# Journal of Nonlinear Mathematical Physics

1499 articles

**Research Article**

## A class of third-order nonlinear evolution equations admitting invariant subspaces and associated reductions

Yujian Ye, Wen-Xiu Ma, Shoufeng Shen, Danda Zhang

Pages: 132 - 148

With the aid of symbolic computation by Maple, a class of third-order nonlinear evolution equations admitting invariant subspaces generated by solutions of linear ordinary differential equations of order less than seven is analyzed. The presented equations are either solved exactly or reduced to finite-dimensional...

**Research Article**

## A Two-Phase Free Boundary Problem for the Nonlinear Heat Equation

S. De Lillo, M.C. Salvatori

Pages: 134 - 140

A two-phase free boundary problem associated with nonlinear heat conduction is cosidered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution...

**Research Article**

## Group Invariant Solution for a Two-Dimensional Turbulent Free Jet described by Eddy Viscosity

D.P. Mason, D.L. Hill

Pages: 134 - 148

The group invariant solution for the stream function and the effective viscosity of a two-dimensional turbulent free jet are derived. Prandtl’s hypothesis is not imposed. When the eddy viscosity is constant across the jet it is found that the mean velocity profile is the same as that of a laminar jet...

**Research Article**

## Exact Pollard-like internal water waves

Mateusz Kluczek

Pages: 133 - 146

In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates...

**Research Article**

## About the Explicit Characterization of Hamiltonians of the Camassa-Holm Hierarchy

Enrique Loubet

Pages: 135 - 143

We present a detailed computation leading to an explicit formula for the fourth Hamitonian in the series of constants of motion with which any flow of the Camassa-Holm hierarchy is equipped, and explain the inherent difficulties in achieving such explicit expressions for invariants higher in the series.

**Research Article**

## On the Non-Inheritance of Symmetries of Partial Differential Equations

Keshlan S. Govinder, Barbara Abraham-Shrauner

Pages: 135 - 142

The inheritance of symmetries of partial differential equations occurs in a different manner from that of ordinary differential equations. In particular, the Lie algebra of the symmetries of a partial differential equation is not sufficient to predict the symmetries that will be inherited by a resulting...

**Research Article**

## The Complex Hamiltonian Systems and Quasi-periodic Solutions in the Hirota Equation

Jinbing Chen, Rong Tong

Pages: 134 - 149

The Hirota equation is reduced to a pair of complex Finite-dimensional Hamiltonian Systems (FDHSs) with real-valued Hamiltonians, which are proven to be completely integrable in the Liouville sense. It turns out that involutive solutions of the complex FDHSs yield finite parametric solutions of the Hirota...

**Research Article**

## Representations of the Conformal Lie Algebra in the Space of Tensor Densities on the Sphere

Pascal Redou

Pages: 136 - 140

Let F(Sn ) be the space of tensor densities on Sn of degree . We consider this space as an induced module of the nonunitary spherical series of the group SO0(n+1, 1) and classify (so(n+1, 1), SO(n+1))-simple and unitary submodules of F(Sn ) as a function of .

**Research Article**

## Existence and Homogenization of the Rayleigh-Bénard Problem

Björn Birnir, Nils Svanstedt

Pages: 136 - 169

The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl number close to one, we prove the existence of...

**Research Article**

## The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Guenbo Hwang, A. S. Fokas

Pages: 135 - 157

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem...

**Research Article**

## Asymptotic symmetries of difference equations on a lattice

Giuseppe Gaeta, Decio Levi, Rosaria Mancinelli

Pages: 137 - 146

It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations (e.g. scaling and/or translations) which are not symmetries of the considered equation. In this note we extend the approach to asymptotic symmetries for...

**Short Communication**

## Conditional Symmetry of Equations of Nonstationary Filtration and of the Nonlinear Heat Equation

Alla Vorobyova

Pages: 137 - 140

Conditional symmetry of the nonlinear gas filtration equation is studied. The operators obtained enabled to constract ansatzes reducing this equation to ordinary differential equations and to obtain its exact solutions.

**Research Article**

## Interchannel Soliton Collisions in Periodic Dispersion Maps in the Presence of Third Order Dispersion

Francisco J. Diaz-Otero, Pedro Chamorro-Posada

Pages: 137 - 143

We study the effects of third order dispersion (TOD) on the collision of wavelength division multiplexed solitons in periodic dispersion maps. The analysis is based on a proposed ODE model obtained using the variational method which takes into account third order dispersion. The impact of TOD on the...

**Research Article**

## A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

Robin Stanley Johnson

Pages: 137 - 160

The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type (“soliton” equations), but with vorticity. The familiar...

**Research Article**

## Integration of Systems of First-Order Equations Admitting Nonlinear Superposition

N. H. Ibragimov

Pages: 137 - 147

Systems of two nonlinear ordinary differential equations of the first order admitting nonlinear superpositions are investigated using Lie’s enumeration of groups on the plane. It is shown that the systems associated with two-dimensional Vessiot–Guldberg–Lie algebras can be integrated by quadrature upon...

**Research Article**

## Closed form Solutions to the Integrable Discrete Nonlinear Schrödinger Equation

Francesco Demontis, Cornelis van der Mee

Pages: 136 - 157

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using...

**Research Article**

## Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

Qiaozhen Zhu, Jian Xu, Engui Fan

Pages: 136 - 165

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices...

**Research Article**

## On a Class of Non Self-Adjoint Quantum Nonlinear Oscillators with Real Spectrum

Emanuela Caliceti, Sandro Graffi

Pages: 138 - 145

We prove reality of the spectrum for a class of PT - symmetric, non self-adjoint quantum nonlinear oscillators of the form H = p2 + P(q) + igQ(q). Here P(q) is an even polynomial of degree 2p positive at infinity, Q(q) an odd polynomial of degree 2r - 1, and the conditions p > 2r, |g| 0 hold.

**Research Article**

## Action-Angle Analysis of Some Geometric Models of Internal Degrees of Freedom

Barbara Gołubowska

Pages: 138 - 144

We derive and discuss equations of motion of infinitesimal affinely-rigid body moving in Riemannian spaces. There is no concept of extended rigid and affinely rigid body in a general Riemannian space. Therefore the gyroscopes with affine degrees of freedom are described as moving bases attached to the...

**Research Article**

## A Weil Representation of *sp*(4) Realized by Differential Operators in the Space of Smooth Functions on *S*^{2} × *S*^{1}

H. Fakhri

Pages: 137 - 144

In the space of complex-valued smooth functions on S2 × S1, we explicitly realize a Weil representation of the real Lie algebra sp(4) by means of differential generators. This representation is a rare example of highest weight irreducible representation of sp(4) all whose weight spaces are 1-dimensional....

**Research Article**

## Beyond Nonlinear Schrödinger Equation Approximation for an Anharmonic Chain with Harmonic Long Range Interactions

D. Grecu, Anca Visinescu, A.S. Cârstea

Pages: 139 - 144

Multi-scales method is used to analyze a nonlinear differential-difference equation. In the order 3 the NLS eq. is found to determine the space-time evolution of the leading amplitude. In the next order this has to satisfy a complex mKdV eq. (the next in the NLS hierarchy) in order to eliminate secular...

**Research Article**

## Symmetry, Singularities and Integrability in Complex Dynamics III: Approximate Symmetries and Invariants

P.G.L. Leach, S. Moyo, S. Cotsakis, R.L. Lemmer

Pages: 139 - 156

The different natures of approximate symmetries and their corresponding first intgrals/invariants are delineated in the contexts of both Lie symmetries of ordinary differential equations and Noether symmetries of the Action Integral. Particular note is taken of the effect of taking higher orders of the...

**Research Article**

## Psi-Series Solutions of the Cubic Hénon-Heiles System and Their Convergence

S. Melkonian

Pages: 139 - 160

The cubic Hénon-Heiles system contains parameters, for most values of which, the system is not integrable. In such parameter regimes, the general solution is expressible in formal expansions about arbitrary movable branch points, the so-called psi-series expansions. In this paper, the convergence of...

**Research Article**

## Formal Linearization and Exact Solutions of Some Nonlinear Partial Differential Equations

V.A. Baikov, K.R. Khusnutdinova

Pages: 139 - 146

**Research Article**

## Intrinsic Characterizations of Orthogonal Separability for Natural Hamiltonians with Scalar Potentials on Pseudo-Riemannian Spaces

Raymond G. McLenaghan, Roman G. Smirnov

Pages: 140 - 151

Orthogonal separability of finite-dimensional Hamiltonians is characterized by using various geometrical concepts, including Killing tensors, moving frames, the Nijehuis tensor, bi-Hamiltonian and quasi-bi-Hamiltonian representations. In addition, a complete classification of separable metrics defined...

**Research Article**

## A Method for Obtaining Darboux Transformations

Baoqun Lu, Yong He, Guangjiong Ni

Pages: 140 - 148

In this paper we give a method to obtain Darboux transformations (DTs) of integrable equations. As an example we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation.

**Research Article**

## Cocycles and stream functions in quasigeostrophic motion

Cornelia Vizman

Pages: 140 - 146

We present a geometric version of the Lie algebra 2-cocycle connected to quasi- geostrophic motion in the beta-plane approximation. We write down an Euler equation for the fluid velocity, corresponding to the evolution equation for the stream function in quasigeostrophic motion.

**Research Article**

## Global Existence and Blow-Up Phenomena for the Periodic Hunter–Saxton Equation with Weak Dissipation

Xuemei Wei, Zhaoyang Yin

Pages: 139 - 149

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that...

**Research Article**

## Geometrical Formulation of the Conformal Ward Identity

Mohamed Kachkachi

Pages: 141 - 150

In this paper we use deep ideas in complex geometry that proved to be very powerful in unveiling the Polyakov measure on the moduli space of Riemann surfaces and lead to obtain the partition function of perturbative string theory for 2, 3, 4 loops. Indeed a geometrical interpretation of the conformal...

**Research Article**

## Superconformal Algebras and Lie Superalgebras of the Hodge Theory

E. Poletaeva

Pages: 141 - 147

We observe a correspondence between the zero modes of superconformal algebras S (2, 1) and W(4) ([8]) and the Lie superalgebras formed by classical operators apearing in the Kähler and hyper-Kähler geometry.

**Research Article**

## The Algebra A ~P (1, 3) Invariants and Their Application to the Theory of Born-Infeld Field

Iryna A. Mishchenko

Pages: 141 - 145

The algebra A ~P (1, 3) invariants were found. These invariants allowed to reduce the Born-Infeld equation. After the reduction some solutions of the equation were found.

**Research Article**

## INCOMPLETE *q*-GAMMA FUNCTION AND TRICOMI EXPANSION

M. Mansour

Pages: 141 - 150

In this paper, we introduce a q-analogue of the Tricomi expansion for the incomplete q-gamma function. A general method is described for converting a power series into an expansion in incomplete q-gamma function. Also, we use the q-Tricomi expansion for giving a formal proof of the relation between the...

**Research Article**

## Parameterless discrete Painlevé equations and their Miura relations

B. Grammaticos, A. Ramani

Pages: 141 - 149

We present a study of discrete Painlevé equations which do not have any parameter, apart from those that can be removed by the appropriate scaling. We find four basic equations of this type as well as several more related to the basic ones by Miura transformations, which we derive explicitly. We obtain...

**Research Article**

## Discretization of Toroidal Soliton Equations

Yasuhiro Ohta

Pages: 143 - 148

We propose a way of discretization for the soliton equations associated with the toroidal Lie algebra based on the direct method. By the discretization, the symetry of the system is modified so that the discrete time evolutions are no longer compatible with the original continuous ones. The solutions...

**Research Article**

## Lie Group Analysis of Moffatt's Model in Metallurgical Industry

N. H. Ibragimov

Pages: 143 - 162

The paper is devoted to the Lie group analysis of a nonlinear equation arising in metallurgical applications of Magnetohydrodynamics. Self-adjointness of the basic equations is investigated. The analysis reveals two exceptional values of the exponent playing a significant role in the model.

**Research Article**

## New C-integrable and S-integrable systems of nonlinear partial differential equations

Francesco Calogero

Pages: 142 - 148

A technique to identify new C-integrable and S-integrable systems of nonlinear partial differential equations is reported, with two representative examples displayed and tersely discussed.

**Research Article**

## A Lie Symmetry Connection between Jacobi's Modular Differential Equation and Schwarzian Differential Equation

L. Rosati, M.C. Nucci

Pages: 144 - 161

In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi's modular differential equation. A six-dimensional Lie symmetry algebra is obtained...

**Research Article**

## Invariant Sets and Explicit Solutions to a Third-Order Model for the Shearless Stratified Turbulent Flow

V.N. Grebenev, B.B. Ilyushin

Pages: 144 - 156

We study dynamics of the shearless stratified turbulent flows. Using the method of differential constraints we find a class of explicit solutions to the problem under consideration and establish that the differential constraint obtained coincides with the well-known ZemanLumley model for stratified...

**Research Article**

## Evolution of Kink Network in Inhomogenous Systems

T. Dobrowolski, P. Tatrocki

Pages: 144 - 154

The purpose of this report is to show the influence of imperfections on creation and evolution of a kink network. Our main finding is a mechanism for reduction of the kinetic energy of kinks which works in both the overdamped and underdamped regimes. This mechanism reduces mobility of kinks and therefore...

**Research Article**

## Extensions of 1-Dimensional Polytropic Gas Dynamics

Boris A. Kupershmidt

Pages: 145 - 157

1-dimensional polytropic gas dynamics is integrable for trivial reasons, having 2

**Research Article**

## Hamiltonian dynamics of planar affinely-rigid body

Agnieszka Martens

Pages: 145 - 150

We discuss the dynamics of an affinely-rigid body in two dimensions. Translational degrees of freedom are neglected. The special stress is laid on completely integrable models solvable in terms of the separation of variables method.

**Research Article**

## Symplectic Structure of the Painlevé Test

Jishan Hu, Min Yan

Pages: 145 - 148

In this note, we present a result to show that the symplectic structures have been naturally encoded into the Painlevé test. In fact, for every principal balance, there is a symplectic change of dependent variables near movable poles.

**Research Article**

## A Vector Fokas-Lenells System from the Coupled Nonlinear Schrödinger Equations

MengXia Zhang, ShaoLing He, ShuQiang Lv

Pages: 144 - 154

With the aid of the spectral gradient method of Fuchssteiner, the compatible pair of Hamiltonian operators for the coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains a vector generalization of Fokas-Lenells system. The vector Fokas-Lenells system is...

**Research Article**

## On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Roberto Camassa, Jingfang Huang, Long Lee

Pages: 146 - 162

An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each "particle" in this method travels along a characteristic curve of...

**Short Communication**

## Reduction and Some Exact Solutions of the Multidimensional Liouville Equation

I.I. Yuryk

Pages: 146 - 148

Exact solutions of the multidimensional Liouville equation are constructed.

**Research Article**

## The nonlinear pendulum always oscillates

M. C. Nucci

Pages: 146 - 156

It is shown that the nonlinear pendulum equation can be transformed into a linear harmonic oscillator in the phase space thanks to Kerner’s method [12]. Moreover, as a mathematical divertissement, the second-order differential equation determining the phase-space trajectories of the nonlinear pendulum...

**Research Article**

## Profiles of Inflated Surfaces

Igor Pak, Jean-Marc Schlenker

Pages: 145 - 157

We study the shape of inflated surfaces introduced in [3] and [12]. More precisely, we analyze profiles of surfaces obtained by inflating a convex polyhedron, or more generally an almost everywhere flat surface, with a symmetry plane. We show that such profiles are in a one-parameter family of curves...

**Research Article**

## Pfaffianization of the q-difference version of the two-dimensional Toda lattice equation

Gegenhasi, Xing-Biao Hu, Hon-Wah Tam

Pages: 147 - 152

**Research Article**

## Symmetry Approach in Boundary Value Problems

I.T. Habibullin

Pages: 147 - 151

The problem of construction of boundary conditions for nonlinear equations compatible with their higher symmetries is considered. Boundary conditions for the sineGordon, ZhiberShabat and KdV equations are discussed. New examples are found for the JS equation.

**Research Article**

## Universal Phenomena in Solution Bifurcations of Some Boundary Value Problems

Alexander Sharkovsky, Andrij Sivak

Pages: 147 - 157

We show that for a class of boundary value problems, the space of initial functions can be stratified dependently on the limit behavior (as the time variable tends to infinity) of solutions. Using known results on universal phenomena appearing in bifurcations of one parameter families of one-dimensional...

**Research Article**

## On application of Liouville type equations to constructing Bäcklund transformations

Dmitry Demskoi

Pages: 147 - 156

It is shown how pseudoconstants of the Liouville-type equations can be exploited as a tool for construction of the Bäcklund transformations. Several new examples of such transformations are found. In particular we obtained the Bäcklund transformations for a pair of three-component analogs of the dispersive...

**Short Communication**

## Fourth-order recursion operators for third-order evolution equations

Marianna Euler

Pages: 147 - 151

We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit fourth-order recursion operators. Under the given assumptions we obtain the complete list of equations, one of which is the well-known Krichever-Novikov equation.

**Research Article**

## Deformations of Modules of Differential Forms

B. Agrebaoui, M. Ben Ammar, N. Ben Fraj, V. Ovsienko

Pages: 148 - 156

We study non-trivial deformations of the natural action of the Lie algebra Vect(Rn ) on the space of differential forms on Rn . We calculate abstractions for integrability of ifinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation...

**Research Article**

## Time-independent Hamiltonians describing systems with friction: the “cyclotron with friction”

Francesco Calogero, François Leyvraz

Pages: 147 - 154

As is well-known, any ordinary differential equation in one dimension can be cast as the Euler–Lagrange equation of an appropriate Lagrangian. Additionally, if the initial equation is autonomous, the Lagrangian can always be chosen to be time-independent. In two dimensions, however, the situation is...

**Research Article**

## The Extension of Integrable Mappings to Non-Commuting Variables

A. Ramani, T. Tamizhmani, B. Grammaticos, K. M. Tamizhmani

Pages: 149 - 165

We present an extension of a family of second-order integrable mappings to the case where the variables do not commute. In every case we introduce a commutation rule which is consistent with the mapping evolution. Through the proper ordering of variables we ensure the existence of an invariant in the...

**Research Article**

## Some Recent Results on Integrable Bilinear Equations

Xing-Biao Hu, Hon-Wah Tam

Pages: 149 - 155

This paper shows that several integrable lattices can be transformed into coupled biliear differential-difference equations by introducing auxiliary variables. By testing the Bäcklund transformations for this type of coupled bilinear equations, a new integrable lattice is found. By using the Bäcklund...

**Research Article**

## Nonlinear Wave Propagation Through Cold Plasma

S.G. Bindu, V.C. Kuriakose

Pages: 149 - 158

Electromagnetic wave propagation through cold collision free plasma is studied using the nonlinear perturbation method. It is found that the equations can be reduced to the modified Kortweg-de Vries equation.

**Short Communication**

## On Classes of Lie Solutions of MHD Equations, Expressed via the General Solution of the Heat Equation

Victor Popovych

Pages: 149 - 151

Large classes of Lie solutions of the MHD equations describing the flows of a viscous homogeneous incompressible fluid of finite electrical conductivity are constructed. These classes contain a number of arbitrary functions of time and the general solutions of the heat equation.

**Research Article**

## Exact Solutions of a Spherically Symmetric Energy Transport Model for Semiconductors

Motlatsi Molati

Pages: 149 - 158

The symmetry classification and reduction of a non–stationary spherically symmetric energy–transport model for semiconductors was investigated by Molati and Wafo Soh (2005). In this work the exact solutions of the reduced model in the stationary case are constructed.

**Research Article**

## A *Symmetry Invariance* Analysis of the Multipliers & Conservation Laws of the Jaulent–Miodek and Some Families of Systems of KdV Type Equations

A. H. Kara

Pages: 149 - 156

In this paper, we study and classify the conservation laws of the Jaulent–Miodek equations and other systems of KdV type equations which arises in, inter alia, shallow water equations. The main focus of the paper is the construction of the conservation laws as a consequence of the interplay between symmetry...

**Research Article**

## Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

Jafar Abedi-Fardad, Adel Rezaei-Aghdam, Ghorbanali Haghighatdoost

Pages: 149 - 170

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi’s...

**Research Article**

## Random Lie-point symmetries

Pedro José Catuogno, Luis Roberto Lucinger

Pages: 149 - 165

We introduce the notion of a random symmetry. It consists of taking the action given by a deterministic flow that maintains the solutions of a given differential equation invariant and replacing it with a stochastic flow. This generates a random action, which we call a random symmetry.

**Research Article**

## Quantization of the planar affinely-rigid body

Agnieszka Martens

Pages: 151 - 156

This paper is a continuation of [1] where the classical model was analyzed. Discussed are some quantization problems of two-dimensional affinely rigid body with the double dynamical isotropy. Considered are highly symmetric models for which the variables can be separated. Some explicit solutions are...

**Research Article**

## A Variational Approach to the Stability of Periodic Peakons

Jonatan Lenells

Pages: 151 - 163

The peakons are peaked traveling wave solutions of an integrable shallow water eqution. We present a variational proof of their stability.

**Research Article**

## Commutator representations of nonlinear evolution equations: Harry-Dym and Kaup-Newell cases

Zhijun Qiao

Pages: 151 - 157

A general structure of commutator representations for the hierarchy of nonlinear evolution equations (NLEEs) is proposed. As two concrete examples, the Harry-Dym and Kaup-Newell cases are discused.

**Research Article**

## Exact Solutions of the Nonlinear Fin Problem with Temperature-dependent Coefficients

Özlem Orhan, Teoman Özer

Pages: 150 - 170

The analytical solutions of a nonlinear fin problem with variable thermal conductivity and heat transfer coefficients are investigated by considering theory of Lie groups and its relations with λ-symmetries and Prelle-Singer procedure. Additionally, the classification problem with respect to different...

**Research Article**

## Reciprocal link for a three-component Camassa-Holm type equation

Nianhua Li

Pages: 150 - 156

A reciprocal transformation is introduced for a three-component Camassa-Holm type equation and it is showed that the transformed system is a reduction of the first negative flow in a generalized MKdV hierarchy.

**Research Article**

## On a q-Analog of ADHMN Construction for Self-Dual Yang-Mills

Atsushi Nakamula

Pages: 152 - 163

It is known that many integrable systems can be reduced from self-dual Yang-Mills equations. The formal solution space to the self-dual Yang-Mills equations is given by the so called ADHM construction, in which the solution space are graded by vector spaces with dimensionality concerning topological...

**Short Communication**

## On the Symmetry of Some Nonlinear Generalization of a Vector Subsystem of the Maxwell Equations

Volodymyr Smalij

Pages: 152 - 154

The problem of studying the maximal Lie symmetry of some nonlinear generalization of the vector subsystem of the Maxwell equations is completely solved.

**Research Article**

## Painleve Analysis and Symmetries of the HirotaSatsuma Equation

A.A. Mohammad, M. Can

Pages: 152 - 155

The singular manifold expansion of Weiss, Tabor and Carnevale [1] has been successfully applied to integrable ordinary and partial differential equations. They yield information such as Lax pairs, Bäcklund transformations, symmetries, recursion operators, pole dynamics, and special solutions. On the...

**Research Article**

## Lagrangian formulation of a generalized Lane-Emden equation and double reduction

C. Masood Khalique, Fazal M. Mahomed, Ben Muatjetjeja

Pages: 152 - 161

We classify the Noether point symmetries of a generalized Lane-Emden equation. We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f (y). One of them is f (y) = ?y r, where r

**Research Article**

## Derivations of the 3-Lie Algebra Realized by *gl*(*n*, ℂ)

Ruipu Bai, Jinxiu Wang, Zhenheng Li

Pages: 151 - 160

This paper studies structures of the 3-Lie algebra M realized by the general linear Lie algebra gl(n, ℂ). We show that M has only one nonzero proper ideal. We then give explicit expressions of both derivations and inner derivations of M. Finally, we investigate substructures of the (inner) derivation...

**Research Article**

## An Approximation of the Kinetic Energy of a Superfluid Film on a Riemann Surface

Chris Petersen Black

Pages: 151 - 160

The flow of a superfluid film adsorbed on a porous medium can be modeled by a meromorphic differential on a Riemann surface of high genus. In this paper, we define the mixed Hodge metric of meromorphic differentials on a Riemann surface and justify using this metric to approximate the kinetic energy...

**Research Article**

## A new family of discrete Painlevé equations and associated linearisable systems

A. Ramani, B. Grammaticos

Pages: 153 - 164

We derive discrete systems which result from a second, not studied up to now, form of the q-PVI equation. The derivation is based on two different procedures: “limits” and “degeneracies”. We obtain several new discrete Painlevé equations along with some linearisable systems. The parallel between the...

**Research Article**

## On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

Vladimir S. Gerdjikov, Georgi G. Grahovski

Pages: 155 - 168

A family of real Hamiltonian forms (RHF) for the special class of affine 1 + dimensional Toda field theories is constructed. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF...

**Research Article**

## A Model of Control of an Equation for Two-Electron Interaction

V.I. Sokolov

Pages: 155 - 160

Quantum Schrödinger equation, describing dynamical spin-interaction of two electrons with external magnetic field, is considered as an object for cybernetic research. Indeed, because of having a possibility to change external magnetic field, we can influence the interaction of particles. The algorithm...

**Research Article**

## Partially Solvable Spin Chains and QES Spin Models

A. Enisco, F. Finkel, A. Gonzalez-Lopez, M.A. Rodriguez

Pages: 155 - 165

In this paper we prove an extension of the usual freezing trick argument which can be applied to a number of quasi-exactly solvable spin models of CalogeroÂSutherland type. In order to illustrate the application of this method we analyze a partially solvable spin chain presenting near-neighbors interactions...

**Research Article**

## Singular Solution of the Liouville Equation under Perturbation

L.A. Kalyakin

Pages: 156 - 160

The Cauchy problem for the Liouville equation with a small perturbation is considered. We are interested in the asymptotics of the perturbed solution under the assumption that one has singularity. The main goal is to study both the asymptotic approximation of the singular lines and the asymptotic approximation...

**Research Article**

## On the Poincaré-Invariant Second-Order Partial Equations for a Spinor Field

Stanislav Spichak

Pages: 156 - 159

The different second-order nonlinear partial equations are found that are invariant under the representation D(1 2, 0) D(0, 1 2) of the Poincaré group P(1, 3) and also under conformal group C(1, 3). The some exact solutions are constructed for the one of these equations.

**Research Article**

## Inverse scattering on the half-line for generalized ZS-AKNS system with general boundary conditions

Mansur I. Ismailov, Bulent Yilmaz

Pages: 155 - 167

The inverse scattering problem for a first order system of three equations on the half-line with nonsingular diagonal matrix multiplying the derivative and general boundary conditions is considered. It is focused the case of two repeated diagonal elements of diagonal matrix. The scattering matrix on...

**Research Article**

## A three-component Camassa-Holm system with cubic nonlinearity and peakons

Baoqiang Xia, Ruguang Zhou, Zhijun Qiao

Pages: 155 - 169

In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peaked solitons (peakons). The 3CH model is proven to be integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system admits peakons and multi-peakon solutions....

**Research Article**

## Classical and Quantized Affine Physics: A Step towards it

Jan J. Slawianowski, Vasyl Kovalchuk

Pages: 157 - 166

The classical and quantum mechanics of systems on Lie groups and their homogeneous spaces are described. The special stress is laid on the dynamics of deformable bodies and the mutual coupling between rotations and deformations. Deformative modes are discretized, i.e., it is assumed that the relevant...

**Research Article**

## Periodic Solutions of a Many-Rotator Problem in the Plane. II. Analysis of Various Motions

F. Calogero, J-P Françoise, M. Sommacal

Pages: 157 - 214

Various solutions are displayed and analyzed (both analytically and numerically) of a recently-introduced many-body problem in the plane which includes both integrable and nonintegrable cases (depending on the values of the coupling constants); in paticular the origin of certain periodic behaviors is...

**Research Article**

## Functional Representation of the AblowitzLadik Hierarchy. II

V.E. Vekslerchik

Pages: 157 - 180

In this paper we continue studies of the functional representation of the Ablowitz Ladik hierarchy (ALH). Using formal series solutions of the zero-curvature condition we rederive the functional equations for the tau-functions of the ALH and obtain some new equations which provide more straightforward...

**Review Article**

## The Maupertuis Principle and Canonical Transformations of the Extended Phase Space

A.V. Tsiganov

Pages: 157 - 182

We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Vrious parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles...

**Research Article**

## Approximate Lie symmetries of the Navier-Stokes equations

V.N. Grebenev, M. Oberlack

Pages: 157 - 163

In the framework of the theory of approximate transformation groups proposed by Baikov, Gaziziv and Ibragimov [1], the first-order approximate symmetry operator is calculated for the Navier-Stokes equations. The symmetries of the coupled system obtained by expanding the dependent variables of the Navier-Stokes...

**Research Article**

## Fiber Bundle Description of Flow and Nonlinear Hydrodynamics on Circles

Andrei Ludu

Pages: 157 - 170

We introduce a differential geometry description of the path lines, stream lines and particles contours in hydrodynamics. We present a generalized form of a Korteweg-de Vries type of equation for the exterior of a circle. Nonlinearities from the boundary conditions, surface tension and the Euler equations...

**Research Article**

## A Novel Riccati Sequence

P. G. L. Leach, N. Euler

Pages: 157 - 164

Hierarchies of evolution partial differential equations have become well-established in the literature over the last thirty years. More recently sequences of ordinary differential equations have been introduced. Of these perhaps the most notable is the Riccati Sequence which has beautiful singularity,...

**Research Article**

## The heptagon-wheel cocycle in the Kontsevich graph complex

Ricardo Buring, Arthemy V. Kiselev, Nina J. Rutten

Pages: 157 - 173

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on n vertices and 2n − 2 edges, induce – under the orientation mapping – infinitesimal symmetries of classical Poisson...

**Short Communication**

## Note on the evolution of compactly supported initial data under the Camassa-Holm flow

Enrique Loubet

Pages: 158 - 162

We clarify and extend some remarks raised in [5] [Constantin A, J. Math. Phys. 46 (2005), 023506] about the evolution of compactly supported initial data under the Camassa-Holm flow.

**Research Article**

## Internal Gravity Waves with Free Upper Surface Over an Obstacle

M.B. Abd-El-Malek, A.H. Tewfick

Pages: 158 - 171

In this paper we discuss a theoretical model for both the free-surface and interfacial profiles of progressive nonlinear waves which result from introducing an obstacle of finite height, in the form of a ramp of gentle slope, attached to the bottom below the flow of a stratified, ideal, two-layer fluid....

**Research Article**

## Symmetry reduction and exact solutions of the Navier-Stokes equations. II

Wilhelm Fushchych, Roman Popowych

Pages: 158 - 188

This article is a direct continuation of our paper which was published in the Journal of Nonlinear Mathematical Physics 1994, V.1, N 1, 75113.

**Research Article**

## Conservation Laws and optimal system of extended quantum Zakharov-Kuznetsov equation

Yao-Lin Jiang, Yi Lu, Cheng Chen

Pages: 157 - 166

In this paper, the (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation is further explored. The equation is shown to be self-adjoint and conserved vector is constructed according to the related theorem. Then the corresponding optimal system of one-dimensional subgroups is determined. Similarity...

**Short Communication**

## What is the Velocity of the Electromagnetic Field?

Wilhelm Fushchych

Pages: 159 - 161

A new definition for the electromagnetic field velocity is proposed. The velocity depends on the physical fields. The question posed by the title of this paper is, surprisingly, not yet answered uniquely today; not even by way of definition. According to modern assumptions the light is the electromagnetic...

**Research Article**

## A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations

Sibusiso Moyo, P. G. L. Leach

Pages: 159 - 164

We study the integrability properties of the hierarchy of a class of nonlinear ordinary differential equations and point out some of the properties of these equations and their connection to the Ermakov-Pinney equation.

**Research Article**

## Diophantine Properties Associated to the Equilibrium Configurations of an Isochronous *N*-Body Problem

Oksana Bihun, Francesco Calogero, Ge Yi

Pages: 158 - 178

Recently a solvable N-body problem featuring several free parameters has been investigated, and conditions on these parameters have been identified which guarantee that this system is isochronous (all its solutions are periodic with a fixed period) and that it possesses equilibria. The N coordinates...

**Research Article**

## Lie Theorem via Rank 2 Distributions (Integration of PDE of Class ω = 1)

Boris Kruglikov

Pages: 158 - 181

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we give a new proof by relating it to the geometry of rank 2 distributions. We find a criterion for...

**Research Article**

## On Representations of *-Algebras in Mathematical Physics

Vasyl' L. Ostrovs'kyĭ, Yurii S. Samoilenko

Pages: 160 - 163

**Research Article**

## Novikov Super-Algebras with Associative Non-Degenerate Super-Symmetric Bilinear Forms

Junna Ni, Zhiqi Chen

Pages: 159 - 166

Novikov super-algebras are related to quadratic conformal super-algebras which correspond to Hamiltonian pairs and play fundamental role in completely integrable systems. In this paper, we focus on quadratic Novikov super-algebras, which are Novikov super-algebras with associative non-degenerate super-symmetric...

**Research Article**

## Nonlinear Gauge Transformation for a Quantum System Obeying an Exclusion-Inclusion Principle

G. Kaniadakis, A. Lavagno, P. Quarati, A.M. Scarfone

Pages: 161 - 165

We introduce a nonlinear and noncanonical gauge transformation which allows the rduction of a complex nonlinearity, contained in a Schrödinger equation, into a real one. This Schrödinger equation describes a canonical system, whose kinetics is governed by a generalized Exclusion-Inclusion Principle....