902 articles

Viktor ABRAMOV

Pages: 1 - 8

Given an associative unital ZN -graded algebra over the complex numbers we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-differential d of the graded q-differential algebra is a homogeneous endomorphism of degree 1 satisfying...

Mats EHRNSTROM

Pages: 1 - 8

We consider a nontrivial symmetric periodic gravity wave on a current with nondcreasing vorticity. It is shown that if the surface profile is monotone between trough and crest, it is in fact strictly monotone. The result is valid for both finite and infinite depth.

Sergey I AGAFONOV

Pages: 1 - 14

It is shown that discrete analogs of zc and log(z), defined via particular "integrable" circle patterns, have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painlevé-II equations, asymptotics of these solutions providing...

V N GREBENEV, M OBERLACK

Pages: 1 - 9

The article is devoted to studying the Millionshtchikov closure model (a particular case of a model introduced by Oberlack [14]) for isotropic turbulence dynamics which appears in the context of the theory of the von K´arm´an-Howarth equation. We write the model in an abstract form that enables us to...

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...

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.

Francesco CALOGERO

Pages: 1 - 6

In two previous papers the quantization was discussed of three one-degree-of-freedom Hamiltonians featuring a constant c, the value of which does not influence at all the corresponding classical dynamics (which is characterized by isochronous solutions, all of them periodic with period 2: "nonlinear...

Claude BREZINSKI

Pages: 1 - 12

In this paper, we compare the degrees and the orders of approximation of vector and matrix Padé approximants for series with matrix coefficients. It is shown that, in this respect, vector Padé approximants have better properties. Then, matrixvector Padé approximants are defined and constructed. Finally,...

Herbert AMANN

Pages: 1 - 11

We discuss the solvability of the time-dependent incompressible NavierStokes equtions with nonhomogeneous Dirichlet data in spaces of low regularity.

Jorge E SOLOMIN, Marcela ZUCCALLI

Pages: 1 - 9

A two cocycle is associated to any action of a Lie group on a symplectic manifold. This allows to enlarge the concept of anomaly in classical dynamical systems considered by F Toppan in [J. Nonlinear Math. Phys. 8, Nr. 3 (2001), 518533] so as to encompass some extensions of Lie algebras related to noncanonical...

Barbara ABRAHAM-SHRAUNER

Pages: 1 - 9

The reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples....

Maciej BLASZAK

Pages: 1 - 13

Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordnates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.

Marek SZYDLOWSKI, Marek BIESIADA

Pages: 1 - 10

Qualitative approach to homogeneous anisotropic Bianchi class A models in terms of dynamical systems reveals a hierarchy of invariant manifolds. By calculating the Kovalevski Exponents according to Adler - van Moerbecke method we discuss how algebraic integrability property is distributed in this class...

S ABENDA, Yu FEDOROV

Pages: 1 - 4

We propose Dirac formalism for constraint Hamiltonian systems as an useful tool for the algebro-geometrical and dynamical characterizations of a class of integrable systems, the so called hyperelliptically separable systems. As a model example, we apply it to the classical geodesic flow on an ellipsoid.

A N LEZNOV

Pages: 1 - 7

The 2n dimensional manifold with two mutually commutative operators of differetiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.

Song-Ju YU, Kouichi TODA

Pages: 1 - 13

We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2 + 1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional...

R. MARTINI, P.K.H. GRAGERT

Pages: 1 - 4

We present a complete proof that solutions of the WDVV equations in Seiberg-Witten theory may be constructed from root systems. A generalization to weight systems is proposed.

Dolan Chapa SEN, A. Roy CHOWDHURY

Pages: 1 - 7

Quantization of BKP type equations are done through the Moyal bracket and the formalism of pseudo-differential operators. It is shown that a variant of the dressing operator can also be constructed for such quantized systems.

J.M. CERVER´O, O. Zurr'on

Pages: 1 - 23

As an example of how to deal with nonintegrable systems, the nonlinear partial differential equation which describes the evolution of long surface waves in a convecting fluid ut + (uxxx + 6uux) + 5uux + (uxxx + 6uux)x = 0, is fully analyzed, including symmetries (nonclassical and contact transformatons),...

David HENRY

Pages: 1 - 7

We show that within the framework of linear theory the particle paths in a periodic gravity-capillary or pure capillary deep-water wave are not closed.

S. Ianus, R. Mazzocco, G.E. Vilcu

Pages: 1 - 8

Rossen I Ivanov

Pages: 1 - 12

The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H 1 and H 1 right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a...

George Bluman

Pages: 1 - 24

Similarity methods include the calculation and use of symmetries and conservation laws for a given partial differential equation (PDE). There exists a variety of software to calculate and use local symmetries and local conservation laws. However, it is often the case that a given PDE admits no useful...

V Aldaya, M Calixto, J Guerrero, F F Lopez-Ruiz

Pages: 1 - 12

We study the dynamics of a particle moving on the SU(2) group manifold. An exact quantization of this system is accomplished by finding the unitary and irreducible representations of a finite-dimensional Lie subalgebra of the whole Poisson algebra in phase space. In fact, the basic position and momentum...

George SVETLICHNY

Pages: 2 - 26

We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to...

N V ALEXEEVA, I V BARASHENKOV, G P TSIRONIS

Pages: 5 - 12

Solitons of the parametrically driven, damped nonlinear Schrödinger equation become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the chaos can be suppressed by introducing localized inhomogeneities in the parameters of the equation. The pinning of the...

Zhimin JIANG

Pages: 5 - 12

An eigenvalue problem with a reference function and the corresponding hierarchy of nonlinear evolution equations are proposed. The bi-Hamiltonian structure of the hierarchy is established by using the trace identity. The isospectral problem is nonlinearized as to be finite-dimensional completely integrable...

W. SARLET, F. CANTRIJN, E. MARTÍNEZ

Pages: 5 - 24

Gerald A. GOLDIN

Pages: 6 - 11

An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equations introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled...

Mohamed KACHKACHI

Pages: 7 - 12

At the quantum level of a bidimensional conformal model, the conformal symmtry is broken by the diffeomorphism anomaly and the conformal covariance is not maintained. Here we interpret geometrically this conformal covariance as an exact holomorphy condition on a two-dimensional Riemann surface on which...

Johan BYSTRÖM

Pages: 8 - 30

In this paper we study homogenization of quasi-linear partial differential equations of the form -div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. Here the sequence (h) tends to 0 as h and the map a (x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions....

M.L. GANDARIAS, M.S. BRUZON

Pages: 8 - 12

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the...

Ismail Naci CANGÜL, Veli KURT, Yilmaz SIMSEK, Hong Kyung PAK, Seog-Hoon RIM

Pages: 8 - 14

The purpose of this paper is to consider q-Euler numbers and polynomials which are q-extensions of ordinary Euler numbers and polynomials by the computations of the p-adic q-integrals due to T. Kim, cf. [1, 3, 6, 12], and to derive the "complete sums for q-Euler polynomials" which are evaluated by using...

V ABRAMOV, O LIIVAPUU

Pages: 9 - 20

We propose a geometric approach to the BRST-symmetries of the Lagrangian of a topological quantum field theory on a four dimensional manifold based on the formalism of superconnections. Making use of a graded q-differential algebra, where q is a primitive N-th root of unity, we also propose a notion...

A S HEGAZI, M MANSOUR

Pages: 9 - 18

In this paper, we define a new q-analogy of the Bernoulli polynomials and the Bernoulli numbers and we deduced some important relations of them. Also, we dduced a q-analogy of the Euler-Maclaurin formulas. Finally, we present a relation between the q-gamma function and the q-Bernoulli polynomials.

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...

PGL LEACH, GP FLESSAS

Pages: 9 - 21

From time to time one finds claims in the literature that first integrals/invariants of Lagrangian systems are nonnoetherian. Such claims diminish the contribution of Noether in the topic of integrability. We provide an explicit demonstration of noethe- rian symmetries associated with the integrals which...

Octavian G MUSTAFA

Pages: 10 - 14

We prove that smooth solutions of the Degasperis-Procesi equation have infinite proagation speed: they loose instantly the property of having compact support.

Zhaoyang YIN

Pages: 10 - 15

We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite time. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate...

K ANDRIOPOULOS, P G L LEACH

Pages: 10 - 23

The complete symmetry groups of systems of linear second order ordinary differential equations are considered in the context of the simple harmonic oscillator. One finds that in general the representation of the complete symmetry group is not unique and in the particular case of a four-dimensional system...

Attilio MACCARI

Pages: 11 - 20

A new integrable class of DaveyStewartson type systems of nonlinear partial diffrential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev Petviashvili equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling....

Nicolas BURQ, Patrick GÉRARD, Nikolay TZVETKOV

Pages: 12 - 27

We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schrödinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrödinger group are used to get global existence in the energy space on arbirary surfaces and three-dimensional manifolds, generalizing...

Tepper L. GILL, James LINDESAY, M.F. MAHMOOD, W.W. ZACHARY

Pages: 12 - 27

We report on a new formulation of classical relativistic Hamiltonian mechanics which is based on a proper-time implementation of special relativity using a transformation from observer proper-time, which is not invariant, to system proper-time which is a canonical contact transformation on extended phase-space....

Stephen C ANCO, Thomas WOLF

Pages: 13 - 31

Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several O(N)-invariant classes of hyperbolic equations Utx = f(U, Ut, Ux) for an N-component vector U(t, x) are considered. In each class we find all scalinhomogeneous equations admitting a higher symmetry of least...

C WAFO SOH, F M MAHOMED

Pages: 13 - 20

The classical reduction of order for scalar ordinary differential equations (ODEs) fails for a system of ODEs. We prove a constructive result for the reduction of order for a system of ODEs that admits a solvable Lie algebra of point symmetries. Applications are given for the case of a system of two...

Peter A CLARKSON, Andrew N W HONE, Nalini JOSHI

Pages: 13 - 26

In this paper we present a method for deriving infinite sequences of difference equations containing well known discrete Painlevé equations by using the Bäcklund transformtions for the equations in the second Painlevé equation hierarchy.

Angel BALLESTEROS, Sergey CHUMAKOV

Pages: 13 - 17

The suq(2) algebra is shown to provide a natural dynamical algebra for some nonlnear models in Quantum Optics. Applications to the computation of eigenvalues and eigenvectors for the Hamiltonian describing second harmonics generation are proposed.

Samir F. RADWAN

Pages: 13 - 34

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical...

Wilhelm FUSHCHYCH, Zoya SYMENOH

Pages: 13 - 22

We study symmetry properties of the Schrödinger equation with the potential as a new dependent variable, i.e., the transformations which do not change the form of the class of equations. We also consider systems of the Schrödinger equations with certain conditions on the potential. In addition we investigate...

Delia Ionesco-Kruse

Pages: 13 - 27

We investigate the particle trajectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending...

Jehad O Alzabut

Pages: 13 - 21

The Banach fixed point theorem is used to prove the existence of a unique( w) periodic solution of a new type of nonlinear impulsive delay differential equation with a small parameter.

Robert CONTE, Micheline MUSETTE

Pages: 14 - 28

A birational transformation is one which leaves invariant an ordinary differential eqution, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation of Okamoto for the mater Painlevé equation P6. Then we improve...

E.V. FERAPONTOV, A.M. GRUNDLAND

Pages: 14 - 21

Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system

Christopher ATHORNE

Pages: 15 - 27

By associating polynomials and power series expansions with sln(C) modules we dscribe the theory of Padé approximants in terms of tensor products of representations and interpret their recurrence relations algebraically. The treatment links with the theory of Hirota derivatives and discrete integrable...

R GOODALL, P G L LEACH

Pages: 15 - 26

Generalised symmetries and point symmetries coincide for systems of first-order odinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the sytem of first-order ordinary differential equations...

Taekyun KIM

Pages: 15 - 27

The main purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials. In particular, by using the q-Volkenborn integration on Zp, we construct p-adic q-Euler numbers and polynomials of higher order. We also define new generating functions of multiple...

Vladimir DORODNITSYN, Roman KOZLOV

Pages: 16 - 50

In this letter we present the set of invariant difference equations and meshes which preserve the Lie group symmetries of the equation ut = (K(u)ux)x +Q(u). All special cases of K(u) and Q(u) that extend the symmetry group admitted by the differential equation are considered. This paper completes the...

Angel BALLESTEROS, Francisco J HERRANZ

Pages: 18 - 22

The two-photon algebra h6 is used to define an infinite class of N-particle Hamiltonian systems having (N -2) additional constants of the motion in involution. By constrution, all these systems are h6-coalgebra invariant. As a straightforward application, a new family of (quasi)integrable N-dimensional...

N BEN FRAJ, S OMRI

Pages: 19 - 33

We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie supealgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced...

Nadezda BAZUNOVA

Pages: 21 - 26

In this article, we generalize a construction of graded q-differential algebra with ternary differential satisfying the property d3 = 0 and q-Leibniz rule on the non-coordinate case, that is on the case where the differentials of generators of underlying algebra do not coincide with generators of bimodule...

Kouichi TAKEMURA

Pages: 21 - 46

A new approach to the finite-gap property for the Heun equation is constructed. The relationship between the finite-dimensional invariant space and the spectral curve is clarified. The monodromies are calculated and are expressed as hyperelliptic integrals. Applications to the spectral problem for the...

S Yu SAKOVICH

Pages: 21 - 25

Integrability of differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordnary differential equations are obtained in this way, which are integrable by quadrtures in spite of very complicated branching of their...

A.M. KOROSTIL

Pages: 22 - 33

A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these solutions in the form of rational functions of the elliptic Weierstrass...

MC Nucci, P.G.L. Leach

Pages: 22 - 34

We apply singularity analysis to a caricature of the simplified multistrain model of Castillo-Chavez and Feng (J Math Biol 35 (1997) 629–656) for the transmission of tuberculosis and the coupled two-stream vector-based model of Feng and Velasco- Hern ?andez (J Math Biol 35 (1997) 523–544) to identify...

Boyka Aneva

Pages: 22 - 33

We consider a lattice driven diffusive system withUq(su(2)) invariance in the bulk. Within the matrix product states approach the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra. Boundary processes amount to the appearance of parameter dependent...

R BEALS, D H SATTINGER, J SZMIGIELSKI

Pages: 23 - 27

Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon.

R. LÉANDRE

Pages: 23 - 40

We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we impose...

T CERQUETELLI, N CICCOLI, M C NUCCI

Pages: 24 - 35

Realizations of four dimensional Lie algebras as vector fields in the plane are explcitly constructed. Fourth order ordinary differential equations which admit such Lie symmetry algebras are derived. The route to their integration is described.

M. LAKSHMANAN, M. SENTHIL VELAN

Pages: 24 - 39

The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik Novikov Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation,...

ANATOLIJ SAMOILENKO, LEON CHUA

Pages: 25 - 40

Sajid Ali, Fazal M Mahomed, Asghar Qadir

Pages: 25 - 35

We present the use of complex Lie symmetries in variational problems by defining a complex Lagrangian and considering its Euler-Lagrange ordinary differential equation. This Lagrangian results in two real “Lagrangians” for the corresponding system of partial differential equations, which satisfy Euler-Lagrange...

M A JAFARIZADEH, S BEHNIA

Pages: 26 - 41

We give a hierarchy of many-parameter families of maps of the interval [0, 1] with an invariant measure and using the measure, we calculate KolmogorovSinai entropy of these maps analytically. In contrary to the usual one-dimensional maps these maps do not possess period doubling or period-n-tupling...

C. BURDIK, S. POSTA, O NAVRATIL

Pages: 27 - 36

We study the exact solvable 3 × 3 matrix model of the type G2. We apply the construction similar to that one, which give the 2 × 2 matrix model. But in the studied case the construction does not give symmetric matrix potential. We conceive that the exact solvable 3 × 3 matrix potential model of the type...

Mats EHRNSTRÖM

Pages: 27 - 30

Given a steady and symmetric deep-water wave we prove that the surface profile and the vorticity distribution determine the wave motion completely throughout the fluid.

Alan K COMMON, Andrew N W HONE, Micheline MUSETTE

Pages: 27 - 40

By considering the Darboux transformation for the third order Lax operator of the Sawada-Kotera hierarchy, we obtain a discrete third order linear equation as well as a discrete analogue of the Gambier 5 equation. As an application of this result, we consider the stationary reduction of the fifth order...

Yury CHAPOVSKY

Pages: 27 - 45

We define a Hopf C -algebra associated with an action of the quantum group SUq(1, 1) on a two-parameter quantum deformation of the unit disc, which has a left comodule structure over this Hopf C -algebra. Mathematics Subject Classification (1991). 81C05.

Mariusz BIALECKI

Pages: 28 - 35

First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation, and next we perform a reduction of dKP to the discrete 1D Toda equation.

Joachim ESCHER

Pages: 28 - 46

It is shown that solutions to the intermediate surface diffusion flow are real analytic in space and time, provided the initial surface is real diffeomorphic to a Euclidean sphere.

I V BIKTASHEVA, V N BIKTASHEV

Pages: 28 - 34

Dynamics of spiral waves in perturbed two-dimensional autowave media can be dscribed asymptotically in terms of Aristotelean dynamics. We apply this general thory to the spiral waves in the Complex GinzburgLandau equation (CGLE). The RFs are found numerically. In this work, we study the dependence of...

George SVETLICHNY

Pages: 28 - 35

We review here the main properties of symmetries of separating hierarchies of nonlinear Schrödinger equations and discuss the obstruction to symmetry liftings from (n)particles to a higher number. We argue that for particles with internal degrees of freedom, new multiparticle effects must appear at each...

Pham Loi VU

Pages: 28 - 43

The paper deals with a problem of developing an inverse-scattering transform for solving the initial-boundary value problem (IBVP) for the Korteweg-de Vries equation on the positive quarter-plane: pt - 6ppx + pxxx = 0, x 0, t 0, (a) with the given initial and boundary conditions: p(x, 0) = p(x), p(x)...

Shuyin Wu, Zhaoyang Yin

Pages: 28 - 49

In the paper, several problems on the periodic Degasperis-Procesi equation with weak dissipation are investigated. At first, the local well-posedness of the equation is established by Kato’s theorem and a precise blow-up scenario of the solutions is given. Then, several critera guaranteeing the blow-up...

Yuri FEDOROV

Pages: 29 - 46

There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra ~gl(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral curves....

K SOKALSKI, T WIETECHA, D SOKALSKA

Pages: 31 - 52

A concept of strong necessary conditions for extremum of functional has been aplied for analysis an existence of dual equations for a system of two nonlinear Partial Differential Equations (PDE) in 1+1 dimensions. We consider two types of the dual equations: the Bäcklund transformations and the Bogomolny...

Pavel GROZMAN

Pages: 31 - 37

Let M be an n-dimensional manifold, V the space of a representation : GL(n)GL(V ). Locally, let T(V ) be the space of sections of the tensor bundle with fiber V over a sufficiently small open set U M, in other words, T(V ) is the space of tensor fields of type V on M on which the group Diff(M) of diffeomorphisms...

K ANDRIOPOULOS, P G L LEACH

Pages: 32 - 42

We investigate the algebraic properties of the time-dependent Schrödinger equations of certain nonlinear oscillators introduced by Calogero and Graffi (Calogero F & Graffi S, On the quantisation of a nonlinear Hamiltonian oscillator Physics Letters A 313 (2003) 356-362; Calogero F, On the quantisation...

Vsevolod E. ADLER

Pages: 34 - 56

The Bäcklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the Bäcklund...

James Atkinson, Frank Nijhoff

Pages: 34 - 42

The Bäcklund transformation (BT) of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Bäcklund parameters) as a 2-cycle of the BT. In...

Piotr BIZON

Pages: 35 - 41

In this brief contribution, which is based on my talk at the conference, I discuss the dynamics of solutions of nonlinear wave equations near the threshold of singularity formation. The heuristic picture of threshold behavior is first presented in a general setting and then illustrated with three examples.

V.A. TROTSENKO

Pages: 35 - 50

A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics [1][6]. In the present paper, using the variational method for solving nonlinear boundary problems of statics of hyper-elastic membranes under the regular...

Yuri Dimitrov BOZHKOV, Igor Leite FREIRE

Pages: 35 - 47

Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group H, carried out in a preceding work, we estab- lish the conservation laws for the critical Kohn-Laplace equations via the Noether’s Theorem.

Martin BOHNER, Christopher C TISDELL

Pages: 36 - 45

The theory of dynamic inclusions on a time scale is introduced, hence accommodating the special cases of differential inclusions and difference inclusions. Fixed point theory for set-valued upper semicontinuous maps, Green's functions, and upper and lower solutions are used to establish existence results...

the Painlevé, the Psi Series

Pages: 36 - 48

The classical (ARS) algorithm used in the Painlevé test picks up only those functions analytic in the complex plane. We complement it with an iterative algorithm giving the leading order and the next terms in all cases. This algorithm works both for an ascending series (about a singularity at finite...

V.A. DANYLENKO, V.A. VLADIMIROV

Pages: 36 - 43

Solutions of the system of dynamical equations of state and equations of the balance of mass and momentum are studied. The system possesses families of periodic, quasiperiodic and soliton-like invariant solutions. Self-similar solutions of this generalized hydrodynamic system are studied. Various complicated...

A. H. Davison, A. H. Kara

Pages: 36 - 43

The method for writing a differential equation or system of differential equations in terms of differential forms and finding their symmetries was devised by Harrison and Estabrook (1971). A modification to the method is demonstrated on a wave equation with variable speed, and the modified method is...

Cestmir BURDIK, Eugen PAAL, Juri VIRKEPU

Pages: 37 - 43

Canonical formalism for plane rotations is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide...

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...

H S DHILLON, F V KUSMARTSEV, K E KÜRTEN

Pages: 38 - 49

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the 4 type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or...