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125134 articles

Matrix Exponential via Clifford Algebras

Rafal ABLAMOWICZ
Pages: 294 - 313
We use isomorphism between matrix algebras and simple orthogonal Clifford algebras C (Q) to compute matrix exponential eA of a real, complex, and quaternionic matrix A. The isomorphic image p = (A) in C (Q), where the quadratic form Q has a suitable signature (p, q), is exponentiated modulo a minimal...

Quantum Differential Forms

Boris A. KUPERSHMIDT
Pages: 245 - 288
Formalism of differential forms is developed for a variety of Quantum and noncommutative situations.

On Integrability of a (2+1)-Dimensional Perturbed KdV Equation

S.Yu. SAKOVICH
Pages: 230 - 233
A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé test for integrability well, and its 4×4 Lax pair with two spectral parameters is found. The results show that the Painlevé classification of coupled KdV equations by A. Karasu...

Some Homogenization and Corrector Results for Nonlinear Monotone Operators

Peter WALL
Pages: 331 - 348
This paper deals with the limit behaviour of the solutions of quasi-linear equations of the form - div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. The sequence (h) tends to 0 and the map a(x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions. It is proved...

Application of the Group-Theoretical Method to Physical Problems

Mina B. ABD-EL-MALEK
Pages: 314 - 330
The concept of the theory of continuous groups of transformations has attracted the attention of applied mathematicians and engineers to solve many physical problems in the engineering sciences. Three applications are presented in this paper. The first one is the problem of time-dependent vertical temperature...

Similarity Reductions for a Nonlinear Diffusion Equation

M.L. GANDARIAS, P. VENERO, J. RAMIREZ
Pages: 234 - 244
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced...

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.

On the Analytical Approach to the N-Fold Bäcklund Transformation of Davey-Stewartson Equation

S.K. PAUL, A. ROY CHOWDHURY
Pages: 349 - 356
N-fold Bäcklund transformation for the Davey-Stewartson equation is constructed by using the analytic structure of the Lax eigenfunction in the complex eigenvalue plane. Explicit formulae can be obtained for a specified value of N. Lastly it is shown how generalized soliton solutions are generated from...

Mode-Coupling and Nonlinear Landau Damping Effects in Auroral Farley-Buneman Turbulence

A.M. HAMZA
Pages: 462 - 470
The fundamental problem of Farley-Buneman turbulence in the auroral E-region has been discussed and debated extensively in the past two decades. In the present paper we intend to clarify the different steps that the auroral E-region plasma has to undergo before reaching a steady state. The mode-coupling...

Resonance Broadening Theory of Farley-Buneman Turbulence in the Auroral E-Region

A.M. HAMZA
Pages: 438 - 461
The conventional theory of resonance broadening for a two-species plasma in a magnetic field is revised, and applied to an ionospheric turbulence case. The assumptions made in the conventional theory of resonance broadening have, in the past, led to replacing the frequency by + ik2 D in the resonant...

On Infinitesimal Symmetries of the Self-Dual Yang-Mills Equations

T.A. IVANOVA
Pages: 396 - 404
Infinite-dimensional algebra of all infinitesimal transformations of solutions of the self-dual Yang-Mills equations is described. It contains as subalgebras the infinitedimensional algebras of hidden symmetries related to gauge and conformal transformations.

On Asymptotic Nonlocal Symmetry of Nonlinear Schrödinger Equations

W.W. ZACHARY, V.M. SHTELEN
Pages: 417 - 437
A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schrödinger equation, discovered by Fushchych and Segeda in 1977, can...

How to Find Discrete Contact Symmetries

Peter E. HYDON
Pages: 405 - 416
This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism...

Solving Simultaneously Dirac and Ricatti Equations

J. CASAHORRÁN
Pages: 371 - 382
We analyse the behaviour of the Dirac equation in d = 1 + 1 with Lorentz scalar potential. As the system is known to provide a physical realization of supersymmetric quantum mechanics, we take advantage of the factorization method in order to enlarge the restricted class of solvable problems. To be precise,...

Fundamental Solution of the Volkov Problem (Characteristic Representation)

A.A. BORGHARDT, D.Ya. KARPENKO
Pages: 357 - 363
The characteristic representation, or Goursat problem, for the Klein-Fock-Gordon equation with Volkov interaction [1] is regarded. It is shown that in this representation the explicit form of the Volkov propagator can be obtained. Using the characteristic representation technique, the Schwinger integral...

Differential Constraints Compatible with Linearized Equations

Ahmet SATIR
Pages: 364 - 370
Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.

Solutions of WDVV Equations in Seiberg-Witten Theory from Root Systems

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.

Neumann and Bargmann Systems Associated with an Extension of the Coupled KdV Hierarchy

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...

On the Fourth-Order Accurate Compact ADI Scheme for Solving the Unsteady Nonlinear Coupled Burgers' Equations

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...

Variational Methods for Solving Nonlinear Boundary Problems of Statics of Hyper-Elastic Membranes

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...

Dynamical Correlation Functions for an Impenetrable Bose Gas with Neumann or Dirichlet Boundary Conditions

Takeo KOJIMA
Pages: 99 - 119
We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions (x1, 0) (x2, t) ±,T . We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case x1 = 0, we express...

Contact Symmetry of Time-Dependent Schrödinger Equation for a Two-Particle System: Symmetry Classification of Two-Body Central Potentials

P. RUDRA
Pages: 51 - 65
Symmetry classification of two-body central potentials in a two-particle Schrödinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem...

Symmetries of a Class of Nonlinear Fourth Order Partial Differential Equations

Peter A. CLARKSON, Thomas J. PRIESTLEY
Pages: 66 - 98
In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations utt = u + u2 xx + uuxxxx + µuxxtt + uxuxxx + u2 xx, (1) where , , , and µ are arbitrary constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm...

On the Nature of the Virasoro Algebra

Boris A. KUPERSHMIDT
Pages: 222 - 245
The multiplication in the Virasoro algebra [ep, eq] = (p - q)ep+q + p3 - p p+q, p, q Z, [, ep] = 0, comes from the commutator [ep, eq] = ep eq - eq ep in a quasiassociative algebra with the multiplication ep eq = q(1 + q)

New Mathematical Models for Particle Flow Dynamics

Denis BLACKMORE, Roman SAMULYAK, Anthony ROSATO
Pages: 198 - 221
A new class of integro-partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of infinite-dimensional dynamical systems models) on the Newtonian...

Exactly Integrable Systems Connected to Semisimple Algebras of Second Rank A2, B2, C2, G2

A.N. LEZNOV
Pages: 181 - 197
Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental representations of the corresponding semisimple groups.

Explode-Decay Dromions in the Non-Isospectral Davey-Stewartson I (DSI) Equation

R. RADHA, S. VIJAYALAKSHMI, M. LAKSHMANAN
Pages: 120 - 126
In this letter, we report the existence of a novel type of explode-decay dromions, which are exponentially localized coherent structures whose amplitude varies with time, through Hirota method for a nonisospectral Davey-Stewartson equation I discussed recently by Jiang. Using suitable transformations,...

Representations of the Infinite Unitary Group from Constrained Quantization

N.P. LANDSMAN
Pages: 161 - 180
We attempt to reconstruct the irreducible unitary representations of the Banach Lie group U0(H) of all unitary operators U on a separable Hilbert space H for which U - I is compact, originally found by Kirillov and Ol'shanskii, through constrained quantization of its coadjoint orbits. For this purpose...

Semiclassical Solutions of the Nonlinear Schrödinger Equation

A.V. SHAPOVALOV, A.Yu. TRIFONOV
Pages: 127 - 138
A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schrödinger equation (NLSE) with an external field. These solutions are considered as multidimensional solitary waves. The center of mass of such a solution is shown to move along with the bicharacteristics...

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

S. MELKONIAN
Pages: 139 - 160
The cubic Hénon-Heiles system contains parameters, for most values of which, the system is not integrable. In such parameter regimes, the general solution is expressible in formal expansions about arbitrary movable branch points, the so-called psi-series expansions. In this paper, the convergence of...

Versal Deformations of a Dirac Type Differential Operator

Anatoliy K. PRYKARPATSKY, Denis BLACKMORE
Pages: 246 - 254
If we are given a smooth differential operator in the variable x R/2Z, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S1 )-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal...

Remarks on Quantization of Classical r-Matrices

Boris A. KUPERSHMIDT
Pages: 269 - 272
If a classical r-matrix r is skewsymmetric, its quantization R can lose the skewsymmetry property. Even when R is skewsymmetric, it may not be unique.

Coadjoint Poisson Actions of Poisson-Lie Groups

Boris A. KUPERSHMIDT, Ognyan S. STOYANOV
Pages: 344 - 354
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.

Coupled KdV Equations of Hirota-Satsuma Type

S.Yu. SAKOVICH
Pages: 255 - 262
It is shown that the system of two coupled Korteweg-de Vries equations passes the Painlevé test for integrability in nine distinct cases of its coefficients. The integrability of eight cases is verified by direct construction of Lax pairs, whereas for one case it remains unknown.

On Certain Classes of Solutions of the Weierstrass-Enneper System Inducing Constant Mean Curvature Surfaces

P. BRACKEN, A. M. GRUNDLAND
Pages: 294 - 313
Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing certain classes of solutions to this system, including potential,...

Viewing the Efficiency of Chaos Control

Philippe CHANFREAU, Hannu LYYJYNEN
Pages: 314 - 331
This paper aims to cast some new light on controlling chaos using the OGY- and the Zero-Spectral-Radius methods. In deriving those methods we use a generalized procedure differing from the usual ones. This procedure allows us to conveniently treat maps to be controlled bringing the orbit to both various...

Algebraic Spectral Relations for Elliptic Quantum Calogero-Moser Problems

L.A. KHODARINOVA, I.A. PRIKHODSKY
Pages: 263 - 268
Explicit algebraic relations between the quantum integrals of the elliptic Calogero­ Moser quantum problems related to the root systems A2 and B2 are found.

Quest for Universal Integrable Models

Partha GUHA
Pages: 273 - 293
In this paper we discuss a universal integrable model, given by a sum of two WessZumino-Witten-Novikov (WZWN) actions, corresponding to two different orbits of the coadjoint action of a loop group on its dual, and the Polyakov-Weigmann cocycle describing their interactions. This is an effective action...

A Note on the Third Family of N = 2 Supersymmetric KdV Hierarchies

F. DELDUC, L. GALLOT
Pages: 332 - 343
We propose a hamiltonian formulation of the N = 2 supersymmetric KP type hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic hamiltonian structure which allows for several reductions of the KP type hierarchy. In particular, the third family of N = 2 KdV hierarchies is recovered. We...

What a Classical r-Matrix Really Is

Boris A. KUPERSHMIDT
Pages: 448 - 488
The notion of classical r-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, ­ where the standard definitions are shown to be deficient, ­ is proposed, the notion of an O-operator. This notion has all the natural properties one would expect form it, but lacks...

Continuous and Discrete Transformations of a One-Dimensional Porous Medium Equation

Christodoulos SOPHOCLEOUS
Pages: 355 - 364
We consider the one-dimensional porous medium equation ut = (un ux)x + µ x un ux. We derive point transformations of a general class that map this equation into itself or into equations of a similar class. In some cases this porous medium equation is connected with well known equations. With the introduction...

r-Matrices for Relativistic Deformations of Integrable Systems

Yuri B. SURIS
Pages: 411 - 447
We include the relativistic lattice KP hierarchy, introduced by Gibbons and Kupershmidt, into the r-matrix framework. An r-matrix account of the nonrelativistic lattice KP hierarchy is also provided for the reader's convenience. All relativistic constructions are regular one-parameter perturbations of...

Poisson Homology of r-Matrix Type Orbits I: Example of Computation

Alexei KOTOV
Pages: 365 - 383
In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang­Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e...

The Nonabelian Liouville-Arnold Integrability by Quadratures Problem: a Symplectic Approach

Anatoliy K. PRYKARPATSKY
Pages: 384 - 410
A symplectic theory approach is devised for solving the problem of algebraic-analytical construction of integral submanifold imbeddings for integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on canonically symplectic phase spaces.

On Exact Solution of a Classical 3D Integrable Model

S.M. SERGEEV
Pages: 57 - 72
We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients...

q-Probability: I. Basic Discrete Distributions

Boris A. KUPERSHMIDT
Pages: 73 - 93
For basic discrete probability distributions, - Bernoulli, Pascal, Poisson, hypergemetric, contagious, and uniform, - q-analogs are proposed.

On the Structure of the Bäcklund Transformations for the Relativistic Lattices

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...

Particles and Strings in a 2 + 1-D Integrable Quantum Model

I.G. KOREPANOV
Pages: 94 - 119
We give a review of some recent work on generalization of the Bethe ansatz in the case of 2 + 1-dimensional models of quantum field theory. As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization of the famous Yang­Baxter equation. We construct...

Lax Pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2 + 1)-Dimensional Equation

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...

Links Between Different Analytic Descriptions of Constant Mean Curvature Surfaces

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