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...
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...
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...
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
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.
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.
Pages: 74 - 85
We give a new derivation and characterisation of the generalised elliptic genus of
Krichever-Höhn by means of a functional equation.
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
) for some
p > n (n = 1 or 3). An extension...
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.
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...
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...
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...
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
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...
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
or U(x) = 1
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...
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.
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:...
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...
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...
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
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...
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...
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...
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.
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...
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.
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...
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...
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
should also consider the decompositions of...
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...
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...
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...
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...
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...
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...
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...
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
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...
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 . Here we show that the equations of motions admit a five-dimensional
Lie symmetry algebra and can be reduced to...
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...
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.
Pages: 535 - 547
In  we have proved a 1-1 correspondence between all separable coordinates on Rn
(according to Kalnins and Miller ) 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...
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.
Pages: 565 - 598
In a previous paper (Regular and Chaotic Dynamics 7 (2002), 351391, Ref. ),
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. . Crucial...
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....
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.
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
and agrees well at Q 1 GeV with the MS s(Q).
Pages: 633 - 647
Time-discretized versions of F. Calogero's rational and hyperbolic "goldfish" systems
are presented, and their exact solutions are given.
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...
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...
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...
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.