# Journal of Nonlinear Mathematical Physics

1499 articles

**Letter to Editor**

## Classical solutions of a massless Wess-Zumino model

Marco Frasca

Pages: 464 - 468

We provide a current expansion for the classical equations of motion of the massles Wess-Zumino model. In the low-energy limit, there appears a massive behavior for bosonic degrees of freedom and, at small coupling, the fermion field shows the same mass and supersymmetry is overall preserved. In the...

**Research Article**

## Bernoulli Numbers and Solitons

Marie-Pierre Grosset, Alexander P. Veselov

Pages: 469 - 474

We present a new formula for the Bernoulli numbers as the following integral B2m = (-1)m-1 22m+1 +( dm-1 dxm-1 sech2 x)2 dx. This formula is motivated by the results of Fairlie and Veselov, who discovered the relation of Bernoulli polynomials with soliton theory. Dedicated to Hermann Flaschka on his...

**Research Article**

## Averaging in Weakly Coupled Discrete Dynamical Systems

Niklas Brännström

Pages: 465 - 487

In [Y. Kifer, Averaging in difference equations driven by dynamical systems, Asterisque287 (2003) 103–123] a general averaging principle for slow-fast discrete dynamical systems was presented. In this paper we extend this method to weakly coupled slow-fast systems. For this setting we obtain sharper...

**Letter to Editor**

## Bäcklund transformations between four Lax-integrable 3D equations

Oleg I. Morozov, Maxim V. Pavlov

Pages: 465 - 468

Recently a classification of contactly-nonequivalent three-dimensional linearly degenerate equations of the second order was presented by E.V. Ferapontov and J. Moss. The equations are Lax-integrable. In our paper we prove that all these equations are connected with each other by appropriate Bäcklund...

**Research Article**

## Symmetry of Equations with Convection Terms

W.I. Fushchych, Z.I. Symenoh

Pages: 470 - 479

We study symmetry properties of the heat equation with convection term (the equation of convection diffusion) and the Schrödinger equation with convection term. We also investigate the symmetry of systems of these equations with additional conditions for potentials. The obtained results are applied to...

**Research Article**

## Linearisable mappings, revisited

B. Grammaticos, A. Ramani, J. Satsuma

Pages: 466 - 473

We examine the growth properties of second-order mappings which are integrable by linearisation and which generically exhibit a linear growth of the homogeneous degree of initial conditions. We show that for Gambier-type mappings for which the growth proceeds generically with a step of 1 there exist...

**Short Communication**

## Least Action Principle for an Integrable Shallow Water Equation

Adrian Constantin, Boris Kolev

Pages: 471 - 474

For an integrable shallow water equation we describe a geometrical approach shoing that any two nearby fluid configurations are successive states of a unique flow minimizing the kinetic energy.

**Research Article**

## Uniqueness Issues on Permanent Progressive Water-Waves

K. Kobayashi, H. Okamoto

Pages: 472 - 479

We consider two-dimensional water-waves of permanent shape with a constant proagation speed. Two theorems concerning the uniqueness of certain solutions are rported. Uniqueness of Crapper's pure capillary waves is proved under a positivity assumption. The proof is based on the theory of positive operators....

**Research Article**

## Quasiperiodic Solutions of the Heisenberg Ferromagnet Hierarchy

Peng Zhao, Engui Fan, Temuerchaolu

Pages: 468 - 482

We present quasi-periodic solutions in terms of Riemann theta functions of the Heisenberg ferromagnet hierarchy by using algebrogeometric method. Our main tools include algebraic curve and Riemann surface, polynomial recursive formulation and a special meromorphic function.

**Letter to Editor**

## Novel differential algorithm to evaluate *all* the zeros of any *generic* polynomial

Francesco Calogero

Pages: 469 - 472

A novel, remarkably neat, differential algorithm is introduced, which is suitable to evaluate all the zeros of a generic polynomial of arbitrary degree N.

**Letter to Editor**

## Properties of the zeros of the sum of three polynomials

Francesco Calogero

Pages: 469 - 474

A system of algebraic equations satisfied by the zeros of the sum of three polynomials are reported.

**Short Communication**

## Symmetry Reduction and Exact Solutions of the Eikonal Equation

Ivan Fedorchuk

Pages: 474 - 477

By means of splitting subgroups of the generalized Poincaré group P(1, 4), ansatzes which reduce the eikonal equation to differential equations with fewer independent variables have been constructed. The corresponding symmetry reduction has been done. By means of the solutions of the reduced equations...

**Research Article**

## The Riemann-Hilbert Formalism For Certain Linear and Nonlinear Integrable PDEs

Dimitrios A. Pinotsis

Pages: 474 - 493

We show that by deforming the Riemann-Hilbert (RH) formalism associated with certain linear PDEs and using the so-called dressing method, it is possible to derive in an algorithmic way nonlinear integrable versions of these equations. In the usual Dressing Method, one first postulates a matrix RH problem...

**Research Article**

## Box-Ball System with Reflecting End

Atsuo Kuniba, Masato Okado, Yasuhiko Yamada

Pages: 475 - 507

A soliton cellular automaton on a one dimensional semi-infinite lattice with a reflecting end is presented. It extends a box-ball system on an infinite lattice associated with the crystal base of Uq(sln). A commuting family of time evolutions are obtained by adapting the K matrices and the double row...

**Research Article**

## On the Lie Symmetries of KeplerErmakov Systems

Ayse Karasu (Kalkanli), Hasan Yildirim

Pages: 475 - 482

In this work, we study the Lie-point symmetries of KeplerErmakov systems presented by C Athorne in J. Phys. A24 (1991), L1385L1389. We determine the forms of arbitrary function H(x, y) in order to find the members of this class possessing the sl(2, R) symmetry and a Lagrangian. We show that these systems...

**Research Article**

## Symmetry, Singularities and Integrability in Complex Dynamics V: Complete Symmetry Groups of Certain Relativistic Spherically Symmetric Systems

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

Pages: 475 - 490

We show that the concept of complete symmetry group introduced by Krause (J. Math. Phys. 35 (1994), 57345748) in the context of the Newtonian Kepler problem has wider applicability, extending to the relativistic context of the Einstein equations dscribing spherically symmetric bodies with certain conformal...

**Research Article**

## On the Darboux integrability of a three–dimensional forced–damped differential system

Jaume Llibre, Regilene Oliveira, Claudia Valls

Pages: 473 - 494

In 2011 Pehlivan proposed a three–dimensional forced–damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability,...

**Research Article**

## Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

Yu Hou, Engui Fan

Pages: 473 - 508

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic...

**Short Communication**

## Symmetry Properties of Generalized Gas Dynamics Equations

Maria Serova

Pages: 478 - 480

We describe a class of generalized gas dynamics equations invariant under the extended Galilei algebra A ~G(1, n).

**Research Article**

## A convenient expression of the time-derivative
zn(k)(t)
, of arbitrary order *k*, of the zero *z*_{n}(*t*) of a time-dependent polynomial *p*_{N}(*z*;*t*) of arbitrary degree *N* in *z*, and solvable dynamical systems

_{n}

_{N}

Mario Bruschi, Francesco Calogero

Pages: 474 - 485

Let pN
(z; t) be a (monic) time-dependent polynomial of arbitrary degree N in z, and let zn ≡ zn (t) be its N zeros:
pN (z;t)=∏n=1N[z-zn(t)]
. In this paper we report a convenient expression of the k-th time-derivative
zn(k)(t)
of the zero zn (t). This formula plays a key role in the...

**Research Article**

## An asymptotic expansion of the q-gamma function q(x)

M. Mansour

Pages: 479 - 483

In this paper, we get an asymptotic expansion of the q-gamma function q(x). Also, we deduced q-analogues of Gauss' multiplication formula and Legendre's relation which give the known results when q tends to 1.

**Research Article**

## 2D Locus Configurations and the Trigonometric Calogero–Moser System

Greg Muller

Pages: 475 - 482

A central hyperplane arrangement in ℂ2 with multiplicity is called a “locus configuration” if it satisfies a series of “locus equations” on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting...

**Research Article**

## An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics

R. S. Johnson

Pages: 475 - 493

We describe a problem that can be tackled more-or-less routinely using the ideas of classical fluid mechanics, but it is a complex flow and even the linearised problem involves considerable algebraic complexity. The presentation here emphasises the approach that we adopt in order to formulate an accessible...

**Letter to Editor**

## On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems

Guillaume Chèze, Thomas Cluzeau

Pages: 475 - 479

We consider generalized Liénard polynomial differential systems of the form ẋ = y, ẏ = -g(x) - f (x) y, with f (x) and g(x) two polynomials satisfying deg(g) ≤ deg(f). In their work, Llibre and Valls have shown that, except in some particular cases, such systems have no Liouvillian first integral. In...

**Research Article**

## Lie Groups and Mechanics: An Introduction

Boris Kolev

Pages: 480 - 498

The aim of this paper is to present aspects of the use of Lie groups in mechanics. We start with the motion of the rigid body for which the main concepts are extracted. In a second part, we extend the theory for an arbitrary Lie group and in a third section we apply these methods for the diffeomorphism...

**Research Article**

## On Symmetries of Chern-Simons and BF Topological Theories

T.A. Ivanova, A.D. Popov

Pages: 480 - 494

We describe constructing solutions of the field equations of Chern-Simons and toplogical BF theories in terms of deformation theory of locally constant (flat) bundles. Maps of flat connections into one another (dressing transformations) are considered. A method of calculating (nonlocal) dressing symmetries...

**Short Communication**

## Lorentz Transformations for the Schrödinger Equation

Vladimir Shtelen

Pages: 480 - 481

We show that the free Schrödinger equation admits Lorentz space-time transformations when corresponding transformations of the -function are nonlocal. Some consequences of this symmetry are discussed.

**Research Article**

## On the Integrability of a Muthuswamy–Chua System

Jaume Llibre, Claudia Valls

Pages: 477 - 488

In this paper we study the integrability of the Muthuswamy–Chua system
x'=y,y'=-x3+y2-yz22,z'=y-αz-yz.
For α = 0 we characterize all its generalized rational first integrals, which contains the Darboux type first integrals. For α ≠ 0 we show that the system has no Darboux type first...

**Research Article**

## Algebraic Extensions of Gaudin Models

Fabio Musso, Matteo Petrera, Orlando Ragnisco

Pages: 482 - 498

We perform a InönüWigner contraction on Gaudin models, showing how the integrbility property is preserved by this algebraic procedure. Starting from Gaudin models we obtain new integrable chains, that we call Lagrange chains, associated to the same linear r-matrix structure. We give a general construction...

**Research Article**

## Invariant Solutions of a Nonlinear System of Differential Equations for Electromagnetic Field

Lyudmyla L. Barannyk

Pages: 482 - 491

Solutions invariant under subalgebras of the affine algebra AIGL(3, R) are found.

**Research Article**

## Nonautonomous symmetries of the KdV equation and step-like solutions

V.E. Adler

Pages: 478 - 493

We study solutions of the KdV equation governed by a stationary equation for symmetries from the non-commutative subalgebra, namely, for a linear combination of the master-symmetry and the scaling symmetry. The constraint under study is equivalent to a sixth order nonautonomous ODE possessing two first...

**Research Article**

## Periodic Solutions of a System of Complex ODEs. II. Higher Periods

F. Calogero, M. Sommacal

Pages: 483 - 516

In a previous paper the real evolution of the system of ODEs ¨zn + zn = N m=1, m=n gnm(zn - zm) -3 , zn zn(t), zn dzn(t) dt , n = 1, . . . , N is discussed in CN , namely the N dependent variables zn, as well as the N(N - 1) (arbitrary!) "coupling constants" gnm, are considered to be complex numbers,...

**Research Article**

## Remarks on Random Evolutions in Hamiltonian Representation

Boris A. Kupershmidt

Pages: 483 - 495

Abstract telegrapher's equations and some random walks of Poisson type are shown to fit into the framework of the Hamiltonian formalism after an appropriate timedependent rescaling of the basic variables has been made.

**Research Article**

## A unique continuation principle for steady symmetric water waves with vorticity

Mats Ehrnström

Pages: 484 - 491

We consider a symmetric, steady, and periodic water wave. It is shown that a locally vanishing vertical velocity component implies a flat or oscillating surface profile.

**Research Article**

*q*-Oscillator Algebra And *d*-Orthogonal Polynomials

Fethi Bouzeffour, Ali Zagouhani

Pages: 480 - 494

In this paper we express the matrix coefficients of the Fock representation of a q-oscillator algebra in terms of the d-orthogonal Al-Salam Carlitz polynomials. Also, we derive a generating functions, recurrence relations and q-difference equations for these d-orthogonal polynomials.

**Research Article**

## A Method for q-Calculus

Thomas Ernst

Pages: 487 - 525

We present a notation for q-calculus, which leads to a new method for computtions and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomals and functions assume a very pleasant form reminding directly...

**Letter to Editor**

## The Transformation between the AKNS Hierarchy and the KN Hierarchy with Self-Consistent Sources

Qi Li, Wen Zhang, Qiu-Yuan Duan, Deng-Yuan Chen

Pages: 483 - 490

It is shown that the AKNS hierarchy with self-consistent sources can transform to KN hierarchy with self-consistent sources through a transformation operator and gauge transformation. Besides, there exists transformation in their conservation laws and Hamiltonian structures.

**Research Article**

## The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

Boling Guo, Nan Liu

Pages: 483 - 508

We derive the long-time asymptotics for the solution of initial-boundary value problem of coupled nonlinear Schrödinger equation whose Lax pair involves 3 × 3 matrix in present paper. Based on a nonlinear steepest descent analysis of an associated 3 × 3 matrix Riemann–Hilbert problem, we can give the...

**Research Article**

## The Quantization of a Fourth-Order Equation without a Second-Order Lagrangian

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

Pages: 485 - 490

We present an equation of the fourth-order which does not possess a second-order Lagrangian and demonstrate by means of the method of reduction of order that one can obtain a first-order Lagrangian for it. This opens the way to quantization through the construction of an Hamiltonian which is suitable...

**Research Article**

## The number of independent Traces and Supertraces on the Symplectic Reflection Algebra *H*_{1,η}(Γ ≀ *S*_{N})

_{N}

S.E. Konstein, I.V. Tyutin

Pages: 485 - 496

Symplectic reflection algebra H1,η(G) has a T(G)-dimensional space of traces whereas, when considered as a superalgebra with a natural parity, it has an S(G)-dimensional space of supertraces. The values of T(G) and S(G) depend on the symplectic reflection group G and do not depend on the parameter η.
In...

**Research Article**

## New *solvable* dynamical systems

Francesco Calogero

Pages: 486 - 493

New solvable dynamical systems are identified and the properties of their solutions are tersely discussed.

**Research Article**

## Random Groups in the Optical Waveguides Theory

G.B. Malykin, V.I. Pozdnyakova, I.A. Shereshevskii

Pages: 491 - 517

We propose a new approach to the mathematical description of light propagation in a single-mode fiber light-guide (SMFLG) with random inhomogeneities. We investgate statistics of complex amplitudes of the electric field of light wave by methods of the random group theory. We have analyzed the behavior...

**Research Article**

## Wavelet Transforms in Quantum Calculus

Ahmed Fitouhi, Néji Bettaibi

Pages: 492 - 506

This paper aims to study the q-wavelets and the q-wavelet transforms, using only the q-Jackson integrals and the q-cosine Fourier transform, for a fix q ]0, 1[. For this purpose, we shall attempt to extend the classical theory by giving their q-analogues.

**Research Article**

## A Dynamical Mapping Method in Nonrelativistic Models of Quantum Field Theory

A.N. Vall, S.E. Korenblit, V.M. Leviant, A.B. Tanaev

Pages: 492 - 503

Exact solutions of Heisenberg equations and two-particle eigenvalue problems for the nonrelativistic four-fermion interaction and N, model are obtained in the framework of a dynamical mapping method. Equivalence of different dynamical mappings is shown.

**Research Article**

## Quasi-Periodic Solutions of the Relativistic Toda Hierarchy

Dong Gong, Xianguo Geng

Pages: 489 - 523

On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the relativistic Toda hierarchy are straightened out using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions...

**Research Article**

## On Nonlocal Symmetries, Nonlocal Conservation Laws and Nonlocal Transformations of Evolution Equations: Two Linearisable Hierarchies

Norbert Euler, Marianna Euler

Pages: 489 - 504

We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal...

**Research Article**

## Q->infinite limit of the quasitriangular WZW model

Ctirad Klimčík

Pages: 494 - 526

We study the 'q-> infinite' limit of the q-deformation of the WZW model on a compact simple and simply connected target Lie group. We show that the commutation rela- tions of the 'q -> infinite' current algebra are underlied by certain affine Poisson structure on the group of holomorphic maps from the...

**Research Article**

## Jordan Manifolds and Dispersionless KdV Equations

I.A.B. Strachan

Pages: 495 - 510

Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation...

**Research Article**

## Propagation of Spikes in Nonresonant Atomic Media: The Reduced Maxwell-Duffing Model

Utpal Roy, Thokala Soloman Raju, Prasanta K. Panigrahi, Ashutosh Rai

Pages: 491 - 499

We obtain spikes or extremely short pulses for reduced Maxwell-Duffing equations under a general boundary condition, connecting the electric field and electron amplitude, using a Möbius transformation. For a nonzero background, this system is shown to admit two families of exact solutions in the form...

**Research Article**

## Lax Pair, Binary Darboux Transformation and New Grammian Solutions of Nonisospectral Kadomtsev–Petviashvili Equation with the Two-Singular-Manifold Method

Shou-Fu Tian, Hong-Qing Zhang

Pages: 491 - 502

In this letter, the two-singular-manifold method is applied to the (2+1)-dimensional nonisospectral Kadomtsev–Petviashvili equation with two Painlevé expansion branches to determine auto-Bäcklund transformation, Lax pairs and Darboux transformation. Based on the two obtained Lax pairs, the binary Darboux...

**Research Article**

## Group analysis of the generalized Burnett equations

Alexander V. Bobylev, Sergey V. Meleshko

Pages: 494 - 508

In this paper group properties of the so-called Generalized Burnett equations are studied. In contrast to the classical Burnett equations these equations are well-posed and therefore can be used in applications. We consider the one-dimensional version of the generalized Burnett equations for Maxwell...

**Research Article**

## On integrability of the Szekeres system. I

Anna Gierzkiewicz, Zdzisław A. Golda

Pages: 494 - 506

The Szekeres system is a four-dimensional system of ﬁrst-order ordinary differential equations with nonlinear but polynomial (quadratic) right-hand side. It can be derived as a special case of the Einstein equations, related to inhomogeneous and nonsymmetrical evolving spacetime. The paper shows how...

**Research Article**

## On Symmetric Water Waves with Constant Vorticity

Florian Kogelbauer

Pages: 494 - 498

We prove that a solution to the gravity water wave problem with constant vorticity, whose wave profile as well as its horizontal velocity component at the free surface are symmetric at any instant of time, is given by a traveling wave. The proof is based on maximum principles and structural properties...

**Research Article**

## Jacobi's Three-Body System Moves like a Free Particle

M.C. Nucci

Pages: 499 - 506

The problem of three bodies which attract each other with forces proportional to the cube of the inverse of their distance and move on a line was reduced to one quadrature by Jacobi [23]. Here we show that the equations of motions admit a five-dimensional Lie symmetry algebra and can be reduced to a...

**Research Article**

## Inverse Spectral Problem for the Periodic Camassa-Holm Equation

Evgeni Korotyaev

Pages: 499 - 507

We consider the direct/inverse spectral problem for the periodic Camassa-Holm eqution. In fact, we survey the direct/inverse spectral problem for the periodic weighted operator Ly = m-1 (-y +1 4 y) acting in the space L2 (R, m(x)dx), where m = uxx-u > 0 is a 1-periodic positive function and u is the...

**Research Article**

## New Properties of the Zeros of Krall Polynomials

Oksana Bihun

Pages: 495 - 515

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall polynomials. Given an orthogonal polynomial family
{pv(x)}v=0∞
, we relate the zeros of the polynomial...

**Research Article**

## Functional representation of the negative DNLS hierarchy

V.E. Vekslerchik

Pages: 495 - 513

This paper is devoted to the negative flows of the derivative nonlinear Schrödinger hierarchy (DNLSH). The main result of this work is the functional representation of the extended DNLSH, composed of both positive (classical) and negative flows. We derive a finite set of functional equations, constructed...

**Research Article**

## Further Riccati Differential Equations with Elliptic Coefficients and Meromorphic Solutions

Adolfo Guillot

Pages: 497 - 508

We exhibit some families of Riccati differential equations in the complex domain having elliptic coefficients and study the problem of understanding the cases where there are no multivalued solutions. We give criteria ensuring that all the solutions to these equations are meromorphic functions defined...

**Research Article**

## Internal equatorial water waves in the *f*−plane

David Henry

Pages: 499 - 506

In this paper we describe an exact, and explicit, three-dimensional nonlinear solution for geophysical internal ocean waves in the Equatorial region which incorporates a transverse-Equatorial meridional current.

**Research Article**

## Exact Solutions of a Quark Plasma Equilibrium in the Abelian Dominance Approximation

Yu.A. Markov, M.A. Markova

Pages: 504 - 515

Stationary kinetic equations for a quark plasma (QP) in the Abelian dominance approximation are reduced to the nonlinear system of A2-periodic Toda chains (with elliptic operator). Using solutions of this system, which are found with the help of the first integrals and Hirota's method, such characteristics...

**Research Article**

## New Integrable Multicomponent Nonlinear Partial Differential-Difference Equations

R. Sahadevan, S. Balakrishnan

Pages: 501 - 518

We report a new three and four coupled nonlinear partial differential-difference equations each admits Lax representation, possess infinitely many generalized (nonpoint) symmetries, conserved quantities and a recursion operator. Hence they are completely integrable both in the sense of Lax and Liouville.

**Research Article**

## Shape Invariant Potentials in "Discrete Quantum Mechanics"

Satoru Odake, Ryu Sasaki

Pages: 507 - 521

Shape invariance is an important ingredient of many exactly solvable quantum mchanics. Several examples of shape invariant "discrete quantum mechanical systems" are introduced and discussed in some detail. They arise in the problem of descriing the equilibrium positions of Ruijsenaars-Schneider type...

**Research Article**

## A geometric interpretation of the spectral parameter for surfaces of constant mean curvature

J.L. Cieśliński

Pages: 507 - 515

Considering the kinematics of the moving frame associated with a constant mean cuvature surface immersed in S3 we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S3 . The application of...

**Research Article**

## Equations Of Long Waves With A Free Surface III. The Multidimensional Case

Boris A. Kupershmidt

Pages: 508 - 517

Long-wave equations for an incompressible inviscid free-surface fluid in N + 1 dimesions are derived and shown to be Hamiltonian and liftable into the space of moments.

**Research Article**

## Traveling Wave Solutions of the Camassa-Holm and Korteweg-de Vries Equations

Jonatan Lenells

Pages: 508 - 520

We show that the smooth traveling waves of the Camassa-Holm equation naturally correspond to traveling waves of the Korteweg-de Vries equation.

**Research Article**

## Nonlinear Stability Analysis of the Emden–Fowler Equation

C. G. Böhmer, T. Harko

Pages: 503 - 516

In this paper, we qualitatively study radial solutions of the semilinear elliptic equation Δu+un = 0 with u(0) = 1 and u′(0) = 0 on the positive real line, called the Emden–Fowler or Lane–Emden equation. This equation is of great importance in Newtonian astrophysics and the constant n is called the polytropic...

**Research Article**

## Global Analytic First Integrals for the Simplified Multistrain/Two-Stream Model for Tuberculosis and Dengue Fever

Jaume Llibre, Clàudia Valls

Pages: 505 - 516

We provide the complete classification of all global analytic first integrals of the simplified multistrain/two-stream model for tuberculosis and dengue fever that can be written as
x˙=x(β1−b−γ1−β1x−(β1−ν)y), y˙=y(β2−b−γ2−(β2−ν)x−β2y),
with β1, β2, b, γ1, γ2, ν ∈ ℝ.

**Research Article**

## Darboux First Integral Conditions and Integrability of the 3D Lotka-Volterra System

Laurent Cairó

Pages: 511 - 531

We apply the Darboux theory of integrability to polynomial ODE's of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3dimensional LotkaVolterra systems with polynomial invariant algebraic solutions linear and quadratic and determine numerous cases...

**Research Article**

## Algebraic entropy, symmetries and linearization of quad equations consistent on the cube

G. Gubbiotti, C. Scimiterna, D. Levi

Pages: 507 - 543

We discuss the non–autonomous nonlinear partial difference equations belonging to Boll classiﬁcation of quad graph equations consistent around the cube. We show how starting from the compatible equations on a cell we can construct the lattice equations, its Bäcklund transformations and Lax pairs. By...

**Research Article**

## Particle paths in Stokes’ edge wave

Raphael Stuhlmeier

Pages: 507 - 515

We treat the particle motion in Stokes’ linear edge wave along a uniformly sloping beach. By a rotation of the coordinate frame, we show that there is no particle motion in the direction orthogonal to the sloping beach, and conclude that particles have a longshore drift in the direction of wave propagation...

**Research Article**

## Global dynamics of a Lotka–Volterra system in ℝ^{3}

Jaume Llibre, Y. Paulina Martínez, Claudia Valls

Pages: 509 - 519

In this work we consider the Lotka–Volterra system in ℝ3
x˙=−x(x−y−z), y˙=−y(−x+y−z), z˙=−z(−x−y+z),
introduced recently in [7], and studied also in [8] and [14]. In the first two papers the authors mainly studied the integrability of this differential system, while in the third paper they...

**Research Article**

## Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

Francesco Calogero, Farrin Payandeh

Pages: 509 - 519

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,
x˜n=F(n)(x1,x2), n=1,2,
which are algebraically solvable. Here ℓ is the “discrete-time” independent variable taking integer values (ℓ = 0, 1, 2,...), xn ≡ xn(ℓ) are 2 dependent variables, and x˜n≡xn(ℓ+1) are...

**Letter to Editor**

## On a surface isolated by Gambier

Runliang Lin, Robert Conte

Pages: 509 - 514

We provide a Lax pair for the surfaces of Voss and Guichard, and we show that such particular surfaces considered by Gambier are characterized by a third Painlevé function.

**Research Article**

## Bi-Hamiltonian structure of multi-component Novikov equation

Hongmin Li, Yuqi Li, Yong Chen

Pages: 509 - 520

In this paper, we present the multi-component Novikov equation and derive it's bi-Hamiltonian structure.

**Research Article**

## On Nonlinear Differential Equations That Describe Localized Processes

Vladimir P. Burskii

Pages: 516 - 522

In this paper we want to characterize nonlinear differential equations that describe processes allowing a localization operation in each subdomain of domain in which we consider the process. We formulate this localization condition by means of visual reresentations and give this operation a mathematical...

**Research Article**

## Representations of the Q-deformed Euclidean Algebra Uq(iso3) and Spectra of their Operators

I.I. Kachurik

Pages: 516 - 524

Representations of the q-deformed Euclidean algebra Uq(iso3), which at q 1 gives the universal enveloping algebra U(iso3) of the Lie algebra iso3 of the Euclidean Lie group ISO(3), are studied. Explicit formulas for operators of irreducible -representations defined by two parameters R and s 1 2 Z are...

**Research Article**

## Tau Functions Associated to Pseudodifferential Operators of Several Variables

Min Ho Lee

Pages: 517 - 529

Pseudodifferential operators of several variables are formal Laurent series in the formal inverses of 1, . . . , n with i = d/dxi for 1 i n. As in the single variable case, Lax equations can be constructed using such pseudodifferential operators, whose solutions can be provided by Baker functions. We...

**Research Article**

## The Asymptotic Behavior of the Solution of Boundary Value Problems for the sine-Gordon Equation on a Finite Interval

Beatrice Pelloni

Pages: 518 - 529

In this article we use thve Fokas transform method to analyze boundary value prolems for the sine-Gordon equation posed on a finite interval. The representation of the solution of this problem has already been derived using this transform method. We interchange the role of the independent variables to...

**Research Article**

## On Anomalies in Classical Dynamical Systems

Francesco Toppan

Pages: 518 - 533

The definition of "classical anomaly" is introduced. It describes the situation in which a purely classical dynamical system which presents both a lagrangian and a hamitonian formulation admits symmetries of the action for which the Noether conserved charges, endorsed with the Poisson bracket structure,...

**Research Article**

## Darboux transformation and novel solutions for the long wave-short wave model

Xin Huang, Boling Guo, Liming Ling

Pages: 514 - 528

Firstly, we establish the relation between the loop group method and gauge transformation method for 3×3 spectral problem. Some novel solutions of long wave-short wave model are obtained by Darboux transformation method. Besides, we give the analysis and classification of solution in detail.

**Letter to Editor**

## Solvable nonlinear discrete-time evolutions and Diophantine findings

Francesco Calogero

Pages: 515 - 517

Certain nonlinearly-coupled systems of N discrete-time evolution equations are identified, which can be solved by algebraic operations; and some remarkable Diophantine findings are thereby obtained. These results might be useful to test the accuracy of numerical routines yielding the N roots of polynomials...

**Review Article**

## On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey

Luc Molinet

Pages: 521 - 533

We survey recent results on well-posedness, blow-up phenomena, lifespan and global existence for the Camassa-Holm equation. Results on weak solutions are also consiered.

**Research Article**

## Lie symmetry analysis, conservation laws and analytical solutions of a time-fractional generalized KdV-type equation*

Xiu-Bin Wang, Shou-Fu Tian, Chun-Yan Qin, Tian-Tian Zhang

Pages: 516 - 530

Under investigation in this work is the time-fractional generalized KdV-type equation, which occurs in different contexts in mathematical physics. Lie group analysis method is presented to explicitly study its vector fields and symmetry reductions. Furthermore, two straightforward methods are employed...

**Research Article**

## Dynamics of the thermocline in the equatorial region of the Pacific ocean

Calin Iulian Martin

Pages: 516 - 522

This paper is devoted to the subsurface current dynamics in equatorial regions, where the hallmark of a strong stratification is a sharp interface (thermocline), separating two layers of different density, and whose depth is dependent upon the strength of the winds above the ocean's surface. We...

**Research Article**

## Universality of Calogero-Moser Model

M.A. Olshanetsky

Pages: 522 - 534

In this review we explain interrelations between the Elliptic Calogero-Moser model, the integrable Elliptic Euler-Arnold top, monodromy preserving equations and the Knizhnik-Zamolodchikov-Bernard equation on a torus.

**Research Article**

## Generalized, Master and Nonlocal Symmetries of Certain Deformed Nonlinear Partial Differential Equations

R. Sahadevan, L. Nalinidevi

Pages: 517 - 538

It is shown that the deformed Nonlinear Schrödinger (NLS), Hirota and AKNS equations with (1 + 1) dimension admit infinitely many generalized (nonpoint) symmetries and polynomial conserved quantities, master symmetries and recursion operator ensuring their complete integrability. Also shown that each...

**Research Article**

## On the Cohomology of the Lie Superalgebra of Contact Vector Fields on S1|2

B. Agrebaoui, N. Ben Fraj, S. Omri

Pages: 523 - 534

We investigate the first cohomology space associated with the embedding of the Lie superalgebra K(2) of contact vector fields on the (1,2)-dimensional supercircle S1|2 in the Lie superalgebra SDO(S1|2 ) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we show...

**Research Article**

## Free-field realizations of the *W*_{𝒜n,N}-algebra

Yanyan Ge, Kelei Tian, Xiaoming Zhu, Dafeng Zuo

Pages: 518 - 527

In this paper, we will construct free-field realizations of the
W𝒜n,N
algebra associated to an
𝒜n
-valued differential operator
=In∂N+UN−1∂N−1+UN−2∂N−2+⋯U0,
where
𝒜n
is a Frobenius algebra with the unit In.

**Research Article**

## Generalization of an Invertible Transformation and Examples of its Applications

M. Bruschi, F. Calogero, F. Leyvraz, M. Sommacal

Pages: 519 - 540

Recently we highlighted the remarkable nature of an explicitly invertible transformation, we reported some generalizations of it and examples of its expediency in several mathematical contexts: algebraic and Diophantine equations, dynamical systems (with continuous and discrete time), nonlinear PDEs,...

**Research Article**

## Constructing discrete Painlevé equations: from E8(1) to A1(1) and back

A. Ramani, B. Grammaticos, R. Willox, T. Tamizhmani

Pages: 520 - 535

The ‘restoration method’ is a novel method we recently introduced for systematically deriving discrete Painlevé equations. In this method we start from a given Painlevé equation, typically with E8(1) symmetry, obtain its autonomous limit and construct all possible QRT-canonical forms of mappings that...

**Research Article**

## Soliton Solutions of the N = 2 Supersymmetric KP Equation

Sasanka Ghosh, Debojit Sarma

Pages: 526 - 538

The N = 2 super-KP equation associated with nonstandard flows is bilinearized using the Hirota method and soliton solutions are obtained. The bilinearization has been done for component fields and its KdV limit is discussed by comparing the soliton solutions obtained by this procedure with those found...

**Letter to Editor**

## On the hierarchies of the fully nonlinear Möbius-invariant and symmetry-integrable evolution equations of order three

Marianna Euler, Norbert Euler

Pages: 521 - 528

This is a follow-up paper to the results published in Studies in Applied Mathematics 143, 139–156 (2019), where we reported a classification of 3rd- and 5th-order semi-linear symmetry-integrable evolution equations that are invariant under the Möbius transformation, which includes a list of fully nonlinear...

**Research Article**

## Integration of some examples of geodesic flows via solvable structures

Diego Catalano Ferraioli, Paola Morando

Pages: 521 - 532

Solvable structures are particularly useful in the integration by quadratures of ordinary differential equations. Nevertheless, for a given equation, it is not always possible to compute a solvable structure. In practice, the simplest solvable structures are those adapted to an already known system of...

**Research Article**

## New solutions of a higher order wave equation of the KdV type

Vangelis Marinakis

Pages: 527 - 533

In this paper we use the Painlev Ìe analysis and study a special case of a water wave equation of the KdV type. More specifically, we use the Pickering algorithm [9] and obtain a new kind of solutions, which constitute of both algebraic and trigonometric (or hyperbolic) functions.

**Research Article**

## On Pollard's wave solution at the Equator

Delia Ionescu-Kruse

Pages: 523 - 530

In this paper we present a dynamical study of the exact nonlinear Pollard wave solution to the geophysical water-wave problem in the f-plane approximation. We deduce an exact dispersion relation and we discuss some properties of this solution.

**Research Article**

## Exact Solutions to Lattice Boussinesq-Type Equations

Wei Feng, Song-Lin Zhao, Da-Jun Zhang

Pages: 524 - 538

In this paper several kinds of exact solutions to lattice Boussinesq-type equations are constructed by means of generalized Cauchy matrix approach, including soliton solutions and mixed solutions. The introduction of the general condition equation set yields that all solutions contain two kinds of plane-wave...

**Research Article**

## Laplacians on Lattices

Wojtek J. Zakrzewski

Pages: 530 - 538

We consider some lattices and look at discrete Laplacians on these lattices. In partiular we look at solutions of the equation (1) = (2)Z, where (1) and (2) denote two such Laplacians on the same lattice. We show that, in one dimension, when (i), i = 1, 2, denote (1) = (i + 1) + (i - 1) - 2(i) and (2)Z...

**Research Article**

## On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices

J. Harnad

Pages: 530 - 550

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space...