Journal of Nonlinear Mathematical Physics

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905 articles

A Remark on Rational Isochronous Potentials

Oleg A CHALYKH, Alexander P VESELOV
Pages: 179 - 183
We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form U(x) = 1 2 2 x2 or U(x) = 1 8 2 x2 + c2 x-2 .

Reductions of Integrable Lattices

B GRAMMATICOS, A RAMANI, J SATSUMA, R WILLOX
Pages: 363 - 371
We present a novel method for the reduction of integrable two-dimensional discrete systems to one-dimensional mappings. The procedure allows for the derivation of nonautonomous systems, which are typically discrete (difference or q) Painlevé equtions, or of autonomous ones. In the latter case we produce...

On a "Quasi" Integrable Discrete Eckhaus Equation

M J ABLOWITZ, C D AHRENS
Pages: 1 - 12
In this paper, a discrete version of the Eckhaus equation is introduced. The discretiztion is obtained by considering a discrete analog of the transformation taking the cotinuous Eckhaus equation to the continuous linear, free Schrödinger equation. The resulting discrete Eckhaus equation is a nonlinear...

Goat cheese for breakfast in Istanbul or Why are certain nonlinear PDEs both widely applicable and integrable? Reminiscences of Francesco Calogero

Robin BULLOUGH
Pages: 124 - 137
It is shown how in the early days of soliton theory 1976-the early 1980's Francesco Calogero maintained a considerable influence on the field and on the work of the athor Robin Bullough in particular. A vehicle to this end was the essentially annual sequence of international conferences Francesco organised...

On the Equilibrium Configuration of the BC-type Ruijsenaars-Schneider System

Jan F VAN DIEJEN
Pages: 689 - 696
It is shown that the ground-state equilibrium configurations of the trigonometric Btype Ruijsenaars-Schneider systems are given by the zeros of Askey-Wilson polynomals.

Functional Equations and the Generalised Elliptic Genus

H W BRADEN, K E FELDMAN
Pages: 74 - 85
We give a new derivation and characterisation of the generalised elliptic genus of Krichever-Höhn by means of a functional equation.

A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)

Darryl D HOLM, Andrew N W HONE
Pages: 380 - 394
We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x). When b = 2 and g is the peakon kernel (i.e. g(x) = exp(-|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained...

Link Invariants and Lie Superalgebras

P GROZMAN, D LEITES
Pages: 372 - 379
Berger and Stassen reviewed skein relations for link invariants coming from the simple Lie algebras g. They related the invariants with decomposition of the tensor square of the g-module V of minimal dimension into irreducible components. (If V V , one should also consider the decompositions of V V and...

Solitons of Wave Equation

Antoni SYM
Pages: 648 - 659
Modulated progressive wave solutions (solitons) to (3 + 1)­dimensional wave equation are discussed within a general geometrical framework. The role of geodesic coordinates defined by hypersurfaces of Riemannian spaces is pointed out in this context. In particular in E3 orthogonal geodesic coordinates...

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

Decio LEVI, Rafael HERNANDEZ HEREDERO
Pages: 440 - 448
In this paper we consider multiple lattices and functions defined on them. We itroduce some slow varying conditions and define a multiscale analysis on the lattice, i.e. a way to express the variation of a function in one lattice in terms of an asymtotic expansion with respect to the other. We apply...

Singular Manifold Method for an Equation in 2 + 1 Dimensions

P G ESTEVEZ, J PRADA
Pages: 266 - 279
The Singular Manifold Method is presented as an excellent tool to study a 2 + 1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1 + 1 reductions of the same equation. Nevertheless these problems are solved when the number of dimensions of the...

New Solvable Nonlinear Matrix Evolution Equations

M BRUSCHI
Pages: 97 - 105
We introduce an extension of the factorization-decomposition technique that allows us to manufacture new solvable nonlinear matrix evolution equations. Several examples of such equations are reported.

Some Water Wave Equations and Integrability

Yishen LI
Pages: 466 - 481
A theory of bidirectional solitons on water is developed by using the classical Boussnesq equation. Moreover, analytical multi-solitons of Camassa-Holm equation are presented.

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

J L CIESLINSKI
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...

G2-Calogero-Moser Lax operators from reduction

Andreas FRING, Nenad MANOJLOVIC
Pages: 467 - 478
We construct a Lax operator for the G2-Calogero-Moser model by means of a double reduction procedure. In the first reduction step we reduce the A6-model to a Bmodel with the help of an embedding of the B3-root system into the A6-root system together with the specification of certain coupling constants....

Vect(S1) Action on Pseudodifferential Symbols on S1 and (Noncommutative) Hydrodynamic Type Systems

Partha GUHA
Pages: 549 - 565
The standard embedding of the Lie algebra V ect(S1 ) of smooth vector fields on the circle V ect(S1 ) into the Lie algebra D(S1 ) of pseudodifferential symbols on S1 identifies vector field f(x) x V ect(S1 ) and its dual as (f(x) x ) = f(x) (u(x)dx2 ) = u(x)-2 . The space of symbols can be viewed as...

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.

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.

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...

Analytic Behaviour of Competition among Three Species

PGL LEACH, J MIRITZIS
Pages: 535 - 548
We analyse the classical model of competition between three species studied by May and Leonard (SIAM J Appl Math 29 (1975) 243-256) with the approaches of singlarity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations...

Provenance of Type II hidden symmetries from nonlinear partial differential equations

Barbara ABRAHAM-SHRAUNER, Keshlan S GOVINDER
Pages: 612 - 622
The provenance of Type II hidden point symmetries of differential equations reduced from nonlinear partial differential equations is analyzed. The hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent...

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.

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...

Distribution of positive type in Quantum Calculus

Akram NEMRI
Pages: 566 - 583
In this paper, we study some remarkable spaces of Sq,(Rq,+) space of the q-tempered distribution introduced by M.A. Olshanetsky and V.B.K. Rogov [14], namely the q-analogue of the pseudo-measure FqL (Rq,+), the q-function of the positive type FqM , and we give a q-version of the Bochner-Shwartz theorem...

The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations

Kouichi TAKEMURA
Pages: 584 - 611
We obtain isomonodromic transformations for Heun's equation by generalizing the Darboux transformation, and we find pairs and triplets of Heun's equation which have the same monodromy structure. By composing generalized Darboux transformations, we establish a new construction of the commuting operator...

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....

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...

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...

Periodic Traveling Water Waves with Isobaric Streamlines

Henrik KALISCH
Pages: 461 - 471
It is shown that in water of finite depth, there are no periodic traveling waves with the property that the pressure in the underlying fluid flow is constant along streamlines. In the case of infinite depth, there is only one such solution, which is due to Gerstner.

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.

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.

Uniqueness for Autonomous Planar Differential Equations and the Lagrangian Formulation of Water Flows with Vorticity

Erik WAHLÉN
Pages: 549 - 555
We prove a uniqueness result for autonomous divergence-free systems of ODE's in the plane and give an application to the study of water flows with vorticity.

On the Spectral Problem Associated with the Camassa-Holm Equation

Christer BENNEWITZ
Pages: 422 - 432
We give a basic uniqueness theorem in the inverse spectral theory for a Sturm-Liouville equation with a weight which is not of one sign. It is shown that the theorem may be applied to the spectral problem associated with the Camassa-Holm integrable system which models shallow water waves.

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...

Deformation quantization for almost-Kähler manifolds

Martin SCHLICHENMAIER
Pages: 49 - 54
On an arbitrary almost-Kähler manifold, starting from a natural affine connection with nontrivial torsion which respects the almost-Kähler structure, in joint work with A. Karabegov a Fedosov-type deformation quantization on this manifold was costructed. This contribution reports on the result and supplies...

On Pauli graded contractions of sl(3, C)

Miloslav HAVLICEK, Jiri PATERA, Edita PELANTOVA, Jiri TOLAR
Pages: 37 - 42
We consider a special fine grading of sl(3, C), where the grading subspaces are geerated by 3 × 3 generalized Pauli matrices. This fine grading decomposes sl(3, C) into eight one­dimensional subspaces. Our aim is to find all contractions of sl(3, C) which preserve this grading. We have found that the...

So. . . what was the question?

Gérard G EMCH
Pages: 1 - 8
An overview of the lectures at the 2002 Bialowiea Workshop is presented. The symbol* after a proper name indicates that a copy of the corresponding contribution to the proceedings was communicated to the author of this summary.

Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

Harald GROSSE, Raimar WULKENHAAR
Pages: 9 - 20
We first review regularization methods based on matrix geometry which provide an ultraviolet cut-off for scalar fields respecting the symmetries. Sections of bundles over the sphere can be quantized, too. This procedure even allows to regularize supesymmetry without violating it. Recently, this work...

The Classification of the Bifurcations Emerging in the case of an Integrable Hamiltonian System with Two Degrees of Freedom when an Isoenergetic Surface is Non-Compact

Galina GOUJVINA
Pages: 122 - 129
On a symplectical manifold M4 consider a Hamiltonian system with two degrees of freedom, integrable with the help of an additional integral f. According to the welknown Liouville theorem, non-singular level surfaces of the integrals H and f can be represented as unions of tori, cylinders and planes....

Quadratic non-Riemannian Gravity

Dmitri VASSILIEV
Pages: 204 - 216
We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature...

Statistical mechanics of a Class of Anyonic Systems. The Rigorous Approach

Roman GIELERAK, Robert RALOWSKI
Pages: 85 - 91
A class of involutive Wick algebras (called anyonic-type Wick algebras) is selected and some its elementary properties are described. In particular, the Fock representtions of the selected anyonic-type commutation relations are described. For the class of so-called r-yonic systems the question of the...

Geodetic Systems on Linear and Affine Groups. Classics and Quantization.

Jan J. SLAWIANOWSKI
Pages: 130 - 137
Described are classical and quantized systems on linear and affine groups. Unlike the traditional models applied in astrophysics, nuclear physics, molecular vibrations and elasticity, our models are not only kinematically ruled by the affine group, but also their kinetic energies are affinely invariant....

On the complexified affine metaplectic representation

Ole RASK
Pages: 191 - 193
We study exponentiability of the infinite polynomials with maximal degree 2 of cration and annihilation operators, which give a Fock Space-representation of the coplexification of the affine symplectic group.

Wigner Quantization on the Circle and R+

G. CHADZITASKOS, J. TOLAR
Pages: 174 - 178
We construct a deformation quantization for two cases of configuration spaces: the multiplicative group of positive real numbers R+ and the circle S1 . In these cases we define the momenta using the Fourier transform. Using the identification of symbols of quantum observables -- real functions on the...

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...

On Poisson Realizations of Transitive Lie Algebroids

Yuri VOROBIEV
Pages: 43 - 48
We show that every transitive Lie algebroid over a connected symplectic manifold comes from an intrinsic Lie algebroid of a symplectic leaf of a certain Poisson structure. The reconstruction of the corresponding Poisson structures from the Lie algebroid is given in terms of coupling tensors.

New Geometrical Applications of the Elliptic Integrals: The Mylar Balloon

Ivaïlo M. MLADENOV
Pages: 55 - 65
An explicit parameterization in terms of elliptic integrals (functions) for the Mylar balloon is found which then is used to calculate various geometric quantities as well as to study all kinds of geodesics on this surface.

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.