Journal of Nonlinear Mathematical Physics

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

1. 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...

2. Some Symmetry Classifications of Hyperbolic Vector Evolution Equations

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...

3. Some Group Theoretical Aspects of Nonlinear Quantal Oscillators

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...

4. Singular Scattering Matrices

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.

5. Tunnelling in Nonlocal Evolution Equations

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.

6. On Dispersionless BKP Hierarchy and its Reductions

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.

7. 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.

8. Lower Bounds for the Spinless Salpeter Equation

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...

9. 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.

10. Hyperelliptic Addition Law

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...

11. 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...

12. On a Class of Non Self-Adjoint Quantum Nonlinear Oscillators with Real Spectrum

Emanuela CALICETI, Sandro GRAFFI
Pages: 138 - 145
We prove reality of the spectrum for a class of PT - symmetric, non self-adjoint quantum nonlinear oscillators of the form H = p2 + P(q) + igQ(q). Here P(q) is an even polynomial of degree 2p positive at infinity, Q(q) an odd polynomial of degree 2r - 1, and the conditions p > 2r, |g| < R for some...

13. On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Roberto CAMASSA, Jingfang HUANG, Long LEE
Pages: 146 - 162
An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each "particle" in this method travels along a characteristic curve...

14. Some Remarks on Materials with Memory: Heat Conduction and Viscoelasticity

Sandra CARILLO
Pages: 173 - 178
Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alterntive to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any...

15. 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 .

16. Extended Prelle-Singer Method and Integrability/Solvability of a Class of Nonlinear nth Order Ordinary Differential Equations

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...

17. A Hamiltonian Formulation for Free Surface Water Waves with Non-Vanishing Vorticity

Adrian CONSTANTIN
Pages: 202 - 211
We describe the derivation of a formalism in the context of the governing equations for two-dimensional water waves propagating over a flat bed in a flow with non-vanishing vorticity. This consists in providing a Hamiltonian structure in terms of two variables which are scalar functions.

18. Explicit integration of the Hénon-Heiles Hamiltonians 1

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:...

19. Integrable and Nonintegrable Initial Boundary Value Problems for Soliton Equations 1

A DEGASPERIS, S V MANAKOV, P M SANTINI
Pages: 228 - 243
It is well-known that the basic difficulty in studying the initial boundary value prolems for linear and nonlinear PDEs is the presence, in any method of solution, of unknown boundary values. In the first part of this paper we review two spectral methods in which the above difficulty is faced in different...

20. On -Function of Conjugate Nets

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...

21. A Haldane­Shastry Spin Chain of BCN Type in a Constant Magnetic Field

A ENCISO, F FINKEL, A GONZALEZ-LOPEZ, M A RODRIGUEZ
Pages: 253 - 265
We compute the spectrum of the trigonometric Sutherland spin model of BCN type in the presence of a constant magnetic field. Using Polychronakos's freezing trick, we derive an exact formula for the partition function of its associated Haldane­Shastry spin chain.

22. 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...

23. Irreducible Characters and Clebsch-Gordan Series for the Exceptional Algebra E6: An Approach through the Quantum Calogero-Sutherland Model

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...

24. The Time Periodic Solution of the Burgers Equation on the Half-Line and an Application to Steady Streaming

A S FOKAS, J T STUART
Pages: 302 - 314
The phenomenon of steady streaming, or acoustic streaming, is an important phyical phenomenon studied extensively in the literature. Its mathematical formulation involves the Navier-Stokes equations, and due to the complexity of these equations is usually studied heuristically using formal perturbation...

25. Isochronous Systems and Perturbation Theory

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.

26. Dimension Increase and Splitting for Poincaré-Dulac Normal Forms

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...

27. Factorization of the Loop Algebras and Compatible Lie Brackets

I Z GOLUBCHIK, V V SOKOLOV
Pages: 343 - 350
It is shown that any decomposition of the loop algebra over a simple Lie algebra into a direct sum of the Taylor series and a complementary subalgebra is defined by a pair of compatible Lie brackets.

28. Periods of the Goldfish Many-Body Problem

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...

29. 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...

30. 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...

31. 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...

32. The Eigenvectors of the Heisenberg Hamiltonian with Elliptic Form of the Exchange Spin Interaction

V I INOZEMTSEV
Pages: 395 - 403
The eigenvectors of the Hamiltonian HN of N-site quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvetors via the solutions to the system of highly transcendental...

33. Canonically Transformed Detectors Applied to the Classical Inverse Scattering Problem

C. JUNG, T H SELIGMAN, J M TORRES
Pages: 404 - 411
The concept of measurement in classical scattering is interpreted as an overlap of a particle packet with some area in phase space that describes the detector. Considering that usually we record the passage of particles at some point in space, a common detector is described e.g. for one-dimensional...

34. Reflection Symmetries of q-Bernoulli Polynomials

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...

35. Gauge Theory Approach Towards an Explicit Solution of the (Classical) Elliptic Calogero-Moser System

Edwin LANGMANN
Pages: 423 - 439
We discuss the relation of the trigonometric Calogero-Moser (CM) system to YanMills gauge theories and its generalization to the elliptic case. This yields a liearization of the time evolution of the elliptic CM system and suggests two promising strategies for finding a fully explicit solution of this...

36. 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...

37. Rigorous Results in the Scaling Theory of Irreversible Aggregation Kinetics

Francois LEYVRAZ
Pages: 449 - 465
The kinetic equations describing irreversible aggregation and the scaling approach dveloped to describe them in the limit of large times and large sizes are tersely reviewed. Next, a system is considered in which aggregates can only react with aggregates of their own size. The existence of a scaling...

38. 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.

39. Algebraic Extensions of Gaudin Models

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...

40. Jacobi's Three-Body System Moves like a Free Particle

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...

41. Shape Invariant Potentials in "Discrete Quantum Mechanics"

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...

42. Universality of Calogero-Moser Model

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.

43. What an Effective Criterion of Separability says about the Calogero Type Systems

Stefan RAUCH-WOJCIECHOWSKI, Claes WAKSJÖ
Pages: 535 - 547
In [15] we have proved a 1-1 correspondence between all separable coordinates on Rn (according to Kalnins and Miller [9]) and systems of linear PDEs for separable potetials V (q). These PDEs, after introducing parameters reflecting the freedom of choice of Euclidean reference frame, serve as an effective...

44. Vortex Trains in Super-Alfvénic Magnetogasdynamics. Application of Reciprocal-Bäcklund Transformations

C ROGERS, W K SCHIEF
Pages: 548 - 564
A multi-parameter class of reciprocal transformations is coupled with the action of a Bäcklund transformation to construct periodic solutions of breather-type in plane, aligned, super-Alfvénic magnetogasdynamics. The constitutive law adopts a genealised K´arm´an-Tsien form.

45. Isometric Reflectionless Eigenfunction Transforms for Higher-order AOs

S N M RUIJSENAARS
Pages: 565 - 598
In a previous paper (Regular and Chaotic Dynamics 7 (2002), 351­391, Ref. [1]), we obtained various results concerning reflectionless Hilbert space transforms arising from a general Cauchy system. Here we extend these results, proving in particular an isometry property conjectured in Ref. [1]. Crucial...

46. Generalised Inverse Scattering for a Linear PDE Associate to KdV

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....

47. Universal Solitonic Hierarchy

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.

48. RG Solutions for s at large Nc in d = 3 + 1 QCD

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).

49. Time Discretization of F. Calogero's "Goldfish" System

Yuri B SURIS
Pages: 633 - 647
Time-discretized versions of F. Calogero's rational and hyperbolic "goldfish" systems are presented, and their exact solutions are given.

50. 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...

51. Quasi-Exactly Solvable Hamiltonians related to Root Spaces

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...

52. Correlation Function of Asymmetric Simple Exclusion Process with Open Boundaries

Masaru UCHIYAMA, Miki WADATI
Pages: 676 - 688
We investigate the correlation functions of the one-dimensional Asymmetric Simple Exclusion Process (ASEP) with open boundaries. The conditions for the boundaries are made most general. The correlation function is expressed in a multifold integral whose behavior we study in detail. We present a phase...

53. 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.