Journal of Nonlinear Mathematical Physics

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1493 articles
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.
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...
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

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

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

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 zn(t) of a time-dependent polynomial pN(z;t) of arbitrary degree N in z, and solvable dynamical systems

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...


Adrian Constantin
Pages: 474 - 474
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

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

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...
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 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...
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 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

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

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

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

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

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

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 H1,η(Γ ≀ SN)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.
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

Global dynamics of a Lotka–Volterra system in ℝ3

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...
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

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

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 W𝒜n,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

Shulin Lyu, James Griffin, Yang Chen
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...
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...
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

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

Vangelis Marinakis
Pages: 527 - 533
In this paper we use the Painlev ́e 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

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...
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

Composition of Lie Group Elements from Basis Lie Algebra Elements

George W. Bluman, Omar Mrani-Zentar, Deshin Finlay
Pages: 528 - 557
It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number...
Research Article

Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body

Alberto Molgado, Adan Rodríguez
Pages: 534 - 547
In this work we investigate a formal mapping between the dynamical properties of the unidimensional relativistic oscillator and the asymmetrical rigid top at a clas- sical level. We study the relativistic oscillator within Yamaleev’s interpretation of Nambu mechanics. Such interpretation is based on...
Research Article

Lp - Lq Decay Estimates for Wave Equations with Time-Dependent Coefficients

Michael Reissig
Pages: 534 - 548
The goal of this survey article is to explain the up-to-date state of the theory of Lp - Lq decay estimates for wave equations with time-dependent coefficients. We explain the influence of oscillations in the coefficients by using a precise classification. Moreover, we will see how mass and dissipation...
Research Article

Integrable Discretizations of Some Cases of the Rigid Body Dynamics

Yuri B. Suris
Pages: 534 - 560
A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra e(n) = so(n) Rn . We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable...
Research Article

The Virasoro action on the tau function for the constrained discrete KP hierarchy

Jipeng Cheng, Maohua Li, Jingsong He
Pages: 529 - 538
With the help of the squared eigenfunction potential, the action of the Virasoro symmetry on the tau function of the constrained discrete KP hierarchy is derived.
Research Article

Minimal surfaces associated with orthogonal polynomials

Vincent Chalifour, Alfred Michel Grundland
Pages: 529 - 549
This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces approach defined by the Enneper-Weierstrass formula for immersion and...