# 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

**Research Article**

## 3. 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...

**Research Article**

## 4. 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...

**Research Article**

## 5. 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...

**Research Article**

## 6. 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.

**Research Article**

## 7. 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.

**Research Article**

## 8. 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.

**Research Article**

## 9. 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.

**Research Article**

## 10. 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 to confining...

**Research Article**

## 11. 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.

**Research Article**

## 12. 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 of...

**Research Article**

## 13. 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...

**Research Article**

## 14. 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| 0 hold.

**Research Article**

## 15. 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 of...

**Research Article**

## 16. 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 linear...

**Short Communication**

## 17. 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 .

**Research Article**

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

V.K. Chandrasekar, M. Senthilvelan, M. Laksmanan

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...

**Research Article**

## 19. 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.

**Research Article**

## 20. 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:...

**Research Article**

## 21. 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...

**Research Article**

## 22. On τ-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 τ-function. We investigate the potential using the nonlocal ∂-dressing method of Manakov and Zakharov and we show that it can be interpreted as...

**Research Article**

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

A. Enisco, 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.

**Research Article**

## 24. Singular Manifold Method for an Equation in 2 + 1 Dimensions

P.G. Estévez, 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 the...

**Research Article**

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

J. Fernández-Núñez, 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...

**Research Article**

## 26. 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...

**Research Article**

## 27. Isochronous Systems and Perturbation Theory

J-P Françoise

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.

**Research Article**

## 28. 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 a...

**Research Article**

## 29. 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.

**Research Article**

## 30. 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...

**Research Article**

## 31. 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...

**Research Article**

## 32. 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 V V and...

**Research Article**

## 33. 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 obtained...

**Research Article**

## 34. 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...

**Research Article**

## 35. 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 systems...

**Research Article**

## 36. 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 classical...

**Research Article**

## 37. 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...

**Research Article**

## 38. 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...

**Research Article**

## 39. Rigorous Results in the Scaling Theory of Irreversible Aggregation Kinetics

François 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...

**Research Article**

## 40. 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.

**Research Article**

## 41. 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...

**Research Article**

## 42. 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 a...

**Research Article**

## 43. 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...

**Research Article**

## 44. 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.

**Research Article**

## 45. 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...

**Research Article**

## 46. 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.

**Research Article**

## 47. 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...

**Research Article**

## 48. 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. LKdV)....

**Research Article**

## 49. 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.

**Research Article**

## 50. 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).

**Research Article**

## 51. 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.

**Research Article**

## 52. 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...

**Research Article**

## 53. 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...

**Research Article**

## 54. 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...

**Research Article**

## 55. 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.