title: Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz

abstract:
  We prove that the solution to a pair of nonlinear integral equations arising
in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent
kernel of the linear integral operator with kernel
exp(-u(theta)-u(theta'))/cosh[(1/2)(theta-theta')]


title:
Nonisospectral symmetries of the KdV equation and the corresponding
  symmetries of the Whitham equations

abstract:
  In our paper we construct a new infinite family of symmetries of the Whitham
equations (averaged Korteveg-de-Vries equation). In contrast with the ordinary
hydrodynamic-type flows these symmetries are nonhomogeneous (i.e. they act
nontrivially at the constant solutions), are nonlocal, explicitly depend upon
space and time coordinates and form a noncommutative algebra, isomorphic to the
algebra of the polynomial vector fields in the complex plane (Virasoro algebra
with the zero central charge).


title: Classical Poisson structures and r-matrices from constrained flows

abstract:
  We construct the classical Poisson structure and $r$-matrix for some finite
dimensional integrable Hamiltonian systems obtained by constraining the flows
of soliton equations in a certain way. This approach allows one to produce new
kinds of classical, dynamical Yang-Baxter structures. To illustrate the method
we present the $r$-matrices associated with the constrained flows of the
Kaup-Newell, KdV, AKNS, WKI and TG hierarchies, all generated by a
2-dimensional eigenvalue problem. Some of the obtained $r$-matrices depend only
on the spectral parameters, but others depend also on the dynamical variables.
For consistency they have to obey a classical Yang-Baxter-type equation,
possibly with dynamical extra terms.


title: Differential substitutions and symmetries of hyperbolic equations

abstract:
  There are considered differential substitutions of the form $v=P(x,u,u_{x})$
for which there exists a differential operator $H=\sum^{k}_{i=0} \alpha_{i}
D^{i}_{x}$ such that the differential substitution maps the equation
$u_{t}=H[s(x,P,D_{x}(P),...,D^{k}_{x}(P))]$ into an evolution equation for any
function $s$ and any nonnegative integer $k$. All differential substitutions of
the form $v=P(x,u,u_{x})$ known to the author have this property. For example,
the well-known Miura transformation $v=u_{x}-u^{2}$ maps any equation of the
form $$u_{t}=(D^{2}_{x}+2uD_{x}+2u_{x})
[s(x,u_{x}-u^{2},D_{x}(u_{x}-u^{2}),...,D^{k}_{x}(u_{x}-u^{2}))]$$ into the
equation $$v_{t}=(D^{3}_{x}+4vD_{x}+2v_{x})[s(x,v,{{\partial v}\over{\partial x
}},...,{{\partial^{k} v}\over{\partial x^{k}}})].$$ The complete classification
of such differential substitutions is given. An infinite set of the pairwise
nonequivalent differential substitutions with the property mentioned above is
constructed. Moreover, a general result about symmetries and invariant
functions of hyperbolic equations is obtained.


title: On Orthogonal and Symplectic Matrix Ensembles

abstract:
  The focus of this paper is on the probability, $E_\beta(0;J)$, that a set $J$
consisting of a finite union of intervals contains no eigenvalues for the
finite $N$ Gaussian Orthogonal ($\beta=1$) and Gaussian Symplectic ($\beta=4$)
Ensembles and their respective scaling limits both in the bulk and at the edge
of the spectrum. We show how these probabilities can be expressed in terms of
quantities arising in the corresponding unitary ($\beta=2$) ensembles. Our most
explicit new results concern the distribution of the largest eigenvalue in each
of these ensembles. In the edge scaling limit we show that these largest
eigenvalue distributions are given in terms of a particular Painlev\'e II
function.


title: BiHamiltonian Formulations of the Bateman Equation

abstract:
  We discuss a class of evolution equations equivalent to the simplest
Universal Field Equation, the so--called Bateman equation, and show that all of
them possess (at least) biHamiltonian structure. The first few conserved
charges are calculated.


title:
The separability and dynamical $r$-matrix for the constrained flows of
  Jaulent-Miodek hierarchy

abstract:
  We show here the separability of Hamilton-Jacobi equation for a hierarchy of
integrable Hamiltonian systems obtained from the constrained flows of the
Jaulent-Miodek hierarchy. The classical Poisson structure for these Hamiltonian
systems is constructed. The associated $r$-matrices depend not only on the
spectral parameters, but also on the dynamical variables and, for consistency,
have to obey the classical Yang-Baxter equations of dynamical type. Some new
solutions of classical dynamical Yang-Baxter equations are presented. Thus
these integrable systems provide examples both for the dynamical $r$-matrix and
for the separable Hamiltonian system not having a natural Hamiltonian form.


title:
Nonsingularity of the direct scattering transform for the KP-2 equation
  with real exponentially decaying at infinity potential

abstract:
  We study the direct spectral transform for the heat equation, associated with
the KP-2 equation. We show, that for real nonsingular exponentially decaying at
infinity potentials the direct problem is nonsingular for arbitrary large
potentials. Earlier this statement was proved only for potentials, satisfying
the ``small norm'' assumption.


title:
Nonlinear Behaviour of Time-Stepping Algorithms for Initial Value
  Problems

abstract:
  Recent advances in nonlinear dynamical systems theory provide a new insight
into numerical properties of discrete algorithms developed to solve nonlinear
initial value problems. Basic features like accuracy and stability are well
pointed out through diagrams or maps of computed asymptotic solutions in a
suitable parametric space. Applying this methodology to a nonlinear test
equation, we compared some numerical features of the well known second-order
Crank-Nicolson solver with those of a recent proposed version which is
fourth-order accurate. The approach gives some useful indication on the
capabilities of familiar and innovative ODE integrators when applied to
nonlinear problems.


title: Peakons, R-Matrix and Toda-Lattice

abstract:
  The integrability of a family of hamiltonian systems, describing in a
particular case the motionof N ``peakons" (special solutions of the so-called
Camassa-Holm equation) is established in the framework of the $r$-matrix
approach, starting from its Lax representation. In the general case, the
$r$-matrix is a dynamical one and has an interesting though complicated
structure. However, for a particular choice of the relevant parameters in the
hamiltonian (the one corresponding to the pure ``peakons" case), the $r$-matrix
becomes essentially constant, and reduces to the one pertaining to the finite
(non-periodic) Toda lattice. Intriguing consequences of such property are
discussed and an integrable time discretisation is derived.


title: On Integrable Doebner-Goldin Equations

abstract:
  We suggest a method for integrating sub-families of a family of nonlinear
{\sc Schr\"odinger} equations proposed by {\sc H.-D.~Doebner} and {\sc
G.A.~Goldin} in the 1+1 dimensional case which have exceptional {\sc Lie}
symmetries. Since the method of integration involves non-local transformations
of dependent and independent variables, general solutions obtained include
implicitly determined functions. By properly specifying one of the arbitrary
functions contained in these solutions, we obtain broad classes of explicit
square integrable solutions. The physical significance and some analytical
properties of the solutions obtained are briefly discussed.


title: Dispersionful analogues of Benney's equations and $N$-wave systems

abstract:
  We recall Krichever's construction of additional flows to Benney's hierarchy,
attached to poles at finite distance of the Lax operator. Then we construct a
``dispersionful'' analogue of this hierarchy, in which the role of poles at
finite distance is played by Miura fields. We connect this hierarchy with
$N$-wave systems, and prove several facts about the latter (Lax representation,
Chern-Simons-type Lagrangian, connection with Liouville equation,
$\tau$-functions).


title: Solution of SRS on the finite interval

abstract:
  The equations of transient stimulated Raman scattering on the finite interval
are solved by the spectral transform method on the semi-line. As the problem
has a free end, the pump and Stokes output at finite distance can be
constructed as the solution of a linear Cauchy-Green integral equation.


title:
Nontrivial scattering of localized solitons in a (2+1)-dimensional
  integrable system

abstract:
  One usually expects localized solitons in integrable systems to interact
trivially. There is an integrable (2+1)-dimensional chiral equation which
admits multi-soliton solutions with trivial dynamics. This paper describes how
to generate explicit solutions representing nontrivial soliton interactions: in
particular, a head-on collision of two solitons resulting in $90^\circ$
scattering.


title: Conserved quantities for integrable chiral equations in 2+1 dimensions

abstract:
  The integrable (2+1)-dimensional chiral equations are related to the
self-dual Yang-Mills equation. Previously-known nonlocal conservation laws do
not yield finite conserved charges, because the relevant spatial integrals
diverge. We exhibit infinite sequences of conserved quantities that do exist,
and have a simple explicit form.


title: Toda Lattice Hierarchy and Zamolodchikov's Conjecture

abstract:
  In this letter, we show that certain Fredholm determinant $D(\lambda;t)$,
introduced by Zamolodchikov in his study of 2D polymers, is a continuum limit
of soliton solution for the Toda lattice hierarchy with 2-periodic reduction
condition.


title: A discrete time relativistic Toda lattice

abstract:
  Four integrable symplectic maps approximating two Hamiltonian flows from the
relativistic Toda hierarchy are introduced. They are demostrated to belong to
the same hierarchy and to examplify the general scheme for symplectic maps on
groups equiped with quadratic Poisson brackets. The initial value problem for
the difference equations is solved in terms of a factorization problem in a
group. Interpolating Hamiltonian flows are found for all the maps.


title: The Orbit Method in the Finite Zone Integration Theory

abstract:
  A construction of integrable hamiltonian systems associated with different
graded realizations of untwisted loop algebras is proposed. Such systems have
the form of Euler - Arnold equations on orbits of loop algebras. The proof of
completeness of the integrals of motion is carried out independently of the
realization of the loop algebra. The hamiltonian systems obtained are shown to
coincide with hierarchies of higher stationary equations for some nonlinear
PDE's integrable by inverse scattering method.
  We apply the general scheme for the principal and homogeneous realizations of
the loop algebra $ sl_3(\R)\otimes{\cal P}(\lambda,\lambda^{-1}) $. The
corresponding equations on the degenerated orbit are interpreted as the
Boussinesq's and two-component modified KDV equations respectively. The scalar
Lax representation for the Boussinesq's equation is found in terms of
coordinates on the orbit applying the Drinfeld - Sokolov reduction procedure.


title: On Integrable Models and their Interrelations

abstract:
  We present an elementary discussion of the Calogero-Moser model. This gives
us an opportunity to illustrate basic concepts of the dynamical integrable
models. Some ideas are also presented regarding interconnections between
integrable models based on the relation established between the Calogero-Moser
model and the truncated KP hierarchy of Burgers-Hopf type.


title: The Gambier Mapping

abstract:
  We propose a discrete form for an equation due to Gambier and which belongs
to the class of the fifty second order equations that possess the Painleve
property. In the continuous case, the solutions of the Gambier equation is
obtained through a system of Riccati equations. The same holds true in the
discrete case also. We use the singularity confinement criterion in order to
study the integrability of this new mapping.


title: Discrete Painleve equations: coalescences, limits and degeneracies

abstract:
  Starting from the standard form of the five discrete Painlev\'e equations we
show how one can obtain (through appropriate limits) a host of new equations
which are also the discrete analogues of the continuous Painlev\'e equations. A
particularly interesting technique is the one based on the assumption that some
simplification takes place in the autonomous form of the mapping following
which the deautonomization leads to a new $n$-dependence and introduces more
new discrete Painlev\'e equations.


title:
A three-by-three matrix spectral problem for AKNS hierarchy and its
  binary Nonlinearization

abstract:
  A three-by-three matrix spectral problem for AKNS soliton hierarchy is
proposed and the corresponding Bargmann symmetry constraint involved in Lax
pairs and adjoint Lax pairs is discussed. The resulting nonlinearized Lax
systems possess classical Hamiltonian structures, in which the nonlinearized
spatial system is intimately related to stationary AKNS flows. These
nonlinearized Lax systems also lead to a sort of involutive solutions to each
AKNS soliton equation.


title: Discrete soliton equations and convergence acceleration algorithms

abstract:
  Some of the well-known convergence acceleration algorithms, when viewed as
two-variable difference equations, are equivalent to discrete soliton
equations. It is shown that the $\eta-$algorithm is nothing but the discrete
KdV equation. In addition, one generalized version of the $\rho-$algorithm is
considered to be integrable discretization of the cylindrical KdV equation.


title: Constructive building of the Lax pair in the non-linear sigma model

abstract:
  A derivation of the Lax pair for the (1+1)-dimensional non-linear sigma-model
is described. Its main benefit is to have a clearer physical origin and to
allow the study of a generalization to higher dimensions.


title:
A Novel Integration Scheme for Partial Differential Equations: an
  Application to the Complex Ginzburg-Landau Equation

abstract:
  A new integration scheme, combining the stability and the precision of usual
pseudo-spectral codes with the locality of finite differences methods, is
introduced. It turns out to be particularly suitable for the study of front and
disturbance propagation in extended systems. An application to the complex
Ginzburg-Landau equation shows the higher precision of this method with respect
to spectral ones.


title: Approximate Lie Group Analysis of Finite-difference Equations

abstract:
  Approximate group analysis technique, that is, the technique combining the
methodology of group analysis and theory of small perturbations, is applied to
finite-difference equations approximating ordinary differential equations.
Finite-difference equations are viewed as a system of algebraic equations with
a small parameter, introduced through the definitions of finite-difference
derivatives. It is shown that application of the approximate invariance
criterion to this algebraic system results in relations that can be viewed as
prolongation formulae and the invariance criterion for the differential
approximation of these finite-difference equations. This allows us to study the
group properties of the finite-difference equations by analyzing the group
properties of their differential approximations, which are the differential
equations with a small parameter. In particular, the question of whether the
group, admitted by the original differential equation, can be corrected by
adding the first-order perturbation to it, so that the resulting group with a
small parameter is approximately admitted by the finite-difference
approximation, is studied. It is shown by examples that, for a given
differential equation, its finite--difference approximation and the group, such
a correction may not always be possible. It is also demonstrated that the
finite--difference approximation can be modified in such a way that the
correction becomes possible.


title:
Explicit and Exact Solutions to a Kolmogorov-Petrovskii-Piskunov
  Equation

abstract:
  Some explicit traveling wave solutions to a Kolmogorov-Petrovskii-Piskunov
equation are presented through two ans\"atze. By a Cole-Hopf transformation,
this Kolmogorov-Petrovskii-Piskunov equation is also written as a bilinear
equation and further two solutions to describe nonlinear interaction of
traveling waves are generated. B\"acklund transformations of the linear form
and some special cases are considered.


title:
Classification of evolutionary equations on the lattice. I. The general
  theory

abstract:
  A modification of the symmetry approach for the classification of integrable
differential-difference equations of the form $$ u_{n,t} = f_n(u_{n-1}, u_n,
u_{n+1}), $$ where $n$ is a discrete integer variable, is presented (the
well-known Volterra and Toda equations can be written in this form). If before,
in the framework of the symmetry approach, only equations similar to $$ u_{n,t}
= f(u_{n-1}, u_n, u_{n+1}), $$ i.e. defined by a function $f$, were considered,
now we have an infinite set $f_n$ of a priori quite different functions.


title: Algebra of Non-Local Charges in Supersymmetric Non-Linear Sigma Models

abstract:
  We propose a graphic method to derive the classical algebra (Dirac brackets)
of non-local conserved charges in the two dimensional supersymmetric non-linear
$O(N)$ sigma model. As in the purely bosonic theory we find a cubic Yangian
algebra. We also consider the extension of graphic methods to other integrable
theories.


title: Properties of equations of the continuous Toda type

abstract:
  We study a modified version of an equation of the continuous Toda type in 1+1
dimensions. This equation contains a friction-like term which can be switched
off by annihilating a free parameter $\ep$. We apply the prolongation method,
the symmetry and the approximate symmetry approach. This strategy allows us to
get insight into both the equations for $\ep =0$ and $\ep \ne 0$, whose
properties arising in the above frameworks are mutually compared. For $\ep =0$,
the related prolongation equations are solved by means of certain series
expansions which lead to an infinite- dimensional Lie algebra. Furthermore,
using a realization of the Lie algebra of the Euclidean group $E_{2}$, a
connection is shown between the continuous Toda equation and a linear wave
equation which resembles a special case of a three-dimensional wave equation
that occurs in a generalized Gibbons-Hawking ansatz \cite{lebrun}. Nontrivial
solutions to the wave equation expressed in terms of Bessel functions are
determined.
  For $\ep\,\ne\,0,$ we obtain a finite-dimensional Lie algebra with four
elements. A matrix representation of this algebra yields solutions of the
modified continuous Toda equation associated with a reduced form of a
perturbative Liouville equation. This result coincides with that achieved in
the context of the approximate symmetry approach. Example of exact solutions
are also provided. In particular, the inverse of the exponential-integral
function turns out to be defined by the reduced differential equation coming
from a linear combination of the time and space translations. Finally, a Lie
algebra characterizing the approximate symmetries is discussed.


title: Remarks on the Whitham equations

abstract:
  We survey some topics involving the Whitham equations, concentrating on the
role of the product of the wave function and its adjoint in averaging and in
producing Cauchy kernels and differentials on Riemann surfaces. There are also
some new results.


title: A discrete time peakons lattice

abstract:
  A discretization of the peakons lattice is introduced, belonging to the same
hierarchy as the continuous--time system. The construction examplifies the
general scheme for integrable discretization of systems on Lie algebras with
$r$--matrix Poisson brackets. An initial value problem for the difference
equations is solved in terms of a factorization problem in a group.
Interpolating Hamiltonian flow is found. A variational (Lagrangian) formulation
is also given.


title: Binary nonlinearization for the Dirac systems

abstract:
  A Bargmann symmetry constraint is proposed for the Lax pairs and the adjoint
Lax pairs of the Dirac systems. It is shown that the spatial part of the
nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Liouville
integrable Hamiltonian system and that under the control of the spatial part,
the time parts of the nonlinearized Lax pairs and adjoint Lax pairs are
interpreted as a hierarchy of commutative, finite dimensional Liouville
integrable Hamiltonian systems whose Hamiltonian functions consist of a series
of integrals of motion for the spatial part. Moreover an involutive
representation of solutions of the Dirac systems exhibits their integrability
by quadratures. This kind of symmetry constraint procedure involving the
spectral problem and the adjoint spectral problem is referred to as a binary
nonlinearization technique like a binary Darboux transformation.


title: On evolution of multiphase nonlinear modulated waves

abstract:
  We present a fundamental solution to an initial value problem for the
KdV-Whitham system in an explicit integral form. Monotonically decreasing
initial data with finite number of breaking points are considered. Generating
function for the commuting flows of the averaged KdV hieararchy producing the
analytical solutions to the KdV-Whitham system is constructed.


title: Estabrook-Wahlquist Prolongations and Infinite-Dimensional Algebras

abstract:
  Detailed mappings between zero-curvture equations for prolongation structures
of nonlinear pde's and Estabrook-Wahlquist algorithms for same are given. The
differences are exemplified by studies of the sine-Gordon equation. An example
where the prolongation structure must be infinite-dimensional is given by the
Robinson-Trautman equation, where the minimal algebra is $K_2$. In general
these algorithms require integration of vector-field valued pde's; solutions of
simultaneous flow equations are given. Applications to coupled systems of flow
equations are given, where the result describes Lie algebras of vector fields
vertical over fibers of pseudopotentials over a jet bundle appropriate for a
given system of pde's; algebras invariant under sl(2,C) are of special
interest.


title:
On Discrete 3-Dimensional Equations Associated with the Local
  Yang-Baxter Relation

abstract:
  The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a
proper generalization to 3 dimensions of the zero curvature relation. Recently,
Korepanov has constructed an infinite set of integrable 3-dimensional lattice
models, and has related them to solutions to the local YBE. The simplest
Korepanov's model is related to the star-triangle relation in the Ising model.
In this paper the corresponding discrete equation is derived. In the continuous
limit it leads to a differential 3d equation, which is symmetric with respect
to all permutations of the three coordinates. A similar analysis of the
star-triangle transformation in electric networks leads to the discrete
bilinear equation of Miwa, associated with the BKP hierarchy. Some related
operator solutions to the tetrahedron equation are also constructed.


title: Integrable Models of the CFT on Hyper-Elliptic Surfaces

abstract:
  In this letter, we continue the work we started at a previous paper and we
propose new series of integrable models in quantum field theory. These models
are obtained as perturbed models of the minimal conformal field theories on the
hyper-elliptic surfaces by particular relevant operators of the untwisted
sector. The quantum group symmetry of the models is also discussed.


title: Discrete time Bogoyavlensky lattices

abstract:
  Discretizations of the Bogoyavlensky lattices are introduced, belonging to
the same hierarchies as the continuous--time systems. The construction
exemplifies the general scheme for integrable discretization of systems on Lie
algebras with $r$--matrix Poisson brackets. An initial value problem for the
difference equations is solved in terms of a factorization problem in a group.
Interpolating Hamiltonian flow is found.


title:
Constrained KP Hierarchies: Darboux-B\"acklund Solutions and Additional
  Symmetries

abstract:
  We illustrate the basic notions of {\em additional non-isospectral
symmetries} and their interplay with the discrete {\em \DB transformations} of
integrable systems at the instance of {\em constrained Kadomtsev-Petviashvili}
(\cKP) integrable hierarchies. As a main application we present the solution of
discrete multi-matrix string models in terms of Wronskian $\t$-functions of
graded $SL(m,1)$ \cKP hierarchies.


title: Hamiltonian Structures for the Generalized Dispersionless KdV Hierarchy

abstract:
  We study from a Hamiltonian point of view the generalized dispersionless KdV
hierarchy of equations. From the so called dispersionless Lax representation of
these equations we obtain three compatible Hamiltonian structures. The second
and third Hamiltonian structures are calculated directly from the r-matrix
approach. Since the third structure is not related recursively with the first
two ones the generalized dispersionless KdV hierarchy can be characterized as a
truly tri-Hamiltonian system.


title:
The Three-Wave Resonant Interaction: Deformation of the Plane-Wave
  Solutions and Darboux Transformations

abstract:
  The plane wave solutions of the three-wave resonant interaction in the plane
are considered. It is shown that rank-one constraints over the right
derivatives of invertible operators on an arbitrary linear space gives
solutions of the three-wave resonant interaction that can be understood as a
Darboux transformation of the plane wave solutions. The method is extended
further to obtain general Darboux transformations: for any solution of the
three-wave interaction problem and vector solutions of the corresponding Lax
pair large families of new solutions, expressed in terms of Grammian type
determinants of these vector solutions, are given.


title:
The Bi-Hamiltonian Structure of the Perturbation Equations of KdV
  Hierarchy

abstract:
  The bi-Hamiltonian structure is established for the perturbation equations of
KdV hierarchy and thus the perturbation equations themselves provide also
examples among typical soliton equations. Besides, a more general
bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a
generalization of the resulting perturbation equations to $1+2$ dimensions.


title:
Bilinear structure and Schlesinger transforms of the $q$-P$_{\rm III}$
  and $q$-P$_{\rm VI}$ equations

abstract:
  We show that the recently derived ($q$-) discrete form of the Painlev\'e VI
equation can be related to the discrete P$_{\rm III}$, in particular if one
uses the full freedom in the implementation of the singularity confinement
criterion. This observation is used here in order to derive the bilinear forms
and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm
VI}$.


title: Nonlinear Discrete Systems with Nonanalytic Dispersion Relations

abstract:
  A discrete system of coupled waves (with nonanalytic dispersion relation) is
derived in the context of the spectral transform theory for the Ablowitz Ladik
spectral problem (discrete version of the Zakharov-Shabat system). This 3-wave
evolution problem is a discrete version of the stimulated Raman scattering
equations, and it is shown to be solvable for arbitrary boundary value of the
two radiation fields and initial value of the medium state. The spectral
transform is constructed on the basis of the D-bar approach.


title: Localized Induction Hierarchy and Weingarten Systems

abstract:
  We describe a method of constructing Weingarten systems of triply orthogonal
coordinates, related to the localized induction equation hierarchy of
integrable geometric evolution equations


title: Some Classes of Solutions to the Toda Lattice Hierarchy

abstract:
  We apply an analogue of the Zakharov-Shabat dressing method to obtain
infinite matrix solutions to the Toda lattice hierarchy. Using an operator
transformation we convert some of these into solutions in terms of integral
operators and Fredholm determinants. Others are converted into a class of
operator solutions to the $l$-periodic Toda hierarchy.


title: Exact Solution of an Octagonal Random Tiling Model

abstract:
  We consider the two-dimensional random tiling model introduced by Cockayne,
i.e. the ensemble of all possible coverings of the plane without gaps or
overlaps with squares and various hexagons. At the appropriate relative
densities the correlations have eight-fold rotational symmetry. We reformulate
the model in terms of a random tiling ensemble with identical rectangles and
isosceles triangles. The partition function of this model can be calculated by
diagonalizing a transfer matrix using the Bethe Ansatz (BA). The BA equations
can be solved providing {\em exact} values of the entropy and elastic
constants.


title: Darboux Coordinates on Coadjoint Orbits of Lie Algebras

abstract:
  The method of constructing spectral Darboux coordinates on finite dimensional
coadjoint orbits in duals of loop algebras is applied to the one pole case,
where the orbit is identified with a coadjoint orbit in the dual of a finite
dimensional Lie algebra. The constructions are carried out explicitly when the
Lie algebra is $\frak{sl}(2,\bold R),\ \frak{sl}(3, \bold R),$ and
$\frak{so}(3, \bold R)$, and for rank two orbits in $\frak{so}(n, \bold R)$. A
new feature that appears is the possibility of identifying spectral Darboux
coordinates associated to ``dynamical" choices of sections of the associated
eigenvector line bundles; i.e. sections that depend on the point within the
given orbit.


title: From Discrete to Continuous Painlev\'e Equations: A Bilinear Approach

abstract:
  We present the bilinear forms of the (continuous) Painlev\'e equations
obtained from the continuous limit of the analogous expresssions for the
discrete ones. The advantage of this method is that it leads to very
symmetrical results. A new and interesting result is the bilinearization of the
P$_{\rm VI}$ equation, something that was missing till now.


title: Cellular Automata and Ultra-Discrete Painlev\'e Equations

abstract:
  Starting from integrable cellular automata we present a novel form of
Painlev\'e equations. These equations are discrete in both the independent
variable and the dependent one. We show that they capture the essence of the
behavior of the Painlev\'e equations organizing themselves into a coalescence
cascade and possessing special solutions. A necessary condition for the
integrability of cellular automata is also presented.


title: Lax representation for two--particle dynamics splitting on two tori

abstract:
  Lax representation in terms of $2\times 2$ matrices is constructed for a
separable multiply--periodic system splitting on two tori. Hyperelliptic
Kleinian functions and their reduction to elliptic functions are used.


title: Hyperelliptic Kleinian functions and applications

abstract:
  We develop the theory of hyperelliptic Kleinian functions. As applications we
consider construction of the explicit matrix realization of the hyperelliptic
Kummer varieties, differential operators to have the hyperelliptic curve as
spectral variety, solution of the KdV equations by Kleinian functions.


title: Dynamical r-matrix for the elliptic Ruijsenaars-Schneider system

abstract:
  The classical r-matrix structure for the generic elliptic
Ruijsenaars-Schneider model is presented. It makes the integrability of this
model as well as of its discrete-time version that was constructed in a recent
paper manifest.


title: The Hubbard model on a complete graph: Exact Analytical results

abstract:
  We derive the analytical expression of the ground state of the Hubbard model
with unconstrained hopping at half filling and for arbitrary lattice sites.


title: Ferromagnetic ground states of the Hubbard model on a complete graph

abstract:
  We use group theory to derive the exact analytical expression of the
ferromagnetic ground states of the Hubbard model on a complete graph for
arbitrary lattice sites f and for arbitrary fillings $N$. We find that for
$t>0$ and for $N=f+1$ the ground state is maximally ferromagnetic with total
spin $S=(f-1)/2$. For $N > f+1$ the ground state is still ferromagnetic but
becomes degenerate with respect to $S$.


title: SO(4) invariant basis functions for strongly correlated Fermi systems

abstract:
  We show how to construct SO(4) invariant functions for strongly correlated
Fermi systems on lattices of finite sizes. We illustrate the method on the case
of the 1D Hubbard chain with four and with six sites.


title:
Approximate Controlability by Control Constraints for Infinite
  Dimensional Systems

abstract:
  For linear infinite systems the approximate controllability problem by
control constraints is considered. Controllability conditions represented via
system parameters are obtained. Partial differential control systems and
control systems with delays are considered as an example.


title:
Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are
  governed by the same r-matrix

abstract:
  We demonstrate that in a certain gauge the elliptic Ruijsenaars--Schneider
models admit Lax representation governed by the same dynamical $r$--matrix as
their non--relativistic counterparts (Calogero--Moser models). This phenomenon
was previously observed for the rational and hyperbolic models.


title:
Non-perturbative non-integrability of non-homogeneous nonlinear lattices
  induced by non-resonance hypothesis

abstract:
  We prove the non-integrability (non-existence of additional analytic
conserved quantities other than Hamiltonian) for Fermi-Pasta-Ulam (FPU)
lattices by virtue of Lyapunov-Kovalevskaya- -Ziglin-Yoshida's monodromy method
about the variational equations. The key to this analysis is that the normal
variational equations along a certain solution happen to be in a type of Lam\'e
equations. We also introduce the classification problem towards non-homogeneous
nonlinear lattices including FPU lattices using non-integrability preserving
transformation.


title: Farey Tree and the Frenkel-Kontorova Model

abstract:
  We solved the Frenkel-Kontorova model with the potential $V(u)= -\frac{1}{2}
|\lambda|(u-{\rm Int}[u]-\frac{1}{2})^2$ exactly. For given $|\lambda|$, there
exists a positive integer $q_c$ such that for almost all values of the tensile
force $\sigma$, the winding number $\omega$ of the ground state configuration
is a rational number in the $q_c$-th level Farey tree. For fixed $\omega=p/q$,
there is a critical $\lambda_c$ when a first order phase transition occurs.
This phase transition can be understood as the dissociation of a large molecule
into two smaller ones in a manner dictated by the Farey tree. A kind of
``commensurate-incommensurate'' transition occurs at critical values of
$\sigma$ when two sizes of molecules co-exist. ``Soliton'' in the usual sense
does not exist but induces a transformation of one size of molecules into the
other.


title: Analytic Structure of the Landau-Ginzburg Equation in 2+1 Dimensions

abstract:
  In this paper, two methods are employed to investigate for which values of
the parameters, if any, the two-dimensional real Landau-Ginzburg equation
possesses the Painleve property. For an ordinary differential equation to have
the Painleve property all of its solutions must be meromorphic but for partial
differential equations there are two inequivalent definitions, one a direct
investigation of a Laurent series expansion and the other indirect and relying
on a knowledge of the continuous symmetry group of the equation. We check both
methods for the Landau-Ginzburg equation in 2+1 dimensions and each one yields
that this equation does not possess the Painleve property for any values of the
parameters.


title: Algebraic properties of the 1+1 dimensional Heisenberg spin field model

abstract:
  The Estabrook-Wahlquist prolongation method is applied to the (compact and
noncompact) continuous isotropic Heisenberg model in 1 + 1 dimensions. Using a
special realization (an algebra of the Kac-Moody type) of the arising
incomplete prolongation Lie algebra, a whole family of nonlinear field
equations containing the original Heisenberg system is generated.


title: Integrable Theory of the Perturbation Equations

abstract:
  An integrable theory is developed for the perturbation equations engendered
from small disturbances of solutions. It includes various integrable properties
of the perturbation equations: hereditary recursion operators, master
symmetries, linear representations (Lax and zero curvature representations) and
Hamiltonian structures etc. and provides us a method to generate hereditary
operators, Hamiltonian operators and symplectic operators starting from the
known ones. The resulting perturbation equations give rise to a sort of
integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP
equation) are carefully carried out.


title:
Painlev\'e Transcendent Describes Quantum Correlation Function of the
  XXZ Antiferromagnet away from the free-fermion point

abstract:
  We consider quantum correlation functions of the antiferromagnetic
spin-$\frac{1}{2}$ Heisenberg XXZ spin chain in a magnetic field. We show that
for a magnetic field close to the critical field $h_c$ (for the critical
magnetic field the ground state is ferromagnetic) certain correlation functions
can be expressed in terms of the solution of the Painlev\'e V transcendent.
This establishes a relation between solutions of Painlev\'e differential
equations and quantum correlation functions in models of {\sl interacting}
fermions. Painlev\'e transcendents were known to describe correlation functions
in models with free fermionic spectra.


title: Integrable discretizations of the spin Ruijsenaars-Schneider models

abstract:
  Integrable discretizations are introduced for the rational and hyperbolic
spin Ruijsenaars--Schneider models. These discrete dynamical systems are
demonstrated to belong to the same integrable hierarchies as their
continuous--time counterparts. Explicit solutions are obtained for arbitrary
flows of the hierarchies, including the discrete time ones.


title: Darboux Transformations for a Lax Integrable System in $2n$-Dimensions

abstract:
  A $2n$-dimensional Lax integrable system is proposed by a set of specific
spectral problems. It contains Takasaki equations, the self-dual Yang-Mills
equations and its integrable hierarchy as examples. An explicit formulation of
Darboux transformations is established for this Lax integrable system. The
Vandermonde and generalized Cauchy determinant formulas lead to a description
for deriving explicit solutions and thus some rational and analytic solutions
are obtained.


title: The Painlev\'e Test of Higher Dimensional KdV Equation

abstract:
  We argue the integrability of the generalized KdV(GKdV) equation using the
Painlev\'e test. For $d( \le 2)$ dimensional space, GKdV equation passes the
Painlev\'e test but does not for $d \geq 3$ dimensional space. We also apply
the Ablowitz-Ramani-Segur's conjecture to the GKdV equation in order to
complement the Painlev\'e test.


title:
Classification of Integrable Evolution Equations of the Form
  $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$

abstract:
  We obtain the classification of integrable equations of the form
$u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ using the formal symmetry method of Mikhailov
et al [A.V.Mikhailov, A.B.Shabat and V.V.Sokolov, in {\it What is
Integrability} edited by V.E. Zakharov (Springer-Verlag, Berlin 1991)]. We show
that all such equations can be transformed to an integrable equation of the
form $v_t=v_{xxx}+f(v,v_x,v_{xx})$ using transformations
$\Phi(x,t,u,v,u_x,v_x)=0$, and the $u_{xx}$ dependence can be eliminated except
for two equations.


title: A stability analysis for the Korteweg-de Vries equation

abstract:
  In this paper the stability of the Korteweg-de Vries (KdV) equation is
investigated. It is shown analytically and numerically that small perturbations
of solutions of the KdV-equation introduce effects of dispersion, hence the
perturbation propagates with a different velocity then the unperturbed
solution. This effect is investigated analytically by formulating a
differential equation for perturbations of solutions of the KdV-equation. This
differential equation is solved generally using an Inverse Scattering Technique
(IST) using the continuous part of the spectrum of the Schr\"{o}dinger
equation. It is shown explicitly that the perturbation consist of two parts.
The first part represents the time-evolution of the perturbation only. The
second part represents the interaction between the perturbation and the
unperturbed solution. It is shown explicitly that singular non-dispersive
solutions of the KdV-equation are unstable.


title: New integrable systems related to the relativistic Toda lattice

abstract:
  New integrable lattice systems are introduced, their different integrable
discretization are obtained. B\"acklund transformations between these new
systems and the relativistic Toda lattice (in the both continuous and discrete
time formulations) are established.


title:
Higher Order SUSY in Quantum Mechanics and Integrability of
  Two-dimensional Hamiltonians

abstract:
  The new method based on the SUSY algebra with supercharges of higher order in
derivatives is proposed to search for dynamical symmetry operators in 2-dim
quantum and classical systems. These symmetry operators arise when closing the
SUSY algebra for a wide set of potentials. In some cases they are of 2-nd order
in derivatives. The particular solutions are obtained also for potentials
accepting symmetry operators of 4-th order. The investigation of quasiclassical
limit of the SUSY algebra yields new classical integrals of motion for a
certain type of systems which are polynomials of 4-th order in momenta. The
general SUSY-inspired algorithm to construct classical systems with additional
integrals of motion is outlined.


title:
On a class of rational and mixed soliton-rational solutions of Toda
  lattice

abstract:
  A class of rational solutions of Toda lattice satisfying certain Backlund
transformations and a class of mixed rational-soliton solutions (quasisolitons)
in wronskian formare obtained using the method of Ablowitz and Satsuma. Also an
extended class of rational solutions are found using an appropriate recursion
relation. They are also solutions of Boussinesq equation and it is conjectured
that there is a larger class of common solutions of both equations.


title: Binary Nonlinearization of Lax Pairs

abstract:
  A kind of Bargmann symmetry constraints involving Lax pairs and adjoint Lax
pairs is proposed for soliton hierarchy. The Lax pairs and adjoint Lax pairs
are nonlinearized into a hierarchy of commutative finite dimensional integrable
Hamiltonian systems and explicit integrals of motion may also be generated. The
corresponding binary nonlinearization procedure leads to a sort of involutive
solutions to every system in soliton hierarchy which are all of finite gap. An
illustrative example is given in the case of AKNS soliton hierarchy.


title: A new integrable system related to the Toda lattice

abstract:
  A new integrable lattice system is introduced, and its integrable
discretizations are obtained. A B\"acklund transformation between this new
system and the Toda lattice, as well as between their discretizations, is
established.


title: On completeness of the Moutard transformations

abstract:
  In this paper we solve positively the problem of (local) density of the
"potentials" M(x,y) of the Moutard equation, u_{xy} = M(x,y) u, u=u(x,y), (used
in many papers for construction of exact solutions of (2+1)-dimensional
integrable systems) obtainable from a given initial potential with consecutive
Moutard transformations.


title:
On completeness of the Ribaucour transformations for triply orthogonal
  curvilinear coordinate systems in R^3

abstract:
  In this paper we solve positively the problem of (local) density of solutions
of the (2+1)-dimentional integrable system describing triply orthogonal
curvilinear coordinates in R^3 (a (2+1)-dimensional generalization of the
3-wave system) obtainable from a given initial solution with consecutive
B\"acklund transformations (called Ribaucour transformations in classical
differential geometry) in the space of all solutions of the system in question.


title:
On superposition of the autoBaecklund transformations for
  (2+1)-dimensional integrable systems

abstract:
  The usual superposition formulas for Baecklund transformations of
(2+1)-dimensional integrable systems include quadratures unlike the well known
case of (1+1)-dimensional inegrable systems where the fourth solution is found
with algebraic operations. In the present paper we show how in the case of
(2+1)-dimensional integrable systems one can find an extended formula of
nonlinear superposition such that the resulting solution will be found uniquely
from the given previous solution with algebraic operations.


title: Symmetries of KdV and Loop Groups

abstract:
  A simple version of the Segal-Wilson map from the SL(2,C) loop group to a
class of solutions of the KdV hierarchy is given, clarifying certain aspects of
this map. It is explained how the known symmetries, including Backlund
transformations, of KdV arise from simple, field independent, actions on the
loop group. A variety of issues in understanding the algebraic structure of
Backlund transformations are thus resolved.


title: Some survey remarks on Whitham theory and EM duality

abstract:
  The nature of the BA function and its adjoint for KP-Toda is traced through
the averaging method in generating the Whitham equations, differentials, and
symplectic forms, with connections to EM duality.


title:
On the geometry of certain isospectral sets in the full Kostant-Toda
  lattice

abstract:
  We use momentum mappings on generalized flag manifolds and their momentum
polytopes to study the geometry of the level sets of the 1-chop integrals of
the full Kostant-Toda lattice in certain isospectral submanifolds of the phase
space. We derive expressions for these integrals in terms of Pl\"ucker
coordinates on the flag manifold in the case that all eigenvalues are zero and
compare the geometry of the base locus of their level set varieties with the
corresponding geometry for distinct eigenvalues. Finally, we illustrate and
extend our results in the context of the full sl(3,C) and sl(4,C) Kostant-Toda
lattices.


title:
Generalized integrable hierarchies and Combescure symmetry
  transformations

abstract:
  Unifying hierarchies of integrable equations are discussed. They are
constructed via generalized Hirota identity. It is shown that the Combescure
transformations, known for a long time for the Darboux system and having a
simple geometrical meaning, are in fact the symmetry transformations of
generalized integrable hierarchies. Generalized equation written in terms of
invariants of Combescure transformations are the usual integrable equations and
their modified partners. The KP-mKP, DS-mDS hierarchies and Darboux system are
considered.


title: Loop algebras, gauge invariants and a new completely integrable system

abstract:
  One fruitful motivating principle of much research on the family of
integrable systems known as ``Toda lattices'' has been the heuristic assumption
that the periodic Toda lattice in an affine Lie algebra is directly analogous
to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper
shows that the analogy is not perfect. A discrepancy arises because the natural
generalization of the structure theory of finite-dimensional simple Lie
algebras is not the structure theory of loop algebras but the structure theory
of affine Kac-Moody algebras. In this paper we use this natural generalization
to construct the natural analog of the nonperiodic Toda lattice. Surprisingly,
the result is not the periodic Toda lattice but a new completely integrable
system on the periodic Toda lattice phase space. This integrable system is
prescribed purely in terms of Lie-theoretic data. The commuting functions are
precisely the gauge-invariant functions one obtains by viewing elements of the
loop algebra as connections on a bundle over $S^1$.


title:
On a generalization of the Fay-Sato identity for KP Baker functions and
  its application to constrained hierarchies

abstract:
  Some new formulas for the KP hierarchy are derived from the differential Fay
identity. They proved to be useful for the $k$-constrained hierarchies
providing a series of determinant identities for them. A differential equation
is introduced which is called ``universal" since it plays an important role for
all the $k$-constrained hierarchies. In the cases $k=1,2$ and 3 explicit
formulas are presented, in all the others recurrence relations are given which
enable one to obtain the identities.


title: A Note on Fractional KdV Hierarchies

abstract:
  We introduce a hierarchy of mutually commuting dynamical systems on a finite
number of Laurent series. This hierarchy can be seen as a prolongation of the
KP hierarchy, or a ``reduction'' in which the space coordinate is identified
with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV
hierarchies are gotten by means of further reductions, obtained by constraining
the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are
discussed in detail.


title:
Exact Solution of Frenkel-Kontorova Models with a Complete Devil's
  Staircase in Higher Dimensions

abstract:
  We solve exactly a class of Frenkel-Kontorova models with piecewise parabolic
potential, which has $d$ sub-wells in a period. With careful analysis, we show
that the phase diagram of the minimum enthalpy configurations exhibits the
structure of a complete $d$-dimensional devil's staircase. The winding number
of a minimum enthalpy configuration is locked to rational values, while the
fraction of atoms in each sub-well is locked to values which are
sub-commensurable with the winding number.


title:
Determinant Structure of the Rational Solutions for the Painlev\'e II
  Equation

abstract:
  Two types of determinant representations of the rational solutions for the
Painlev\'e II equation are discussed by using the bilinear formalism. One of
them is a representation by the Devisme polynomials, and another one is a
Hankel determinant representation. They are derived from the determinant
solutions of the KP hierarchy and Toda lattice, respectively.


title:
Coupled Integrable Systems Associated with a Polynomial Spectral Problem
  and their Virasoro Symmetry Algebras

abstract:
  An isospectral hierarchy of commutative integrable systems associated with a
polynomial spectral problem is proposed. The resulting hierarchy possesses a
recursion structure controlled by a hereditary operator. The nonisospectral
flows generate the time first order dependent symmetries of the isospectral
hierarchy, which constitute Virasoro symmetry algebras together with
commutative symmetries.


title: A Convexity Theorem in the Scattering Theory for the Dirac Operator

abstract:
  The Dirac operator enters into zero curvature representation for the cubic
nonlinear Schr\"{o}dinger equation. We introduce and study a conformal map from
the upper half-plane of the spectral parameter of the Dirac operator into
itself. The action variables turn out to be limiting boundary values of the
imaginary part of this map. We describe the image of the momentum map
(convexity theorem) in the simplest case of a potential from the Schwartz
class. We apply this description to the invariant manifolds for the nonlinear
Schr\"{o}dinger equation.


title:
Some further curiosities from the world of integrable lattice systems
  and their discretizations

abstract:
  Unexpected relations are found between the Toda lattice, the relativistic
Toda lattice and the Bruschi--Ragnisco lattice, as well as between their
integrable discretizations.


title:
A Local Asymptotic Analysis of the First Discrete Painlev\'e Equation as
  the Discrete Independent Variable Approaches Infinity

abstract:
  The first discrete Painlev\'e equation (dPI), which appears in a model of
quantum gravity, is an integrable nonlinear nonautonomous difference equation
which yields the well known first Painlev\'e equation (PI) in a continuum
limit. The asymptotic study of its solutions as the discrete time-step
$n\to\infty$ is important both for physical application and for checking the
accuracy of its role as a numerical discretization of PI. Here we show that the
asymptotic analysis carried out by Boutroux (1913) for PI as its independent
variable approaches infinity can also be achieved for dPI as its discrete
independent variable approaches the same limit.


title:
The Radius of Convergence and the Well-Posedness of the Painlev\'e
  Expansions of the Korteweg-deVries equation

abstract:
  In this paper we obtain explicit lower bounds for the radius of convergence
of the Painlev\'e expansions of the Korteweg-de-Vries equation around a movable
singularity manifold ${\Cal S}$ in terms of the sup norms of the arbitrary
functions involved. We use this estimate to prove the well-posedness of the
singular Cauchy problem on ${\Cal S}$ in the form of continuous dependence of
the meromorphic solution on the arbitrary data.


title: Gibbs' States for Moser-Calogero Potentials

abstract:
  We present two independent approaches for computing the thermodynamics for
classical particles interacting via the Moser--Calogero potential. Combining
the results we propose the form of equation of state or, what is equivalent,
the asymptotics of the Jacobian between volume elements corresponding two
symplectic structures on the phase space.


title:
On the dynamics of rational solutions for 1-D generalized Volterra
  system

abstract:
  The Hirota bilinear formalism and soliton solutions for a generalized
Volterra system is presented. Also, starting from the soliton solutions, we
obtain a class of nonsingular rational solutions using the "long wave" limit
procedure of Ablowitz and Satsuma, and appropriate "gauge" transformations.
Their properties are also discussed and it is shown that these solutions
interact elastically with no phase shift.


title: Equations with an infinite number of explicit Conservation Laws

abstract:
  A large class of first order partial nonlinear differential equations in two
independent variables which possess an infinite set of polynomial conservation
laws derived from an explicit generating function is constructed. The conserved
charge densities are all homogeneous polynomials in the unknown functions which
satisfy the differential equations in question.
  The simplest member of the class of equations is related to the Born-Infeld
equation in two dimensions. It is observed that some members of this class
possess identical charge densities. This enables the construction of a set of
multivariable equations with an infinite number of conservation laws.


title: Binary Nonlinearization of Lax pairs of Kaup-Newell Soliton Hierarchy

abstract:
  Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different
from the original ones. Binary nonlinearization procedure corresponding to the
Bargmann symmetry constraint is carried out for those Lax pairs. The proposed
Lax pairs together with adjoint Lax pairs are constrained as a hierarchy of
commutative, finite dimensional integrable Hamiltonian systems in the Liouville
sense, which also provides us with new examples of finite dimensional
integrable Hamiltonian systems. A sort of involutive solutions to the
Kaup-Newell hierarchy are exhibited through the obtained finite dimensional
integrable systems and the general involutive system engendered by binary
nonlinearization is reduced to a specific involutive system generated by
mono-nonlinearization.


title: The Calogero-Sutherland Model and Generalized Classical Polynomials

abstract:
  Multivariable generalizations of the classical Hermite, Laguerre and Jacobi
polynomials occur as the polynomial part of the eigenfunctions of certain
Schr\"odinger operators for Calogero-Sutherland-type quantum systems. For the
generalized Hermite and Laguerre polynomials the multidimensional analogues of
many classical results regarding generating functions, differentiation and
integration formulas, recurrence relations and summation theorems are obtained.
We use this and related theory to evaluate the global limit of the ground state
density, obtaining in the Hermite case the Wigner semi-circle law, and to give
an explicit solution for an initial value problem in the Hermite and Laguerre
case.


title: Fusion Hierarchies with Open Boundaries and Exactly Solvable Models

abstract:
  The formulation of integrable models with open boundary conditions and the
functional relations of fused transfer matrices are discussed. It is shown that
finite-size corrections to the transfer matrices and unitarity relations of
free energies can be obtained from the functional relations. Unitarity
relations of surface free energies presented in previous papers are also
reviewed.


title: On Simplest Hamiltonian Systems

abstract:
  Simple Hamiltonian systems, such as mathematical pendulum or Euler equations
for rigid body, are solved without computation. It is nothing but a joke but
maybe you will find it nice.


title: A geometrical method towards first integrals for dynamical systems

abstract:
  We develop a method, based on Darboux' and Liouville's works, to find first
integrals and/or invariant manifolds for a physically relevant class of
dynamical systems, without making any assumption on these elements' form. We
apply it to three dynamical systems: Lotka--Volterra, Lorenz and Rikitake.


title: Discrete version of the Chazy class III equation

abstract:
  We study the discretisation of the Chazy class III equation by two means: a
discrete Painlev\'e test, and the preservation of a two-parameter solution to
the continuous equation. We get that way a best discretisation scheme.


title: Common Algebraic Structure for the Calogero-Sutherland Models

abstract:
  We investigate common algebraic structure for the rational and trigonometric
Calogero-Sutherland models by using the exchange-operator formalism. We show
that the set of the Jack polynomials whose arguments are Dunkl-type operators
provides an orthogonal basis for the rational case.

categories: solv-int cond-mat hep-th math.QA nlin.SI q-alg


title: Nambu tensors and commuting vector fields

abstract:
  Takhtajan has recently studied the consistency conditions for Nambu brackets,
and suggested that they have to be skew-symmetric, and satisfy Leibnitz rule
and the Fundamental Identity (FI, it is a generalization of the Jacobi
identity). If the n'th order Nambu brackets in dimension N is written in terms
of the Nambu tensor \eta, the FI implies two conditions on it, one algebraic
and one differential. The algebraic part of FI implies decomposability of \eta
and in this letter we show that the Nambu bracket can then be written in terms
of the usual totally antisymmetric n-dimensional tensor and n vector fields D.
Our main result is that the differential part of the FI is satisfied iff the
vector fields D commute. Examples are provided by integrable Hamiltonian
systems. It turns out that then the Nambu bracket itself guarantees that the
motions stays on the manifold defined by the constants of motion of the
integrable system, while the n-1 Nambu Hamiltonians determine the (possibly
non-integrable) motion on this manifold.


title:
Toda lattices with indefinite metric II: Topology of the iso-spectral
  manifolds

abstract:
  We consider the iso-spectral real manifolds of tridiagonal Hessenberg
matrices with real eigenvalues. The manifolds are described by the iso-spectral
flows of indefinite Toda lattice equations introduced by the authors [Physica,
91D (1996), 321-339]. These Toda lattices consist of $2^{N-1}$ different
systems with hamiltonians $H = (1/2) \sum_{k=1}^{N} y_k^2 + \sum_{k=1}^{N-1}
s_ks_{k+1} \exp(x_k-x_{k+1})$, where $s_i=\pm 1$. We compactify the manifolds
by adding infinities according to the Toda flows which blow up in finite time
except the case with all $s_is_{i+1}=1$. The resulting manifolds are shown to
be nonorientable for $N>2$, and the symmetric group is the semi-direct product
of $(\ZZ_2)^{N-1}$ and the permutation group $S_N$. These properties identify
themselves with ``small covers'' introduced by Davis and Januszkiewicz [Duke
Mathematical Journal, 62 (1991), 417-451]. As a corollary of our construction,
we give a formula on the total numbers of zeroes for a system of exponential
polynomials generated as Hankel determinant.


title: Nonlinear Spectral Characterization of Discrete Data

abstract:
  The explicit analytical expression of the Nonlinear Fourier Transform (NFT)
of a finite set of data is provided. Then a simple recursion relation for the
NFT is constructed as a function of the spectral parameter. These tools provide
a complete characterization of the nonlinear coherent structures (solitons,
breathers, ...) present in numerical or experimental data representing the
solution, at a given value of time, of a nonlinear evolution equation (e.g. of
the nonlinear Schroedinger family).


title:
Quasi-point separation of variables for the Henon-Heiles system and a
  system with quartic potential

abstract:
  We examine the problem of integrability of two-dimensional Hamiltonian
systems by means of separation of variables. The systematic approach to
construction of the special non-pure coordinate separation of variables for
certain natural two-dimensional Hamiltonians is presented. The relations with
SUSY quantum mechanics are discussed.


title:
Symmetries of a class of Nonlinear Third Order Partial Differential
  Equations

abstract:
  In this paper we study symmetry reductions of a class of nonlinear third
order partial differential equations $u_t -\epsilon u_{xxt} +2\kappa u_x= u
u_{xxx} +\alpha u u_x +\beta u_x u_{xx}$ where $\epsilon$, $\kappa$, $\alpha$
and $\beta$ are arbitrary constants. Three special cases of equation (1) have
appeared in the literature, up to some rescalings. In each case the equation
has admitted unusual travelling wave solutions: the Fornberg-Whitham equation,
for the parameters $\epsilon=1$, $\alpha=-1$, $\beta=3$ and $\kappa=\tfr12$,
admits a wave of greatest height, as a peaked limiting form of the travelling
wave solution; the Rosenau-Hyman equation, for the parameters $\epsilon=0$,
$\alpha=1$, $\beta=3$ and $\kappa=0$, admits a ``compacton'' solitary wave
solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters
$\epsilon=1$, $\alpha=-3$ and $\beta=2$, has a ``peakon'' solitary wave
solution. A catalogue of symmetry reductions for equation (1) is obtained using
the classical Lie method and the nonclassical method due to Bluman and Cole.


title:
Application of Uniform Asymptotics to the Second Painlev{\'e}
  Transcendent

abstract:
  In this work we propose a new method for investigating connection problems
for the class of nonlinear second-order differential equations known as the
Painlev{\'e} equations. Such problems can be characterized by the question as
to how the asymptotic behaviours of solutions are related as the independent
variable is allowed to pass towards infinity along different directions in the
complex plane. Connection problems have been previously tackled by a variety of
methods. Frequently these are based on the ideas of isomonodromic deformation
and the matching of WKB solutions. However, the implementation of these methods
often tends to be heuristic in nature and so the task of rigorising the process
is complicated. The method we propose here develops uniform approximations to
solutions. This removes the need to match solutions, is rigorous, and can lead
to the solution of connection problems with minimal computational effort. Our
method is reliant on finding uniform approximations of differential equations
of the generic form ${d^2\phi}/{d\eta^2} = - \xi^2F(\eta,\xi)\phi$ as the
complex-valued parameter $\xi \to \infty.$ The details of the treatment rely
heavily on the locations of the zeros of the function $F$ in this limit. If
they are isolated then a uniform approximation to solutions can be derived in
terms of Airy functions of suitable argument. On the other hand, if two of the
zeros of $F$ coalesce as $|\xi| \to \infty$ then an approximation can be
derived in terms of parabolic cylinder functions. In this paper we discuss both
cases, but illustrate our technique in action by applying the parabolic
cylinder case to the ``classical'' connection problem associated with the
second Painlev{\'e} transcendent. Future papers will show how the technique can
be applied with very little change to the other Painlev{\'e} equations, and to
the wider problem of the asymptotic behaviour of the general solution to any of
these equations.


title: Virasoro Symmetry Algebra of Dirac Soliton Hierarchy

abstract:
  A hierarchy of first-degree time-dependent symmetries is proposed for Dirac
soliton hierarchy and their commutator relations with time-dependent symmetries
are exhibited. Meantime, a hereditary structure of Dirac soliton hierarchy is
elucidated and a Lax operator algebra associated with Virasoro symmetry algebra
is given.


title: The Kowalewski top: a new Lax representation

abstract:
  The 2x2 monodromy matrices for the Kowalewski top on the Lie algebras e(3),
so(4) and so(3,1) are presented. The corresponding quadratic R-matrix structure
is the dynamical deformation of the standard R-matrix algebras. Some tops and
Toda lattices related to the Kowalewski top are discussed.


title: N=2 KP and KdV hierarchies in extended superspace

abstract:
  We give the formulation in extended superspace of an $N=2$ supersymmetric KP
hierarchy using chirality preserving pseudo-differential operators. We obtain
two quadratic hamiltonian structures, which lead to different reductions of the
KP hierarchy. In particular we find two different hierarchies with the $N=2$
classical super-${\cal W}_n$ algebra as a hamiltonian structure. The relation
with the formulation in $N=1$ superspace is carried out.


title:
Analytic-bilinear approach to integrable hierarchies. I.Generalized KP
  hierarchy

abstract:
  Analytic-bilinear approach for construction and study of integrable
hierarchies, in particular, the KP hierarchy is discussed. It is based on the
generalized Hirota identity. This approach allows to represent generalized
hierarchies of integrable equations in a condensed form of finite functional
equations. Resolution of these functional equations leads to the
$\tau$-function and addition formulae to it. General discrete transformations
of the $\tau$-function are presented in the determinant form. Closed one-form
and other formulae also arise naturally within the approach proposed.
Generalized KP hierarchy written in terms of different invariants of Combescure
symmetry transformations coincides with the usual KP hierarchy and the mKP
hierarchy.


title: The Calogero-Sutherland Model and Polynomials with Prescribed Symmetry

abstract:
  The Schr\"odinger operators with exchange terms for certain
Calogero-Sutherland quantum many body systems have eigenfunctions which factor
into the symmetric ground state and a multivariable polynomial. The polynomial
can be chosen to have a prescribed symmetry (i.e. be symmetric or
antisymmetric) with respect to the interchange of some specified variables. For
four particular Calogero-Sutherland systems we construct an eigenoperator for
these polynomials which separates the eigenvalues and establishes
orthogonality. In two of the cases this involves identifying new operators
which commute with the corresponding Schr\"odinger operators. In each case we
express a particular class of the polynomials with prescribed symmetry in a
factored form involving the corresponding symmetric polynomials.


title:
Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky
  lattices

abstract:
  A new Lax representation for the Bogoyavlensky lattice is found, its
$r$--matrix interpretation is elaborated. The $r$--matrix structure turns out
to be related to a highly nonlocal quadratic Poisson structure on a direct sum
of associative algebras. The theory of such nonlocal structures is developed,
the Poisson property of the monodromy map is worked out in the most general
situation. Some problems concerning the duality of Lax representations are
raised.


title: Darboux Coverings and Rational Reductions of the KP Hierarchy

abstract:
  We use the method of Darboux coverings to discuss the invariant submanifolds
of the KP equations, presented as conservation laws in the space of monic
Laurent series in the spectral parameter (the space of the Hamiltonian
densities). We identify a special class of these submanifolds with the rational
invariant submanifolds entering matrix models of $2D$--gravity, recently
characterized by Dickey and Krichever. Four examples of the general procedure
are provided.


title: Automorphisms of sl(2) and dynamical r-matrices

abstract:
  Two outer automorphisms of infinite-dimensional representations of $sl(2)$
algebra are considered. The similar constructions for the loop algebras and
yangians are presented. The corresponding linear and quadratic $R$-brackets
include the dynamical $r$-matrices.


title: Contractions of Integrable Equations

abstract:
  The contraction is applied to obtaining of integrable systems associated with
nonsemisimple algebras. The effect of contraction is splitting off some
components from initial system without loss of integrability.


title:
Bilinearization of Discrete Soliton Equations and Singularity
  Confinement

abstract:
  Bilinear forms for some nonlinear partial difference equations(discrete
soliton equations) are derived based on the results of singularity confinement.
Using the bilinear forms, the N-soliton and algebraic solutions of the discrete
potential mKdV equation are constructed.


title: On the solution of normality equations for the dimension $n\geq 3$

abstract:
  The normality equations for the Newtonian dynamical systems on an arbitrary
Riemannian manifold of the dimension $n \geq 3$ are considered. Locally the
solution of such equations reduces to three possible cases: in two of them the
solution is written out explicitly, and in the third case the equations of
normality are reduced to an ordinary differential equation of the second order.
Some new examples of explicit solutions of normality equations are constructed.


title: A new method to test discrete Painlev\'e equations

abstract:
  Necessary discretization rules to preserve the Painlev\'e property are
stated. A new method is added to the discrete Painlev\'e test, which perturbs
the continuous limit and generates infinitely many no-log conditions.


title:
Lax Representations and Zero Curvature Representations by Kronecker
  Product

abstract:
  It is showed that Kronecker product can be applied to construct not only new
Lax representations but also new zero curvature representations of integrable
models. Meantime a different characteristic between continuous and discrete
zero curvature equations is pointed out.


title: The exact solution of an octagonal rectangle triangle random tiling

abstract:
  We present a detailed calculation of the recently published exact solution of
a random tiling model possessing an eight-fold symmetric phase. The solution is
obtained using Bethe Ansatz and provides closed expressions for the entropy and
phason elastic constants. Qualitatively, this model has the same features as
the square-triangle random tiling model. We use the method of P. Kalugin, who
solved the Bethe Ansatz equations for the square-triangle tiling, which were
found by M. Widom.


title: An orthogonal basis for the $B_N$-type Calogero model

abstract:
  We investigate algebraic structure for the $B_N$-type Calogero model by using
the exchange-operator formalism. We show that the set of the Jack polynomials
whose arguments are Dunkl-type operators provides an orthogonal basis.


title:
Solutions of Discretized Affine Toda Field Equations for A_{n}^{(1)},
  B_{n}^{(1)}, C_{n}^{(1)}, D_{n}^{(1)}, A_{n}^{(2)} and D_{n+1}^{(2)}

abstract:
  It is known that a family of transfer matrix functional equations, the
T-system, can be compactly written in terms of the Cartan matrix of a simple
Lie algebra. We formally replace this Cartan matrix of a simple Lie algebra
with that of an affine Lie algebra, and then we obtain a system of functional
equations different from the T-system. It may be viewed as an X_{n}^{(a)} type
affine Toda field equation on discrete space time. We present, for A_{n}^{(1)},
B_{n}^{(1)}, C_{n}^{(1)}, D_{n}^{(1)}, A_{n}^{(2)} and D_{n+1}^{(2)}, its
solutions in terms of determinants or Pfaffians.


title:
Integrabilities of the long range t-J model of twisted boundary
  condition

abstract:
  The integrability of the one-dimensional long range supersymmetric t-J model
has previously been established for both open systems and those closed by
periodic boundary conditions through explicit construction of its integrals of
motion. Recently the system has been extended to include the effect of magnetic
flux, which gives rise to a closed chain with twisted boundary conditions.
While the t-J model with twisted boundary conditions has been solved for the
ground state and full energy spectrum, proof of its integrability has so far
been lacking. In this letter we extend the proof of integrability of the long
range supersymmetric t-J model and its SU(m|n) generalization to include the
case of twisted boundary conditions.


title: Complex Blow-Up in Burgers' Equation: an Iterative Approach

abstract:
  We show that for a given holomorphic noncharacteristic surface S in
two-dimensional complex space, and a given holomorphic function on S, there
exists a unique meromorphic solution of Burgers' equation which blows up on S.
This proves the convergence of the formal Laurent series expansion found by the
Painlev\'e test. The method used is an adaptation of Nirenberg's iterative
proof of the abstract Cauchy-Kowalevski theorem.


title: The Coalescence Limit of the Second Painlev\'E Equation

abstract:
  In this paper, we study a well known asymptotic limit in which the second
Painlev\'e equation (P_II) becomes the first Painlev\'e equation (P_I). The
limit preserves the Painlev\'e property (i.e. that all movable singularities of
all solutions are poles). Indeed it has been commonly accepted that the movable
simple poles of opposite residue of the generic solution of P_{II} must
coalesce in the limit to become movable double poles of the solutions of P_I,
even though the limit naively carried out on the Laurent expansion of any
solution of P_{II} makes no sense. Here we show rigorously that a coalescence
of poles occurs. Moreover we show that locally all analytic solutions of P_I
arise as limits of solutions of P_{II}.


title:
Small-amplitude excitations in a deformable discrete nonlinear
  Schroedinger equation

abstract:
  A detailed analysis of the small-amplitude solutions of a deformed discrete
nonlinear Schr\"{o}dinger equation is performed. For generic deformations the
system possesses "singular" points which split the infinite chain in a number
of independent segments. We show that small-amplitude dark solitons in the
vicinity of the singular points are described by the Toda-lattice equation
while away from the singular points are described by the Korteweg-de Vries
equation. Depending on the value of the deformation parameter and of the
background level several kinds of solutions are possible. In particular we
delimit the regions in the parameter space in which dark solitons are stable in
contrast with regions in which bright pulses on nonzero background are
possible. On the boundaries of these regions we find that shock waves and
rapidly spreading solutions may exist.


title: Some explicit solutions of the Lam\'e and Bourlet type equations

abstract:
  Some special solutions to the multidimensional Lam\'e and Bourlet type
equations are constructed in an explicit form.


title: Spinless Calogero-Sutherland model with twisted boundary condition

abstract:
  In this work, the spinless Calogero-Sutherland model with twisted boundary
condition is studied. The ground state wavefunctions, the ground state
energies, the full energy spectrum are provided in details.


title: On the integrability of the square-triangle random tiling model

abstract:
  It is shown that the square-triangle random tiling model is equivalent to an
asymmetric limit of the three colouring model on the honeycomb lattice. The
latter model is known to be the O(n) model at T=0 and corresponds to the
integrable model connected to the affine $A_2^{(1)}$ Lie algebra. Thus it is
shown that the weights of the square-triangle random tiling satisfy the
Yang-Baxter equation, albeit in a singular limit of a more general model. The
three colouring model for general vertex weights is solved by algebraic Bethe
Ansatz.


title:
The orthogonal eigenbasis and norms of eigenvectors in the Spin
  Calogero-Sutherland Model

abstract:
  Using a technique based on the Yangian Gelfand-Zetlin algebra and the
associated Yangian Gelfand-Zetlin bases we construct an orthogonal basis of
eigenvectors in the Calogero-Sutherland Model with spin, and derive
product-type formulas for norms of these eigenvectors.


title: The L-Matrix for the Massive Thirring Model

abstract:
  As the new results for the massive Thirring model the L-matrix and the
algebraic relations for its action angle variables are given. So it is shown
most directly that this model which describes self-interacting relativistic
Fermions in one-dimensional space is a quantum integrable system.


title:
The conservation of the Hamiltonian structures in Whitham's method of
  averaging

abstract:
  The work is devoted to the proof of the conservation of local
field-theoretical Hamiltonian structures in Whitham's method of averaging. The
consideration is based on the procedure of averaging of local Poisson bracket,
proposed by B.A.Dubrovin and S.P.Novikov. Using the Dirac procedure of
restriction of the Poisson bracket on the submanifold in the functional space,
it is shown in the generic case that the Poisson bracket, constructed by method
of Dubrovin and Novikov, satisfies the Jacobi identity. Besides that, the
invariance of this bracket with respect to the choice of the set of local
conservation laws, used in this procedure, is proved.


title: Representations of Two-Colour BWM Algebras and Solvable Lattice Models

abstract:
  Many of the known solutions of the Yang-Baxter equation, which are related to
solvable lattice models of vertex- and IRF-type, yield representations of the
Birman-Wenzl-Murakami algebra. From these, representations of a two-colour
generalization of the Birman-Wenzl-Murakami algebra can be constructed, which
in turn are used to derive trigonometric solutions to the Yang-Baxter equation.
In spirit, this construction resembles the fusion procedure, in the sense that
starting from known solutions of the Yang-Baxter equation new solutions can be
obtained.


title: Correlators of the phase model

abstract:
  We introduce the phase model on a lattice and solve it using the algebraic
Bethe ansatz. Time-dependent temperature correlation functions of phase
operators and the "darkness formation probability" are calculated in the
thermodynamical limit. These results can be used to construct integrable
equations for the correlation functions and to calculate there asymptotics.


title: Solutions of (2+1)-dimensional spin systems

abstract:
  We use the methods of group theory to reduce the equations of motion of two
spin systems in (2+1) dimensions to sets of coupled ordinary differential
equations. We present solutions of some classes of these sets and discuss their
physical significance.


title:
Riemannian Manifolds with Diagonal Metric. The Lam\'e and Bourlet
  Systems

abstract:
  We discuss a Lie algebraic and differential geometry construction of
solutions to some multidimensional nonlinear integrable systems describing
diagonal metrics on Riemannian manifolds, in particular those of zero and
constant curvature. Here some special solutions to the Lam\'e and Bourlet type
equations, determining by n arbitrary functions of one variable are obtained in
an explicit form. For the case when the sum of the diagonal elements of the
metric is a constant, these solutions are expressed as a product of the Jacobi
elliptic functions and are determined by 2n arbitrary constants.


title:
Difference Operator Approach to the Moyal Quantization and Its
  Application to Integrable Systems

abstract:
  Inspired by the fact that the Moyal quantization is related with nonlocal
operation, I define a difference analogue of vector fields and rephrase quantum
description on the phase space. Applying this prescription to the theory of the
KP-hierarchy, I show that their integrability follows to the nature of their
Wigner distribution. Furthermore the definition of the ``expectation value''
clarifies the relation between our approach and the Hamiltonian structure of
the KP-hierarchy. A trial of the explicit construction of the Moyal bracket
structure in the integrable system is also made.


title: Geometric Discretisation of the Toda System

abstract:
  The Laplace sequence of the discrete conjugate nets is constructed. The
invariants of the nets satisfy, in full analogy to the continuous case, the
system of difference equations equivalent to the discrete version of the
generalized Toda equation.


title: Multidimensional Quadrilateral Lattices are Integrable

abstract:
  The notion of multidimensional quadrilateral lattice is introduced. It is
shown that such a lattice is characterized by a system of integrable discrete
nonlinear equations. Different useful formulations of the system are given. The
geometric construction of the lattice is also discussed and, in particular, it
is clarified the number of initial--boundary data which define the lattice
uniquely.


title:
On the two-magnon bound states for the quantum Heisenberg chain with
  variable range exchange

abstract:
  The spectrum of finite-difference two-magnon operator is investigated for
quantum S=1/2 chain with variable range exchange of the form $h(j-k)\propto
\sinh^{-2}a(j-k)$. It is found that usual bound state appears for some values
of the total pseudomomentum of two magnons as for the Heisenberg chain with
nearest-neighbor spin interaction. Besides this state, a new type of bound
state with oscillating wave function appears at larger values of the total
pseudomomentum.


title:
Irreducible Representations of an Algebra underlying Hidden Symmetries
  of a class of Quasi Exactly Solvable Systems of Equations

abstract:
  The set of linear, differential operators preserving the vector space of
couples of polynomials of degrees n and n-2 in one real variable leads to an
abstract associative graded algebra A(2). The irreducible, finite dimensional
representations of this algebra are classified into five infinite discrete sets
and one exceptional case. Their matrix elements are given explicitely. The
results are related to the theory of quasi exactly solvable equations.


title: The Whitham equations revisited

abstract:
  We survey some topics involving the Whitham equations, concentrating on the
role of the Baker Akhiezer function in averaging. Some connections to
symplectic geometry and Seiberg-Witten theory are indicated.


title: Some kernels on a Riemann surface

abstract:
  We discuss certain kernels on a Riemann surface, constructed mainly via Baker
Akhiezer functions, and indicate relations to dispersionless theory.


title: Bispectral Operators of Rank 1 and Dual Isomonodromic Deformations

abstract:
  A comparison is made between bispectral operator pairs and dual pairs of
isomonodromic deformation equations. Through examples, it is shown how
operators belonging to rank one bispectral algebras may be viewed equivalently
as defining 1-parameter families of rational first order differential operators
with matricial coefficients on the Riemann sphere, whose monodromy is trivial.
By interchanging the r\^oles of the two variables entering in the bispectral
pair, a second 1-parameter family of operators with trivial monodromy is
obtained, which may be viewed as the dual isomonodromic deformation system.


title:
Leading Order Temporal Asymptotics of the Modified Non-Linear
  Schrodinger Equation: Solitonless Sector

abstract:
  Using the matrix Riemann-Hilbert factorisation approach for non-linear
evolution equations (NLEEs) integrable in the sense of the inverse scattering
method, we obtain, in the solitonless sector, the leading-order asymptotics as
$t$ tends to plus and minus infinity of the solution to the Cauchy
initial-value problem for the modified non-linear Schrodinger equation: also
obtained are analogous results for two gauge-equivalent NLEEs; in particular,
the derivative non-linear Schrodinger equation.

categories: solv-int nlin.PS nlin.SI patt-sol physics.plasm-ph


title: Painleve equations in terms of entire functions

abstract:
  In these lectures we discuss how the Painleve equations can be written in
terms of entire functions, and then in the Hirota bilinear (or multilinear)
form. Hirota's method, which has been so useful in soliton theory, is reviewed
and connections from soliton equations to Painleve equations through similarity
reductions are discussed from this point of view. In the main part we discuss
how singularity structure of the solutions and formal integration of the
Painleve equations can be used to find a representation in terms of entire
functions. Sometimes the final result is a pair of Hirota bilinear equations,
but for $P_{VI}$ we need also a quadrilinear expression. The use of discrete
versions of Painleve equations is also discussed briefly. It turns out that
with discrete equations one gets better information on the singularities, which
can then be represented in terms of functions with a simple zero.


title: Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

abstract:
  The small t asymptotics of a class of solutions to the 2D cylindrical Toda
equations is computed. The solutions, q_k(t), have the representation q_k(t) =
log det(I-lambda K_k) - log det(I-lambda K_{k-1}) where K_k are integral
operators. This class includes the n-periodic cylindrical Toda equations. For
n=2 our results reduce to the previously computed asymptotics of the 2D radial
sinh-Gordon equation and for n=3 (and with an additional symmetry contraint)
they reduce to earlier results for the radial Bullough-Dodd equation.


title: Separation of variables for the Ruijsenaars system

abstract:
  We construct a separation of variables for the classical n-particle
Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser
system). The separated coordinates appear as the poles of the properly
normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax
matrix. Two different normalisations of the BA functions are analysed. The
canonicity of the separated variables is verified with the use of r-matrix
technique. The explicit expressions for the generating function of the
separating canonical transform are given in the simplest cases n=2 and n=3.
Taking nonrelativistic limit we also construct a separation of variables for
the elliptic Calogero-Moser system.


title:
Crum Transformations and Wronskian Type Solutions for Supersymmetric KdV
  equation

abstract:
  Darboux transformation is reconsidered for the supersymmetric KdV system. By
iterating the Darboux transformation, a supersymmetric extension of the Crum
transformation is obtained for the Manin-Radul SKdV equation, in doing so one
gets Wronskian superdeterminant representations for the solutions. Particular
examples provide us explicit supersymmetric extensions, super solitons, of the
standard soliton of the KdV equation. The KdV soliton appears as the body of
the super soliton.


title: Aspects of Solitons in Affine Integrable Hierarchies

abstract:
  We argue that one of the basic ingredients for the appearance of soliton
solutions in integrable hierarchies, is the existence of ``vacuum solutions''
corresponding to Lax operators lying in some abelian subalgebra of the
associated affine Kac-Moody algebra. Using the dressing transformation we
construct the solutions in the orbit of those vacuum solutions, and conjecture
that the solitons correspond to some special points in those orbits. The
generalized tau-function for those hierarchies are defined for integrable
highest weight representations. It applies for any level of the representation.
We illustrate our methods with the recently proposed non abelian Toda models
coupled to matter fields. A very special class of such theories possess a U(1)
Noether charge that is proportional to a topological charge. That leads to a
mechanism that confines the matter fields inside the solitons.


title: Darboux Transformation for the Manin-Radul Supersymmetric KdV equation

abstract:
  In this paper we present a vectorial Darboux transformation, in terms of
ordinary determinants, for the supersymmetric extension of the Korteweg-de
Vries equation proposed by Manin and Radul. It is shown how this transformation
reduces to the Korteweg-de Vries equation. Soliton type solutions are
constructed by dressing the vacuum and we present some relevant plots.


title:
$P_\infty$ algebra of KP, free fermions and 2-cocycle in the Lie algebra
  of pseudodifferential operators

abstract:
  The symmetry algebra $P_\infty = W_\infty \oplus H \oplus I_\infty$ of
integrable systems is defined. As an example the classical Sophus Lie point
symmetries of all higher KP equations are obtained. It is shown that one
(``positive'') half of the point symmetries belongs to the $W_\infty$
symmetries while the other (``negative'') part belongs to the $I_\infty$ ones.
The corresponing action on the tau-function is obtained for the positive part
of the symmetries. The negative part can not be obtained from the free fermion
algebra. A new embedding of the Virasoro algebra into $gl(\infty )$n describes
conformal transformations of the KP time variables. A free fermion algebra
cocycle is described as a PDO Lie algebra cocycle.


title: Separation of variables for the Dn type periodic Toda lattice

abstract:
  We prove separation of variables for the most general (Dn type) periodic Toda
lattice with 2x2 Lax matrix. It is achieved by finding proper normalisation for
the corresponding Baker-Akhiezer function. Separation of variables for all
other periodic Toda lattices associated with infinite series of root systems
follows by taking appropriate limits.


title:
A note on the integrable discretization of the nonlinear Schr\"odinger
  equation

abstract:
  We revisit integrable discretizations for the nonlinear Schr\"odinger
equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the
non-locality, can be overcome. Namely, we factorize the non-local difference
scheme into the product of local ones. This must improve the performance of the
scheme in the numerical computations dramatically. Using the equivalence of the
Ablowitz--Ladik and the relativistic Toda hierarchies, we find the
interpolating Hamiltonians for the local schemes and show how to solve them in
terms of matrix factorizations.


title: Twisted Quantum Lax Equations

abstract:
  We give the construction of twisted quantum Lax equations associated with
quantum groups. We solve these equations using factorization properties of the
corresponding quantum groups. Our construction generalizes in many respects the
Adler-Kostant-Symes construction for Lie groups and the construction of M. A.
Semenov Tian-Shansky for the Lie-Poisson case.


title:
Some comments on the bi(tri)-Hamiltonian structure of Generalized AKNS
  and DNLS hierarchies

abstract:
  We give the correct prescriptions for the terms involving the inverse of the
derivative of the delta function, in the Hamiltonian structures of the AKNS and
DNLS systems, in order for the Jacobi identities to hold. We also establish
that the sl(2) AKNS and DNLS systems are tri-Hamiltonians and construct two
compatible Hamiltonian structures for the sl(3) AKNS system. We also give a
derivation of the recursion operator for the sl(n+1) DNLS system.


title: The structures underlying soliton solutions in integrable hierarchies

abstract:
  We point out that a common feature of integrable hierarchies presenting
soliton solutions is the existence of some special ``vacuum solutions'' such
that the Lax operators evaluated on them, lie in some abelian subalgebra of the
associated Kac-Moody algebra. The soliton solutions are constructed out of
those ``vacuum solitons'' by the dressing transformation procedure.


title: The Fourier method for the linearized Davey-Stewartson I equation

abstract:
  The linearized Davey-Stewartson equation with varing coefficients is solved
by Fourier method. The approach uses the inverse scattering transform for the
Davey-Stewartson equation.


title: The Yangian Symmetry in the Spin Calogero Model and its Applications

abstract:
  By using the non-symmetric Hermite polynomials and a technique based on the
Yangian Gelfand-Zetlin bases, we decompose the space of states of the Calogero
model with spin into irreducible Yangian modules, construct an orthogonal basis
of eigenvectors and derive product-type formulas for norms of these
eigenvectors.


title:
Some eigenstates for a model associated with solutions of tetrahedron
  equation

abstract:
  Here we present some eigenstates for a 2+1-dimensional model associated with
a solution of the tetrahedron equation. The eigenstates include those
"particle-like" (namely one-particle and two-particle ones), constructed in
analogy with the usual 1+1-dimensional Bethe ansatz, and some simple
"string-like" ones.


title:
Method of Squared Eigenfunction Potentials in Integrable Hierarchies of
  KP Type

abstract:
  The method of squared eigenfunction potentials (SEP) is developed
systematically to describe and gain new information about
Kadomtsev-Petviashvili (KP) hierarchy and its reductions. Interrelation to the
tau-function method is discussed in detail. The principal result, which forms
the basis of our SEP method, is the proof that any eigenfunction of the general
KP hierarchy can be represented as a spectral integral over the Baker-Akhiezer
(BA) wave function with a spectral density expressed in terms of SEP. In fact,
the spectral representations of the (adjoint) BA functions can, in turn, be
considered as defining equations for the KP hierarchy. The SEP method is
subsequently used to show how the reduction of the full KP hierarchy to the
constrained KP hierarchies can be given entirely in terms of linear constraint
equations on the pertinent tau-functions. The concept of SEP turns out to be
crucial in providing a description of constrained KP hierarchies in the
language of universal Sato Grassmannian and finding the non-isospectral
Virasoro symmetry generators acting on the underlying tau-functions. The SEP
method is used to write down generalized binary Darboux-Backlund
transformations for constrained KP hierarchies whose orbits are shown to
correspond to a new Toda model on a square lattice. As a result, we obtain a
series of new determinant solutions for the tau-functions generalizing the
known Wronskian (multi-soliton) solutions. Finally, applications to random
matrix models in condensed matter physics are briefly discussed.


title:
Symplectic Structures and Volume Elements in the Function Space for the
  Cubic Schrodinger Equation

abstract:
  We consider various trace formulas for the cubic Schrodinger equation in the
space of infinitely smooth functions subject to periodic boundary conditions.
The formulas relate conventional integrals of motion to the periods of some
Abelian differentials (holomorphic one-forms) on the spectral curve. We show
that the periods of Abelian differentials are global coordinates on the moduli
space of spectral curves. The exterior derivatives of the holomorphic one-forms
are the basic and higher symplectic structures on the phase space. We write
explicitly these symplectic structures in $QP$ coordinates. We compute the
ratio of two symplectic volume elements in the infinite genus limit.


title: The Solution of the N=2 Supersymmetric f-Toda Chain with Fixed Ends

abstract:
  The integrability of the recently introduced N=2 supersymmetric f-Toda chain,
under appropriate boundary conditions, is proven. The recurrent formulae for
its general solutions are derived. As an example, the solution for the simplest
case of boundary conditions is presented in explicit form.


title: The Discrete Symmetries of the N=2 Supersymmetric GNLS Hierarchies

abstract:
  The discrete symmetry transformations of the N=2 supersymmetric (n,m)-GNLS
hierarchy are constructed. Their bosonic limit is analyzed and new discrete
symmetries of the modified GNLS hierarchy are derived. The explicit relations
connecting the integrable hierarchy, produced by the junction of the Lax
operators for the N=2 supersymmetric a=4 KdV and (n-1,m)-GNLS hierarchies, to
the N=2 supersymmetric (n,m)-GNLS hierarchy are established.


title:
Is the classical Bukhvostov-Lipatov model integrable? A Painlev\'e
  analysis

abstract:
  In this work we apply the Weiss, Tabor and Carnevale integrability criterion
(Painlev\'e analysis) to the classical version of the two dimensional
Bukhvostov-Lipatov model. We are led to the conclusion that the model is not
integrable classically, except at a trivial point where the theory can be
described in terms of two uncoupled sine-Gordon models.


title:
Coupled Nonlinear Schr\"{o}dinger equation and Toda equation (the Root
  of Integrability)

abstract:
  We consider the relation between the discrete coupled nonlinear
Schr\"{o}dinger equation and Toda equation. Introducing complex times we can
show the intergability of the discrete coupled nonlinear Schr\"{o}dinger
equation. In the same way we can show the integrability in coupled case of dark
and bright equations. Using this method we obtain several integrable equations.


title: Rational Solutions for the Discrete Painlev\'e II Equation

abstract:
  The rational solutions for the discrete Painlev\'e II equation are
constructed based on the bilinear formalism. It is shown that they are
expressed by the determinant whose entries are given by the Laguerre
polynomials. Continuous limit to the Devisme polynomial representation of the
rational solutions for the Painlev\'e II equation is also discussed.