Journal of Non-linear Mathematical Physics

ISSN: 1402-9251
Volume 12, Issue Supplement 1, January 2005

Special Issue in Honour of Francesco Calogero on the Occasion of His 70th Birthday

Foreword
Francesco, a First Class Scientist and a Committed Intellectual
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...
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...
David ATKINSON
Pages: 43 - 49
A nonlinear integrodifferential equation is solved by the methods of S-matrix theory. The technique is shown to be applicable to situations in which the effective potential is singular.
Giovanni BELLETTINI, Anna DE MASI, Errico PRESUTTI
Pages: 50 - 63
We study "tunnelling" in a one-dimensional, nonlocal evolution equation by assigning a penalty functional to orbits which deviate from solutions of the evolution equation. We discuss the variational problem of computing the minimal penalty for orbits which connect two stable, stationary solutions.
L V BOGDANOV, B G KONOPELCHENKO
Pages: 64 - 73
Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is consiered. Addition formula for the -function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry constraints for the dBKP hierarchy are studied.
Fabian BRAU
Pages: 86 - 96
We obtain lower bounds on the ground state energy, in one and three dimensions, for the spinless Salpeter equation (Schrödinger equation with a relativistic kinetic energy operator) applicable to potentials for which the attractive parts are in Lp (Rn ) for some p > n (n = 1 or 3). An extension...
Victor BUCHSTABER, Dmitry LEYKIN
Pages: 106 - 123
Given a family of genus g algebraic curves, with the equation f(x, y, ) = 0, we cosider two fiber-bundles U and X over the space of parameters . A fiber of U is the Jacobi variety of the curve. U is equipped with the natural groupoid structure that induces the canonical addition on a fiber. A fiber...
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...
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 .
V K CHANDRASEKAR, M SENTHILVELAN, M LAKSHMANAN
Pages: 184 - 201
We discuss a method of solving nth order scalar ordinary differential equations by extending the ideas based on the Prelle-Singer (PS) procedure for second order ordnary differential equations. We also introduce a novel way of generating additional integrals of motion from a single integral. We illustrate...
Robert CONTE, Micheline MUSETTE, Caroline VERHOEVEN
Pages: 212 - 227
We consider the cubic and quartic Hénon-Heiles Hamiltonians with additional inverse square terms, which pass the Painlevé test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties:...
Adam DOLIWA
Pages: 244 - 252
We study a potential introduced by Darboux to describe conjugate nets, which within the modern theory of integrable systems can be interpreted as a -function. We investigate the potential using the nonlocal ¯-dressing method of Manakov and Zkharov and we show that it can be interpreted as the Fredholm...
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...
J FERNANDEZ-NUNEZ, W GARCIA-FUERTES, A M PERELOMOV
Pages: 280 - 301
We re-express the quantum Calogero-Sutherland model for the Lie algebra E6 and the particular value of the coupling constant = 1 by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan...
J-P FRANCOISE
Pages: 315 - 326
This article displays examples of planar isochronous systems and discuss the new techniques found by F. Calogero with these examples. A sufficient condition is found to keep track of some periodic orbits for perturbations of isochronous systems.
Giuseppe GAETA, Sebastian WALCHER
Pages: 327 - 342
Integration of nonlinear dynamical systems is usually seen as associated to a symmetry reduction, e.g. via momentum map. In Lax integrable systems, as pointed out by Kazhdan, Kostant and Sternberg in discussing the Calogero system, one proceeds in the opposite way, enlarging the nonlinear system to...
David GOMEZ-ULLATE, Matteo SOMMACAL
Pages: 351 - 362
Calogero's goldfish N-body problem describes the motion of N point particles subject to mutual interaction with velocity-dependent forces under the action of a constant magnetic field transverse to the plane of motion. When all coupling constants are equal to one, the model has the property that for...
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...
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...
Boris A KUPERSHMIDT
Pages: 412 - 422
A large part of the theory of classical Bernoulli polynomials Bn(x)'s follows from their reflection symmetry around x = 1/2: Bn(1 - x) = (-1)n Bn(x). This symmetry not only survives quantization but has two equivalent forms, classical and quantum, depending upon whether one reflects around 1/2 the...
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...
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...
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...
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.
P C SABATIER
Pages: 599 - 613
Inverse Scattering methods for solving integrable nonlinear p.d.e. found their limits as soon as one tried to solve with them new boundary value problems. However, some of these problems, e.g. the quarter-plane problem, can be solved (e.g. by Fokas linear methods), for related linear p.d.e., (e.g....
Alexei SHABAT
Pages: 614 - 624
We describe recent results on the construction of hierarchies of nonlinear evolution equations associated to generalized second order spectral problems. The first results in this subject had been presented by Francesco Calogero.
Yu A SIMONOV
Pages: 625 - 632
Solutions of RG equations for () and (Q) are found in the class of meromorphic functions satisfying asymptotic conditions at large Q (resp. small ), and analyticity properties in the Q2 plane. The resulting R(Q) is finite in the Euclidean Q2 region and agrees well at Q 1 GeV with the MS s(Q).
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...
Alexander V TURBINER
Pages: 660 - 675
sl(2)-Quasi-Exactly-Solvable (QES) generalization of the rational An, BCn, G2, F4, E6,7,8 Olshanetsky-Perelomov Hamiltonians including many-body Calogero Hamiltonian is found. This generalization has a form of anharmonic perturbations and it appears naurally when the original rational Hamiltonian is...