Journal of Nonlinear Mathematical Physics

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905 articles

A fuzzy difference equation of a rational form

G STEFANIDOU, G PAPASCHINOPOULOS
Pages: 300 - 315
In this paper, we prove some effects concerning a Fuzzy Difference Equation of a rational form.

Non-isospectral lattice hierarchies in 2 + 1 dimensions and generalized discrete Painlevé hierarchies

P R GORDOA, A PICKERING, Z N ZHU
Pages: 180 - 196
In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of discrete equations, which we obtain once again...

Bispectrality for deformed Calogero­Moser­Sutherland systems

Misha FEIGIN
Pages: 95 - 136
We prove bispectral duality for the generalized Calogero­Moser­Sutherland systems related to configurations An,2(m), Cn(l, m). The trigonometric axiomatics of the Baker­Akhiezer function is modified, the dual difference operators of rational Madonald type and the Baker­Akhiezer functions related to both...

A unitary joint eigenfunction transform for the AOs exp(ia±d/dz) + exp(2z/a)

S N M RUIJSENAARS
Pages: 253 - 294
For positive parameters a+ and a- the commuting difference operators exp(ia±d/dz) + exp(2z/a), acting on meromorphic functions f(z), z = x + iy, are formally self-adjoint on the Hilbert space H = L2 (R, dx). Volkov showed that they admit joint eigenfunctions. We prove that the joint eigenfunctions for...

Searching for CAC-maps

Jarmo HIETARINTA
Pages: 223 - 230
For two-dimensional lattice equations the standard definition of integrability is that it should be possible to extend the map consistently to three dimensions, i.e., that it is "consistent around a cube" (CAC). Recently Adler, Bobenko and Suris conducted a search based on this principle, together with...

Integrable 1D Toda cellular automata

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.

On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

Vladimir S GERDJIKOV, Georgi G GRAHOVSKI
Pages: 155 - 168
A family of real Hamiltonian forms (RHF) for the special class of affine 1 + dimensional Toda field theories is constructed. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF...

Asymptotic behavior of discrete holomorphic maps zc and log(z)

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...

Twisted Volterra equation

Sergei D SILVESTROV
Pages: 295 - 299
In this paper an extension of the q-deformed Volterra equation associated with linear rescaling to the general non-linear rescaling is obtained.

A factorization for Z × Z-matrices yielding solutions of Toda-type hierarchies

Gerardus Franciscus HELMINCK
Pages: 206 - 222
In this paper one considers the problem of finding solutions to a number of Todtype hierarchies. All of them are associated with a commutative subalgebra of the k×k-matrices. The first one is formulated in terms of upper triangular Z×Z-matrices, the second one in terms of lower triangular ones and the...

On a special two-dimensional lattice by Blaszak and Szum: pfaffianization and molecule solutions

Guo-Fu YU, Chun-Xia LI, Jun-Xiao ZHAO
Pages: 316 - 332
In this paper, we first present the Casorati and grammian determinant solutions to a special two-dimensional lattice by Blaszak and Szum. Then, by using the pfaffianiztion procedure of Hirota and Ohta, a new integrable coupled system is generated from the special lattice. Moreover, gram-type pfaffian...

Asymptotic symmetries of difference equations on a lattice

Giuseppe GAETA, Decio LEVI, Rosaria MANCINELLI
Pages: 137 - 146
It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations (e.g. scaling and/or translations) which are not symmetries of the considered equation. In this note we extend the approach to asymptotic symmetries for...

Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples

Mirta M CASTRO, F Alberto GRUNBAUM
Pages: 63 - 76
We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices,...

Integrable flows and Bäcklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)

Yuri N FEDOROV
Pages: 77 - 94
We show that the m-dimensional Euler­Manakov top on so (m) can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety ¯V(k, m), and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic...

Second Order Dynamic Inclusions

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 Relation between a 2D Lotka-Volterra equation and a 2D Toda Lattice

Claire R GILSON, Jonathan J C NIMMO
Pages: 169 - 179
It is shown that the 2-discrete dimensional Lotka-Volterra lattice, the two dmensional Toda lattice equation and the recent 2-discrete dimensional Toda lattice equation of Santini et al can be obtained from a 2-discrete 2-continuous dimensional Lotka-Volterra lattice.

Darboux transformations for q-discretizations of 2D second order differential equations

P MALKIEWICZ, M NIESZPORSKI
Pages: 231 - 238
We present q-discretizations of a second order differential equation in two independent variables that not only go to the differential counterpart as q goes to 1 but admit Moutard-Darboux transformations as well.

Bäcklund transformations for the rational Lagrange chain

Fabio MUSSO, Matteo PETRERA, Orlando RAGNISCO, Giovanni SATTA
Pages: 240 - 252
We consider a long­range homogeneous chain where the local variables are the geerators of the direct sum of N e(3) interacting Lagrange tops. We call this classical integrable model rational "Lagrange chain" showing how one can obtain it starting from su(2) rational Gaudin models. Moreover we construct...

Classification of Fully Nonlinear Integrable Evolution Equations of Third Order

Rafael HERNANDEZ HEREDERO
Pages: 567 - 585
A fully nonlinear family of evolution equations is classified. Nine new integrable equtions are found, and all of them admit a differential substitution into the Korteweg-de Vries or Krichever-Novikov equations. One of the equations contains hyperelliptic functions, but it is transformable into the Krichever-Novikov...

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...

On A Group Of Automorphisms Of The Noncommutative Burgers Hierarchy

Boris A KUPERSHMIDT
Pages: 539 - 549
Bäcklund transformations are constructed for the noncommutative Burgers hierarchy, generalizing the commutative ones of Weiss, Tabor, Carnevale, and Pickering. These transformations are shown to be invertible and form a group.

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...

Asymptotic Approximations in Quantum Calculus

Ahmed FITOUHI, Kamel BRAHIM, Néji BETTAIBI
Pages: 586 - 606
This paper aims to study the asymptotic approximation of some functions defined by the q-Jackson integrals, for a fix q ]0, 1[. For this purpose, we shall attempt to extend the classical methods by giving their q-analogues. In particular, a q-analogue of the Watson's lemma is discussed and new asymptotic...

Seiberg-Witten-like Equations on 7-Manifolds

Nedim DEGIRMENCI, Nülifer ÖZDEMIR
Pages: 457 - 461
The Seiberg-Witten equations are of great importance in the study of topology of smooth four-dimensional manifolds. In this work, we propose similar equations for 7-dimensional compact manifolds with G2-structure.

Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

Giuseppe GAETA, Rosaria MANCINELLI
Pages: 550 - 566
We analyze asymptotic scaling properties of a model class of anomalous reactiodiffusion (ARD) equations. Numerical experiments show that solutions to these have, for large t, well defined scaling properties. We suggest a general framework to anlyze asymptotic symmetry properties; this provides an analytical...

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...

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.

Bernoulli Numbers and Solitons

Marie-Pierre GROSSET, Alexander P VESELOV
Pages: 469 - 474
We present a new formula for the Bernoulli numbers as the following integral B2m = (-1)m-1 22m+1 +( dm-1 dxm-1 sech2 x)2 dx. This formula is motivated by the results of Fairlie and Veselov, who discovered the relation of Bernoulli polynomials with soliton theory. Dedicated to Hermann Flaschka on his...

On the Integrability of a Class of Nonlinear Dispersive Wave Equations

Ivanov ROSSEN
Pages: 462 - 468
We investigate the integrability of a class of 1+1 dimensional models describing nolinear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.

Compactly Supported Solutions of the Camassa-Holm Equation

David HENRY
Pages: 342 - 347
We give a simple proof that for any non-zero initial data, the solution of the CamassHolm equation loses instantly the property of being compactly supported.

Symmetries of Modules of Differential Operators

H GARGOUBI, P MATHONET, V OVSIENKO
Pages: 348 - 380
Let F(S1 ) be the space of tensor densities of degree (or weight) on the circle S1 . The space Dk ,µ(S1 ) of k-th order linear differential operators from F(S1 ) to Fµ(S1 ) is a natural module over Diff(S1 ), the diffeomorphism group of S1 . We determine the algebra of symmetries of the modules Dk ,µ(S1...

Implicit Solutions to Some Lorentz Invariant Nonlinear Equations Revisited

David B FAIRLIE
Pages: 449 - 456
An implicit solution to the vanishing of the so-called Universal Field Equation, or Bordered Hessian, which dates at least as far back as 1935 [1] is revived, and derived from a much later form of the solution. A linear ansatz for an implicit solution of second order partial differential equations, previously...

Nonclassical Contact Symmetries and Charpit's Method of Compatibility

Daniel J ARRIGO
Pages: 321 - 329
Charpit's method of compatibility and the method of nonclassical contact symmetries for first order partial differential equation are considered. It is shown that these two methods are equivalent as Charpit's method leads to the determining equations arising from the method of nonclassical contact symmetries....

The Stimulated Scattering of Solitons on a Resonance

Sergei GLEBOV, Oleg KISELEV
Pages: 330 - 341
We investigate a propagation of solitons for nonlinear Schrödinger equation under small driving force. The driving force passes through the resonance. The process of scattering on the resonance leads to changing of number of solitons. After the resonance the number of solitons depends on the amplitude...

Functional Representation of the Volterra Hierarchy

V E VEKSLERCHIK
Pages: 409 - 431
In this paper I study the functional representation of the Volterra hierarchy (VH). Using the Miwa's shifts I rewrite the infinite set of Volterra equations as one functional equation. This result is used to derive a formal solution of the associated linear problem, a generating function for the conservation...

The Structure of Gelfand-Levitan-Marchenko Type Equations for Delsarte Transmutation Operators of Linear Multi-Dimensional Differential Operators and Operator Pencils. Part 2

Jolanta GOLENIA, Anatolij K PRYKARPATSKY, Yarema A PRYKARPATSKY
Pages: 381 - 408
The differential-geometric and topological structure of Delsarte transmutation opertors their associated Gelfand-Levitan-Marchenko type equations are studied making use of the de Rham-Hodge-Skrypnik differential complex. The relationships with spetral theory and special Berezansky type congruence properties...

A Three State Hard-Core Model on a Cayley Tree

James MARTIN, Utkir ROZIKOV, Yuri SUHOV
Pages: 432 - 448
We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state (x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson...

Jacobi's Last Multiplier and the Complete Symmetry Group of the Ermakov-Pinney Equation

M C NUCCI, P G L LEACH
Pages: 305 - 320
The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in...

Symmetry Reductions of a Hamilton-Jacobi-Bellman Equation Arising in Financial Mathematics

V NAICKER, K ANDRIOPOULOS, PGL LEACH
Pages: 268 - 283
We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We...

Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

José F. CARINENA, Manuel F RAÑADA, Mariano SANTANDER
Pages: 230 - 252
We characterize and completely describe some types of separable potentials in twdimensional spaces, S2 [1]2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere...

Jacobi, Ellipsoidal Coordinates and Superintegrable Systems

E G KALNINS, J M KRESS, W. Miller
Pages: 209 - 229
We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems...

A Lie Symmetry Connection between Jacobi's Modular Differential Equation and Schwarzian Differential Equation

L ROSATI, M C NUCCI
Pages: 144 - 161
In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi's modular differential equation. A six-dimensional Lie symmetry algebra is obtained...

Subgroup Type Coordinates and the Separationof Variables in Hamilton-Jacobi and Schrödinger Equations

E G KALNINS, Z THOMOVA, P WINTERNITZ
Pages: 178 - 208
Separable coordinate systems are introduced in complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically) in terms of subgroup chains. Finally the explicit solutions...

On Properties of Elliptic Jacobi Functions and Applications

A Raouf CHOUIKHA
Pages: 162 - 169
In this paper we are interested in developments of the elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53-57)...

Triangular Newton Equations with Maximal Number of Integrals of Motion

Fredrik PERSSON, Stefan RAUCH-WOJCIECHOWSKI
Pages: 253 - 267
We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic intgrals of motion. The main idea is to exploit the fact that the first component M1(q1) of a triangular force depends on one variable only. By...

Separabilty in Hamilton­Jacobi Sense in Two Degrees of Freedom and the Appell Hypergeometric Functions

Vladimir Dragovi
Pages: 170 - 177
A huge family of separable potential perturbations of integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid is given through the Appell hypegeometric functions F4 of two variables, leading to an interesting connection between two classical theories: separability in Hamilton­Jacobi...

Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

M C NUCCI
Pages: 284 - 304
After giving a brief account of the Jacobi last multiplier for ordinary differential equtions and its known relationship with Lie symmetries, we present a novel application which exploits the Jacobi last multiplier to the purpose of finding Lie symmetries of first-order systems. Several illustrative...

Uniqueness of Steady Symmetric Deep-Water Waves with Vorticity

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.