290

We present a clarification of a recent inverse scattering algorithm developed for the Camassa-Holm equation.

**Research Article**

## Explicit Solution Processes for Nonlinear Jump-Diffusion Equations

Gazanfer Ünal, Hasret Turkeri, Chaudry Masood Khalique

Pages: 281 - 292

Recent studies have shown that the nonlinear jump-diffusion models give results which are in agreement with financial data. Here we provide linearization criteria together with transformations which linearize the nonlinear jump-diffusion models with compound Poisson processes. Furthermore, we introduce...

**Research Article**

## Differential Equations Invariant Under Conditional Symmetries

Decio Levi, Miguel A. Rodríguez, Zora Thomova

Pages: 281 - 293

Nonlinear PDE’s having given conditional symmetries are constructed. They are obtained starting from the invariants of the conditional symmetry generator and imposing the extra condition given by the characteristic of the symmetry. Series of examples starting from the Boussinesq and including non-autonomous...

**Research Article**

## Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

M.C. Nucci

Pages: 284 - 304

After giving a brief account of the Jacobi last multiplier for ordinary differential equtions and its known relationship with Lie symmetries, we present a novel application which exploits the Jacobi last multiplier to the purpose of finding Lie symmetries of first-order systems. Several illustrative...

**Research Article**

## Asymptotic Solitons of the Johnson Equation

Igor Anders, Anne Boutet de Monvel

Pages: 284 - 302

We prove the existence of non-decaying real solutions of the Johnson equation, vaishing as x +. We obtain asymptotic formulas as t for the solutions in the form of an infinite series of asymptotic solitons with curved lines of constant phase and varying amplitude and width.

**Research Article**

## Symmetries and invariants for the 2D-Ricci flow model

Rodica Cimpoiasu, Radu Constantinescu

Pages: 285 - 292

The paper investigates some special Lie type symmetries and associated invariant quantities which appear in the case of the 2D Ricci flow equation in conformal gauge. Starting from the invariants some simple classes of solutions will be determined.

**Research Article**

## Invariant Linearization Criteria for Systems of Cubically Nonlinear Second-Order Ordinary Differential Equations

F. M. Mahomed, Asghar Qadir

Pages: 283 - 298

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches...

**Research Article**

## Generalized Self-Duality for the Supersymmetric Yang-Mills Theory with a Scalar Multiplet

V.A. Yatsun, A.M. Pavlyuk

Pages: 286 - 290

Generalized self-duality equations for the supersymmetric Yang-Mills theory with a scalar multiplet are presented in terms of component fields and superfields as well.

**Research Article**

## An Analogue of Holstein-Primakoff and Dyson Realizations for Lie Superalgebras. The Lie Superalgebra sl(1/n)

T.D. Palev

Pages: 287 - 292

An analogue of the Holstein-Primakoff and Dyson realizations for the Lie superalgebra sl(1/n) is written down. Expressions are the same as for the Lie algebra sl(n + 1), however in the latter, Bose operators have to be replaced with Fermi operators.

**Research Article**

## Soliton Propagation in Homogeneous and Inhomogeneous Models for DNA Torsion Dynamics

Mariano Cadoni, Roberto de Leo, Sergio Demelio

Pages: 287 - 319

The existence of solitonic excitations is a generic feature of a broad class of homogeneous models for nonlinear DNA internal torsional dynamics, but many properties of solitonic propagation depend on the actual model one is considering. In this paper we perform a detailed and comparative numerical investigation...

**Letter to Editor**

## Invariance of the Kaup–Kupershmidt Equation and Triangular Auto-Bäcklund Transformations

Marianna Euler, Norbert Euler

Pages: 285 - 291

We report triangular auto-Bäcklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup–Kupershmidt equation derived by E. G. Reyes in his paper titled “Nonlocal symmetries and the Kaup–Kupershmidt equation” [J. Math. Phys....

**Research Article**

## Matrix Integrals and the Geometry of Spinors

Johan van de Leur

Pages: 288 - 310

We obtain the collection of symmetric and symplectic matrix integrals and the colletion of Pfaffian tau-functions, recently described by Peng and Adler and van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector of a fermionic Fock space. This fermionic Fock space is the same...

**Research Article**

## Hopf Bifurcations in a Watt Governor with a Spring

Jorge Sotomayor, Luis Fernando Mello, Denis de Carvalho Braga

Pages: 288 - 299

This paper pursues the study carried out in [10], focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor system. Here are studied Hopf bifurcations of codimensions two, three and four and the pertinent Lyapunov stability coefficients and bifurcation diagrams. This allows to...

**Research Article**

## The Influence of Quantum Field Fluctuations on Chaotic Dynamics of Yang-Mills System II. The Role of the Centrifugal Term

V.I. Kuvshinov, A.V. Kuzmin, V.A. Piatrou

Pages: 289 - 293

We have considered SU(2) U(1) gauge field theory describing electroweak interations. We have demonstrated that centrifugal term in model Hamiltonian increases the region of regular dynamics of Yang-Mills and Higgs fields system at low densities of energy. Also we have found analytically the approximate...

**Research Article**

## A Dynamical System: Mars and its Satellite

Piotr Wąż

Pages: 289 - 293

A description of the most accurate analytical theory of the motion of Phobos, so far constructed, is presented. Several elements of the gravitational field of Mars, gravitational interactions between Phobos and Mars, Deimos and Jupiter, as well as tidal effects due to the interaction between the Sun...

**Research Article**

## A Solvable Many-Body Problem in the Plane

F. Calogero

Pages: 289 - 293

A solvable many-body problem in the plane is exhibited. It is characterized by rotation-invariant Newtonian ("acceleration equal force") equations of motion, featuring one-body ("external") and pair ("interparticle") forces. The former depend quadratically on the velocity, and nonlinearly on the coordinate,...

**Research Article**

## The Riccati and Ermakov-Pinney hierarchies

Marianna Euler, Norbert Euler, Peter Leach

Pages: 290 - 310

The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from...

**Research Article**

## On Poincaré-Invariant Reduction and Exact Solutions of the Yang-Mills Equations

Victor Lahno

Pages: 291 - 295

Classical ideas and methods developed by Sophus Lie provide us with a powerful tool for constructing exact solutions of partial differential equations (PDE) (see, e.g., [14]). In the present paper we apply the above methods to obtain new explicit solutions of the nonlinear Yang-Mills equations (YME).

**Research Article**

## Fordy-Kulish model and spinor Bose-Einstein condensate

V.A. Atanasov, V.S. Gerdjikov, G.G. Grahovski, N.A. Kostov

Pages: 291 - 298

A three-component nonlinear Schrodinger-type model which describes spinor Bose-Einstein condensate (BEC) is considered. This model is integrable by the inverse scattering method and using Zakharov-Shabat dressing method we obtain three types of soliton solutions. The multi-component nonlinear Schr¨odinger...

**Research Article**

## Madelung Representation for Complex Nonlinear D'Alembert Equations in n-Dimensional Minkowski Space

N. Euler, M. Euler

Pages: 292 - 300

**Research Article**

## Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System

Fengfeng Dong, Lingjun Zhou

Pages: 290 - 308

In this paper, we are concerned with the explicit construction of peakon solutions of the integrable twocomponent system with cubic non-linearity. We establish the spectral and inverse spectral problem associated to the Lax pairs of the system. The inverse problem is solved by the classical results of...

**Research Article**

## Interpolation of entire functions, product formula for basic sine function

Fethi Bouzeffour

Pages: 293 - 301

We solve the problem of constructing entire functions where ln M(r; f) grows like ln2 r from their values at q-n , for 0

**Research Article**

## Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP Equations

Wen-Xiu Ma, R.K. Bullough, P.J. Caudrey

Pages: 293 - 309

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with m arbitrary time-dependent coefficients...

**Research Article**

## Exact Solutions of Classical Scalar Field Equations

Marco Frasca

Pages: 291 - 297

We give a class of exact solutions of quartic scalar field theories. These solutions prove to be interesting as are characterized by the production of mass contributions arising from the nonlinear terms while maintaining a wave-like behavior. So, a quartic massless equation has a nonlinear wave solution...

**Short Communication**

## A Note on Fermionic Flows of the N=(1|1) Supersymmetric Toda Lattice Hierarchy

O. Lechtenfeld, A.S. Sorin

Pages: 294 - 296

We extend the Sato equations of the N=(1|1) supersymmetric Toda lattice hierachy by two new infinite series of fermionic flows and demonstrate that the algebra of the flows of the extended hierarchy is the Borel subalgebra of the N=(2|2) loop superalgebra.

**Research Article**

## A Pivotal Model for the (1 + 1)-Dimensional Heisenberg and Sigma Models

A.E. Winn

Pages: 294 - 299

An integrable interpolative (Pivotal) model for the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models is proposed and some solutions classifiable by an integer winding number examined.

**Research Article**

## On Certain Classes of Solutions of the Weierstrass-Enneper System Inducing Constant Mean Curvature Surfaces

P. Bracken, A.M. Grundland

Pages: 294 - 313

Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing certain classes of solutions to this system, including potential,...

**Research Article**

## Matrix Exponential via Clifford Algebras

Rafał Abłamowicz

Pages: 294 - 313

We use isomorphism between matrix algebras and simple orthogonal Clifford algebras C (Q) to compute matrix exponential eA of a real, complex, and quaternionic matrix A. The isomorphic image p = (A) in C (Q), where the quadratic form Q has a suitable signature (p, q), is exponentiated modulo a minimal...

**Letter to Editor**

## Conservation Laws for the Schrödinger–Newton Equations

G. Gubbiotti, M. C. Nucci

Pages: 292 - 299

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined...

**Research Article**

## The Orthogonal and Symplectic Schur Functions, Vertex Operators and Integrable Hierarchies

Linjie Shi, Na Wang, Minru Chen

Pages: 292 - 302

In this paper, we first construct an integrable system whose solutions include the orthogonal Schur functions and the symplectic Schur functions. We find that the orthogonal Schur functions and the symplectic Schur functions can be obtained by one kind of Boson-Fermion correspondence which is slightly...

**Research Article**

## Twisted Volterra equation

Sergei D. Silvestrov

Pages: 295 - 299

In this paper an extension of the q-deformed Volterra equation associated with linear rescaling to the general non-linear rescaling is obtained.

**Research Article**

## On Linear and Non-Linear Representations of the Generalized Poincaré Groups in the Class of Lie Vector Fields

Wilhelm Fushchych, Renat Zhdanov, Victor Lahno

Pages: 295 - 308

We study representations of the generalized Poincaré group and its extensions in the class of Lie vector fields acting in a space of n + m independent and one dependent variables. We prove that an arbitrary representation of the group P(n, m) with max {n, m} 3 is equivalent to the standard one, while...

**Research Article**

## On the Mapping of Jet Spaces

Václav Tryhuk, Veronika Chrastinová

Pages: 293 - 310

Any locally invertible morphism of a finite-order jet space is either a prolonged point transformation or a prolonged Lie's contact transformation (the Lie–Bäcklund theorem). We recall this classical result with a simple proof and moreover determine explicit formulae even for all (not necessarily...

**Research Article**

## On Unique Symmetry of Two Nonlinear Generalizations of the Schrödinger Equation

Wilhelm Fushchych, Roman Cherniha, Volodymyr Chopyk

Pages: 296 - 301

We prove that two nonlinear generalizations of the nonlinear Schrödinger equation are invariant with respect to a Lie algebra that coincides with the invariance algebra of the Hamilton-Jacobi equation.

**Research Article**

*N* = 2 Supercomplexification of the Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt Equations

Ziemowit Popowicz

Pages: 294 - 312

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated....

**Research Article**

## Weierstrass integrability for a class of differential systems

Jaume Llibre, Claudia Valls

Pages: 294 - 307

We characterize the differential equations of the form
x′=y, y′=an(x)yn+an-1(x)yn-1+⋯+a1(x)y+a0(x), n≥2, an(0)≠0,
where aj(x) are meromorphic functions in the variable x for j = 0,…,n that admits either a Weierstrass first integral or a Weierstrass inverse integrating factor.

**Research Article**

## A Holomorphic Point of View about Geodesic Completeness

Claudio Meneghini

Pages: 297 - 324

We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with their complex counterparts, and of Clifton-Pohl torus, to show that...

**Research Article**

## Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

Isaac A. García, Benito Hernández-Bermejo

Pages: 295 - 307

A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the bifurcation phenomena of periodic orbits as a result of these perturbations...

**Research Article**

## Klein operator and the Number of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

S.E. Konstein, R. Stekolshchik

Pages: 295 - 308

In the Coxeter group W (ℛ) generated by the root system ℛ, let T (ℛ) be the number of conjugacy classes having no eigenvalue +1 and let S (ℛ) be the number of conjugacy classes having no eigenvalue −1. The algebra HW (ℛ) of observables of the rational Calogero model based on the root system ℛ possesses...

**Research Article**

## Polynomial integrals for third- and fourth-order ordinary difference equations

R. Sahadevan, C. Uma Maheswari

Pages: 299 - 315

A direct method to construct polynomial integrals for third order ordinary difference equation (OE) w(n + 3) = F(w(n),w(n + 1),w(n + 2)) and fourth order OE w(n+4) = F(w(n),w(n+1),w(n+2),w(n+3)) is presented. The effectiveness of the method to construct more than one polynomial integral for N-th order...

**Research Article**

## A fuzzy difference equation of a rational form

G. Stefanidou, G. Papaschinopoulos

Pages: 300 - 315

In this paper, we prove some effects concerning a Fuzzy Difference Equation of a rational form.

**Research Article**

## Quasi-Bi-Hamiltonian Systems Obtained from Constrained Flows

Yunbo Zeng

Pages: 300 - 304

An infinite number of families of quasi-bi-Hamiltonian (QBH) systems can be costructed from the constrained flows of soliton equations. The Nijenhuis coordinates for the QBH systems are proved to be exactly the same as the separation variables introduced by the Lax matrices for the constrained flows.

**Research Article**

## Integrating Factors and λ—Symmetries

C. Muriel, J. L. Romero

Pages: 300 - 309

We investigate the relationship between integrating factors and λ-symmetries for ordinary differential equations of arbitrary order. Some results on the existence of λ-symmetries are used to prove an independent existence theorem for integrating factors. A new method to calculate integrating...

**Research Article**

## The Bilinear Integrability, N-soliton and Riemann-theta function solutions of B-type KdV Equation

Jianqin Mei, Lijuan Wu

Pages: 298 - 307

In this paper, the bilinear integrability for B-type KdV equation have been explored. According to the relation to tau function, we find the bilinear transformation and construct the bilinear form with an auxiliary variable of the B-type KdV equation. Based on the truncation form, the Bäcklund transformation...

**Research Article**

## On Lie Reduction of the Navier-Stokes Equations

Roman Popovych

Pages: 301 - 311

Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is described.

**Research Article**

## Binary Darboux Transformation and Quasideterminant Solutions of The Chiral Field

Bushra Haider, M. Hassan, U. Saleem

Pages: 299 - 321

The standard binary Darboux transformation is composed and is used to obtain exact multisoliton solutions of the chiral field model in two dimensions. The solutions are expressed in terms of quasideterminants. It has been shown that the standard binary Darboux transformation is equivalent to the elementary...

**Research Article**

## Conservation Laws for Heated Laminar Radial Liquid and Free Jets

R. Naz, D. P. Mason

Pages: 299 - 309

The conserved quantities for the heated radial liquid jet and the heated radial free jet are established by using conservation laws. The flow in a heated radial jet is described by Prandtl's momentum boundary layer equation, the continuity equation and the energy equation. Viscous dissipation is...

**Research Article**

## Hamiltonian formalism of the Landau-Lifschitz equation for a spin chain with full anisotropy

Nian-Ning Huang, Hao Cai, Tian Yan, Fan-Rong Xu

Pages: 302 - 314

The Hamiltonian formalism of the Landau-Lifschitz equation for a spin chain with full anisotropy is formulated completely, which constructs a stable base for further investigations.

**Research Article**

## The Time Periodic Solution of the Burgers Equation on the Half-Line and an Application to Steady Streaming

A.S. Fokas, J.T. Stuart

Pages: 302 - 314

The phenomenon of steady streaming, or acoustic streaming, is an important phyical phenomenon studied extensively in the literature. Its mathematical formulation involves the Navier-Stokes equations, and due to the complexity of these equations is usually studied heuristically using formal perturbation...

**Research Article**

## Gauge Classification, Lie Symmetries and Integrability of a Family of Nonlinear Schrödinger Equations

P. Nattermann, H.-D. Doebner

Pages: 302 - 310

In this contribution we review and summarize recent articles on a family of nonlinear Schrödinger equations proposed by G.A. Goldin and one of us (HDD) [J. Phys. A. 27, 1994, 17711780], dealing with a gauge description of the family, a classification of its Lie symmetries in terms of gauge invariants...

**Research Article**

## Solvable and/or Integrable and/or Linearizable N-Body Problems in Ordinary (Three-Dimensional) Space. I

M. Bruschi, F. Calogero

Pages: 303 - 386

Several N -body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion ("acceleration equal force;" in most cases, the forces are velocity-dependent) and are amenable to exact treatment ("solable" and/or "integrable" and/or "linearizable")....

**Research Article**

## Billiard Algebra, Integrable Line Congruences, and Double Reflection Nets

Vladimir Dragović, Milena Radnović

Pages: 300 - 317

Billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrability...

**Research Article**

## A Solvable *N*-body Problem of Goldfish Type Featuring *N*^{2} Arbitrary Coupling Constants

Francesco Calogero

Pages: 300 - 305

A N-body problem “of goldfish type” is introduced, the Newtonian (“acceleration equal force”) equations of motion of which describe the motion of N pointlike unit-mass particles moving in the complex z-plane. The model—for arbitrary N—is solvable, namely its configuration (positions and velocities of...

**Research Article**

## Stability Analysis of Some Integrable Euler Equations for SO(n)

L. Fehér, I. Marshall

Pages: 304 - 317

A family of special cases of the integrable Euler equations on so(n) introduced by Manakov in 1976 is considered. The equilibrium points are found and their stability is studied. Heteroclinic orbits are constructed that connect unstable equilibria and are given by the orbits of certain 1-parameter subgroups...

**Research Article**

## Jacobi's Last Multiplier and the Complete Symmetry Group of the Ermakov-Pinney Equation

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

Pages: 305 - 320

The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in...

**Research Article**

## Dynamics of Soliton-Like Excitations in a Chain of a Polymer Crystal: Influence of Neighbouring Chains Mobility

Elena A. Zubova, N.K. Balabaev

Pages: 305 - 311

We investigate influence of mobility of neighbouring chains on dynamics of soliton-like excitations in a chain of the simplest polymer crystal (polyethylene in the "united atoms" approximation) using molecular dynamics simulation. We present results for point-like structural defects: static and moving...

**Letter to Editor**

## Multipotentializations and nonlocal symmetries: Kupershmidt, Kaup-Kupershmidt and Sawada-Kotera equations

Marianna Euler, Norbert Euler, Enrique G Reyes

Pages: 303 - 314

In this letter we report a new invariant for the Sawada-Kotera equation that is obtained by a systematic potentialization of the Kupershmidt equation. We show that this result can be derived from nonlocal symmetries and that, conversely, a previously known invariant of the Kaup-Kupershmidt equation can...

**Research Article**

## A New Case of Separability in a Quartic Hénon-Heiles System

Nicola Sottocornola

Pages: 303 - 308

There are four quartic integrable Hénon-Heiles systems. Only one of them has been separated in the generic form while the other three have been solved only for particular values of the constants. We consider two of them, related by a canonical transformation, and we give their separation coordinates...

**Research Article**

## Direct Method of Finding First Integrals of Finite Dimensional Systems and Construction of Nondegenerate Poisson Structures

A. Annamalai, K.M. Tamizhmani

Pages: 309 - 330

We present a novel method of finding first integrals and nondegenerate Poisson structures for a given system. We consider the given system as a system of differential

**Research Article**

## Darboux transformations of the Supersymmetric BKP hierarchy

Chuanzhong Li

Pages: 306 - 313

In this paper, we construct Darboux transformations of the supersymmetric BKP(SBKP) hierarchy. These Darboux transformations can generate new solutions from seed solutions by using bosonic eigenfunctions.

**Research Article**

## Transformation Properties of x'' + f_1(t)x' + f2(t)x + f3(t)x^n = 0

Norbert Euler

Pages: 310 - 337

In this paper, we consider a general anharmonic oscillator of the form ¨x + f1(t) x + f2(t)x+f3(t)xn = 0, with n Q. We seek the most general conditions on the functions f1, f2 and f3, by which the equation may be integrable, as well as conditions for the existence of Lie point symmetries. Time-dependent...

**Research Article**

## The Sato Grassmannian and the CH Hierarchy

Gregorio Falqui, Giovanni Ortenzi

Pages: 310 - 322

We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmannian. We discuss the geometry of an extension of the negative flows of the CH hierarchy, recover the well-known CH equations, and frame the problem within the theory of pseudo-differential operators.

**Research Article**

## Bounds for the Threshold Amplitude for Plane Couette Flow

Mattias Liefvendahl, Gunilla Kreiss

Pages: 311 - 324

We prove nonlinear stability for finite amplitude perturbations of plane Couette flow. A bound of the solution of the resolvent equation in the unstable complex half-plane is used to estimate the solution of the full nonlinear problem. The result is a lower bound, including Reynolds number dependence,...

**Addendum**

## Addendum to: "Coupled KdV Equations of Hirota-Satsuma Type" (J. Nonlin. Math. Phys. Vol. 6, No.3 (1999), 255262)

S. Yu. Sakovich

Pages: 311 - 312

It is shown that one system of coupled KdV equations, found in J. Nonlin. Math. Phys.,

**Research Article**

## A Symmetry Connection Between Hyperbolic and Parabolic Equations

Peter Basarab-Horwath

Pages: 311 - 318

We give ansatzes obtained from Lie symmetries of some hyperbolic equations which reduce these equations to the heat or Schrödinger equations. This enables us to construct new solutions of the hyperbolic equations using the Lie and conditional symmetries of the parabolic equations. Moreover, we note that...

**Research Article**

## Variational derivation of the Camassa-Holm shallow water equation

Delia Ionescu-Kruse

Pages: 311 - 320

We describe the physical hypothesis in which an approximate model of water waves is obtained. For an irrotational unidirectional shallow water flow, we derive the Camassa- Holm equation by a variational approach in the Lagrangian formalism.

**Research Article**

## Examples of Simple Vectorial Lie Algebras in Characteristic 2

Uma N. Iyer, Dimitry Leites, Mohamed Messaoudene, Irina Shchepochkina

Pages: 311 - 374

The classification of simple finite dimensional modular Lie algebras over algebraically closed fields of characteristic p > 3 (described by the generalized Kostrikin–Shafarevich conjecture) being completed due to Block, Wilson, Premet and Strade (with contributions from other researchers) the next...

**Research Article**

## Solutions of the constrained mKP hierarchy by Boson-Fermion correspondence

Huizhan Chen, Lumin Geng, Jipeng Cheng

Pages: 308 - 323

In this paper, the Hirota bilinear equation of the constrained modified KP hierarchy is expressed as the vacuum expectation values of Clifford operators by using the free fermions method of mKP hierarchy. Then we mainly use the Boson-Fermion correspondence to solve the Hirota bilinear equation of the...

**Research Article**

## Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

Chaohong Pan, Yating Yi

Pages: 308 - 320

In this paper, by using the bifurcation method of dynamical systems, we derive the traveling wave solutions of the nonlinear equation UUτyy − UyUτy + U2Uτ + 3Uy = 0. Based on the relationship of the solutions between the Novikov equation and the nonlinear equation, we present the parametric representations...