title: The Quantum Inverse Scattering Method for Hubbard-like Models

abstract:
  This work is concerned with various aspects of the formulation of the quantum
inverse scattering method for the one-dimensional Hubbard model. We first
establish the essential tools to solve the eigenvalue problem for the transfer
matrix of the classical ``covering'' Hubbard model within the algebraic Bethe
Ansatz framework. The fundamental commutation rules exhibit a hidden 6-vertex
symmetry which plays a crucial role in the whole algebraic construction. Next
we apply this formalism to study the SU(2) highest weights properties of the
eigenvectors and the solution of a related coupled spin model with twisted
boundary conditions. The machinery developed in this paper is applicable to
many other models, and as an example we present the algebraic solution of the
Bariev XY coupled model.


title: The solution to the q-KdV equation

abstract:
  Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The
purpose of this paper is to show that any KdV solution leads effectively to a
solution of the q-approximation of KdV. Two different q-KdV approximations were
proposed, one by Frenkel and a variation by Khesin et al. We show there is a
dictionary between the solutions of q-KP and the 1-Toda lattice equations,
obeying some special requirement; this is based on an algebra isomorphism
between difference operators and D-operators, where $Df(x)=f(qx)$. Therefore,
every notion about the 1-Toda lattice can be transcribed into q-language.


title: Toda-Darboux maps and vertex operators

abstract:
  The purpose of this paper is to study Toda-Darboux transforms, i.e., Darboux
transforms for operators L(t) flowing according to the Toda lattice. Each
element of the null-space $L(t)-z$ specifies a factorization for all t and thus
a Toda-Darboux transform on $L(t)$. The Toda-Darboux map induces a
transformation on the tau-vectors, given by a certain vertex operator, and on
eigenfunctions, given by a Wronskian. .


title: Transformations of Quadrilateral Lattices

abstract:
  Motivated by the classical studies on transformations of conjugate nets, we
develop the general geometric theory of transformations of their discrete
analogues: the multidimensional quadrilateral lattices, i.e. lattices x: Z^N ->
R^M, whose elementary quadrilaterals are planar. Our investigation is based on
the discrete analogue of the theory of the rectilinear congruences, which we
also present in detail. We study, in particular, the discrete analogues of the
Laplace, Combescure, Levy, radial and fundamental transformations and their
interrelations. The composition of these transformations and their
permutability is also investigated from a geometric point of view. The deep
connections between "transformations" and "discretizations" is also
investigated for quadrilateral lattices. We finally interpret these results
within the D-bar formalism.


title: Sigma Models and Minimal Surfaces

abstract:
  The correspondance is established between the sigma models, the minimal
surfaces and the Monge-Ampere equation. The Lax -Pairs of the minimality
condition of the minimal surfaces and the Monge-Ampere equations are given.
Existance of infinitely many nonlocal conservation laws is shown and some
Backlund transformations are also given.


title:
Quadratically integrable geodesic flows on the torus and on the Klein
  bottle

abstract:
  In the present paper we prove, that if the geodesic flow of a metric G on the
torus T is quadratically integrable, then the torus T isometrically covers a
torus with a Liouville metric on it, and describe the set of quadratically
integrable geodesic flows on the Klein bottle.


title: Polarization scattering by soliton-soliton collisions

abstract:
  Collision of two solitons of the Manakov system is analytically studied.
Existence of a complete polarization mode switching regime is proved and the
parameters of solitons prepared for polarization switching are found.


title:
Asymptotics of Solutions to the Modified Nonlinear Schr\"{o}dinger
  Equation: Solitons on a Non-Vanishing Continuous Background

abstract:
  Using the matrix Riemann-Hilbert factorization approach for nonlinear
evolution systems which take the form of Lax-pair isospectral deformations and
whose corresponding Lax operators contain both discrete and continuous spectra,
the leading-order asymptotics as $t \to \pm \infty$ of the solution to the
Cauchy problem for the modified nonlinear Schr\"{o}dinger equation, $i
\partial_{t} u + {1/2} \partial_{x}^{2} u + | u |^{2} u + i s \partial_{x} (| u
|^{2} u) = 0$, $s \in \Bbb R_{>0}$, which is a model for nonlinear pulse
propagation in optical fibers in the subpicosecond time scale, are obtained:
also derived are analogous results for two gauge-equivalent nonlinear evolution
equations; in particular, the derivative nonlinear Schr\"{o}dinger equation, $i
\partial_{t} q + \partial_{x}^{2} q - i \partial_{x} (| q |^{2} q) = 0$. As an
application of these asymptotic results, explicit expressions for position and
phase shifts of solitons in the presence of the continuous spectrum are
calculated.


title:
A Symmetric Generalization of Linear B\"acklund Transformation
  associated with the Hirota Bilinear Difference Equation

abstract:
  The Hirota bilinear difference equation is generalized to discrete space of
arbitrary dimension. Solutions to the nonlinear difference equations can be
obtained via B\"acklund transformation of the corresponding linear problems.


title:
N Soliton Solutions to The Bogoyavlenskii-Schiff Equation and A Quest
  for The Soliton Solution in (3 + 1) Dimensions

abstract:
  We study the integrable systems in higher dimensions which can be written not
by the Hirota's bilinear form but by the trilinear form. We explicitly discuss
about the Bogoyavlenskii-Schiff(BS) equation in (2 + 1) dimensions. Its
analytical proof of multi soliton solution and a new feature are given. Being
guided by the strong symmetry, we also propose a new equation in (3 + 1)
dimensions.


title: Temperature correlators in the two-component one-dimensional gas

abstract:
  The quantum nonrelativistic two-component Bose and Fermi gases with the
infinitely strong point-like coupling between particles in one space dimension
are considered. Time and temperature dependent correlation functions are
represented in the thermodynamic limit as Fredholm determinants of integrable
linear integral operators.


title:
Statistical Mechanics of Non-stretching Elastica in Three Dimensional
  Space

abstract:
  Recently I proposed a new calculation scheme of a partition function of an
immersion object using path integral method and theory of soliton (to appear in
J.Phys.A). I applied the scheme to problem of elastica in two-dimensional space
and Willmore surface in three dimensional space. In this article, I will apply
the scheme to elastica in three dimensional space as a more physical model in
polymer science. Then orbit space of the nonlinear Schrodinger and complex
modified Korteweg-de Vries equations can be regarded as the functional space of
the partition function. By investigation of the partition function, I gives a
conjecture of the relation of these soliton equations.


title:
Dirac Operator of a Conformal Surface Immersed in R^4: Further
  Generalized Weierstrass Relation

abstract:
  In the previous report (J. Phys. A (1997) 30 4019-4029), I showed that the
Dirac operator defined over a conformal surface immersed in R^3 is identified
with the Dirac operator which is generalized the Weierstrass- Enneper equation
and Lax operator of the modified Novikov-Veselov (MNV) equation. In this
article, I determine the Dirac operator defined over a conformal surface
immersed in R^4, which is reduced to the Lax operators of the nonlinear
Schrodinger and the MNV equations by taking appropriate limits. Thus the Dirac
operator might be the Lax operator of (2+1)- dimensional soliton equation.


title: General vorticity conservation

abstract:
  The motion of an incompressible fluid in Lagrangian coordinates involves
infinitely many symmetries generated by the left Lie algebra of group of volume
preserving diffeomorphisms of the three dimensional domain occupied by the
fluid. Utilizing a 1+3-dimensional Hamiltonian setting an explicit realization
of this symmetry algebra and the related Lagrangian and Eulerian conservation
laws are constructed recursively. Their Lie algebraic structures are inherited
from the same construction. The laws of general vorticity and helicity
conservations are formulated globally in terms of invariant differential forms
of the velocity field.


title: Asymptotics of a class of Fredholm determinants

abstract:
  In this expository article we describe the asymptotics of certain Fredholm
determinants which provide solutions to the cylindrical Toda equations, and we
explain how these asymptotics are derived. The connection with Fredholm
determinants arising in the theory of random matrices, and their asymptotics,
are also discussed.


title: Systems of PDEs obtained from factorization in loop groups

abstract:
  We propose a generalization of a Drinfeld-Sokolov scheme of attaching
integrable systems of PDEs to affine Kac-Moody algebras. With every affine
Kac-Moody algebra $\gg$ and a parabolic subalgebra $\gp$, we associate two
hierarchies of PDEs. One, called positive, is a generalization of the KdV
hierarchy, the other, called negative, generalizes the Toda hierarchy. We prove
a coordinatization theorem, which establishes that the number of functions
needed to express all PDEs of the the total hierarchy equals the rank of $\gg$.
The choice of functions, however, is shown to depend in a noncanonical way on
$\gp$. We employ a version of the Birkhoff decomposition and a ``2-loop''
formulation which allows us to incorporate geometrically meaningful solutions
to those hierarchies. We illustrate our formalism for positive hierarchies with
a generalization of the Boussinesq system and for the negative hierarchies with
the stationary Bogoyavlenskii equation.


title: Integrable Systems and Isomonodromy Deformations

abstract:
  We analyze in detail three classes of isomondromy deformation problems
associated with integrable systems. The first two are related to the scaling
invariance of the $n\times n$ AKNS hierarchies and the Gel'fand-Dikii
hierarchies. The third arises in string theory as the representation of the
Heisenberg group by $[(L^{k/n})_+,L]=I$ where $L$ is an $n^{th}$ order scalar
differential operator. The monodromy data is constructed in each case; the
inverse monodromy problem is solved as a Riemann-Hilbert problem; and a simple
proof of the Painlev\'e property is given for the general case


title: Factorization and the Dressing Method for the Gel'fand-Dikii Hierarch

abstract:
  The isospectral flows of an $n^{th}$ order linear scalar differential
operator $L$ under the hypothesis that it possess a Baker-Akhiezer function
were originally investigated by Segal and Wilson from the point of view of
infinite dimensional Grassmanians, and the reduction of the KP hierarchy to the
Gel'fand-Dikii hierarchy. The associated first order systems and their formal
asymptotic solutions have a rich Lie algebraic structure which was investigated
by Drinfeld and Sokolov. We investigate the matrix Riemann-Hilbert
factorizations for these systems, and show that different factorizations lead
respectively to the potential, modified, and ordinary Gel'fand-Dikii flows. Lie
algebra decompositions (the Adler-Kostant-Symes method) are obtained for the
modified and potential flows. For $n>3$ the appropriate factorization for the
Gel'fand-Dikii flows is not a group factorization, as would be expected; yet a
modification of the dressing method still works.
  A direct proof, based on a Fredholm determinant associated with the
factorization problem, is given that the potentials are meromorphic in $x$ and
in the time variables. Potentials with Baker-Akhiezer functions include the
multisoliton and rational solutions, as well as potentials in the scattering
class with compactly supported scattering data. The latter are dense in the
scattering class.


title: Dynamical boundary conditions for integrable lattices

abstract:
  Some special solutions to the reflection equation are considered. These
boundary matrices are defined on the common quantum space with the other
operators in the chain. The relations with the Drinfeld twist are discussed.


title: Particles and strings in a (2+1)-D integrable quantum model

abstract:
  We give a review of some recent work on generalization of the Bethe ansatz in
the case of $2+1$-dimensional models of quantum field theory. As such a model,
we consider one associated with the tetrahedron equation, i.e. the
$2+1$-dimensional generalization of the famous Yang--Baxter equation. We
construct some eigenstates of the transfer matrix of that model. There arise,
together with states composed of point-like particles analogous to those in the
usual $1+1$-dimensional Bethe ansatz, new string-like states and
string-particle hybrids.


title: Asymptotics of perturbed soliton for Davey--Stewartson II equation

abstract:
  It is shown that, under a small perturbation of lump (soliton) for
Davey--Stewartson (DS-II) equation, the scattering data gain the nonsoliton
structure. As a result, the solution has the form of Fourier type integral.
Asymptotic analysis shows that, in spite of dispertion, the principal term of
the asymptotic expansion for the solution has the solitary wave form up to
large time.


title: Functional Tetrahedron Equation

abstract:
  We describe a scheme of constructing classical integrable models in
2+1-dimensional discrete space-time, based on the functional tetrahedron
equation - equation that makes manifest the symmetries of a model in local
form. We construct a very general "block-matrix model" together with its
algebro-geometric solutions, study its various particular cases, and also
present a remarkably simple scheme of quantization for one of those cases.


title: Perturbation theory for the modified nonlinear Schr{\"o}dinger solitons

abstract:
  The perturbation theory based on the Riemann-Hilbert problem is developed for
the modified nonlinear Schr{\"o}dinger equation which describes the propagation
of femtosecond optical pulses in nonlinear single-mode optical fibers. A
detailed analysis of the adiabatic approximation to perturbation-induced
evolution of the soliton parameters is given. The linear perturbation and the
Raman gain are considered as examples.


title: The classical r-matrix in a geometric framework

abstract:
  We use a Riemannian (or pseudo-Riemannian) geometric framework to formulate
the theory of the classical r-matrix for integrable systems. In this picture
the r-matrix is related to a fourth rank tensor, named the r-tensor, on the
configuration space. The r-matrix itself carries one connection type index and
three tensorial indices. Being defined on the configuration space it has no
momentum dependence but is dynamical in the sense of depending on the
configuration variables. The tensorial nature of the r-matrix is used to derive
its transformation properties. The resulting transformation formula turns out
to be valid for a general r-matrix structure independently of the geometric
framework. Moreover, the entire structure of the r-matrix equation follows
directly from a simple covariant expression involving the Lax matrix and its
covariant derivative. Therefore it is argued that the geometric formulation
proposed here helps to improve the understanding of general r-matrix
structures. It is also shown how the Jacobi identity gives rise to a
generalized dynamical classical Yang-Baxter equation involving the Riemannian
curvature.


title: Nonlinear dynamical systems and classical orthogonal polynomials

abstract:
  It is demonstrated that nonlinear dynamical systems with analytic
nonlinearities can be brought down to the abstract Schr\"odinger equation in
Hilbert space with boson Hamiltonian. The Fourier coefficients of the expansion
of solutions to the Schr\"odinger equation in the particular occupation number
representation are expressed by means of the classical orthogonal polynomials.
The introduced formalism amounts a generalization of the classical methods for
linearization of nonlinear differential equations such as the Carleman
embedding technique and Koopman approach.


title: Matrix Formulation of Hamiltonian Structures of Constrained KP Hierarchy

abstract:
  We give a matrix formulation of the Hamiltonian structures of constrained KP
hierarchy. First, we derive from the matrix formulation the Hamiltonian
structure of the one-constraint KP hierarchy, which was originally obtained by
Oevel and Strampp. We then generalize the derivation to the multi-constraint
case and show that the resulting bracket is actually the second Gelfand-Dickey
bracket associated with the corresponding Lax operator. The matrix formulation
of the Hamiltonian structure of the one-constraint KP hierarchy in the form
introduced in the study of matrix model is also discussed


title: Universal formats for nonlinear dynamical systems

abstract:
  It is demonstrated that very general nonlinear dynamical systems covering all
cases arising in practice can be brought down to rate equations of chemical
kinetics


title:
Supersymmetric KP Hierarchy: ``Ghost'' Symmetry Structure, Reductions
  and Darboux-Backlund Solutions

abstract:
  This paper studies Manin-Radul supersymmetric KP hierarchy (MR-SKP) in three
related aspects: (i) We find an infinite set of additional (``ghost'') symmetry
flows spanning the same (anti-)commutation algebra as the ordinary MR-SKP
flows; (ii) The latter are used to construct consistent reductions of the
initial unconstrained MR-SKP hierarchy which involves a nontrivial modification
for the fermionic flows; (iii) For the simplest constrained MR-SKP hierarchy we
show that the orbit of Darboux-Backlund transformations lies on a
supersymmetric Toda lattice being a square-root of the standard one-dimensional
Toda lattice, and also we find explicit Wronskian-ratio solutions for the
super-tau function.


title:
Vacuum curves of elliptic L-operators and representations of Sklyanin
  algebra

abstract:
  An algebro-geometric approach to representations of Sklyanin algebra is
proposed. To each 2 \times 2 quantum L-operator an algebraic curve
parametrizing its possible vacuum states is associated. This curve is called
the vacuum curve of the L-operator. An explicit description of the vacuum curve
for quantum L-operators of the integrable spin chain of XYZ type with arbitrary
spin $\ell$ is given. The curve is highly reducible. For half-integer $\ell$ it
splits into $\ell +{1/2}$ components isomorphic to an elliptic curve. For
integer $\ell$ it splits into $\ell$ elliptic components and one rational
component. The action of elements of the L-operator to functions on the vacuum
curve leads to a new realization of the Sklyanin algebra by difference
operators in two variables restricted to an invariant functional subspace.


title: Computation of conservation laws for nonlinear lattices

abstract:
  An algorithm to compute polynomial conserved densities of polynomial
nonlinear lattices is presented. The algorithm is implemented in Mathematica
and can be used as an automated integrability test. With the code diffdens.m,
conserved densities are obtained for several well-known lattice equations. For
systems with parameters, the code allows one to determine the conditions on
these parameters so that a sequence of conservation laws exist.


title:
Invariants and Symmetries for Partial Differential Equations and
  Lattices

abstract:
  Methods for the computation of invariants and symmetries of nonlinear
evolution, wave, and lattice equations are presented. The algorithms are based
on dimensional analysis, and can be implemented in any symbolic language, such
as Mathematica. Invariants and symmetries are shown for several well-known
equations.
  Our Mathematica package allows one to automatically compute invariants and
symmetries. Applied to systems with parameters, the package determines the
conditions on these parameters so that a sequence of invariants or symmetries
exists. The software can thus be used to test the integrability of model
equations for wave phenomena.


title:
On the Lakshmanan and gauge equivalent counterpart of the
  Myrzakulov-VIII equation

abstract:
  The Lakshmanan equivalent counterparts of the some Myrzakulov equations are
found.


title: Correlation Functions of Finite XXZ model with Boundaries

abstract:
  The finite XXZ model with boundaries is considered. We use the Matrix Product
Ansatz (MPA), which was originally developed in the studies on the asymmetric
simple exclusion process and the quantum antiferromagnetic spin chain. The MPA
tells that the eigenstate of the Hamiltonian is constructed by the
Zamolodchikov-Faddeev algebra (ZF-algebra) and the boundary states. We adopt
the type I vertex operator of $U_q(\hat{sl}_2)$ as the ZF-algebra and realize
the boundary states in the bosonic $U_q(\hat{sl}_2)$ form. The correlation
functions are given by the product of the vertex operators and the bosonic
boundary states. We express them in the integration forms.


title:
Braid Structure and Raising-Lowering Operator Formalism in Sutherland
  Model

abstract:
  We algebraically construct the Fock space of the Sutherland model in terms of
the eigenstates of the pseudomomenta as basis vectors. For this purpose, we
derive the raising and lowering operators which increase and decrease
eigenvalues of pseudomomenta. The operators exchanging eigenvalues of two
pseudomomenta have been known. All the eigenstates are systematically produced
by starting from the ground state and multiplying these operators to it.


title: Chiral Solitons in Generalized Korteweg-de Vries Equations

abstract:
  Generalizations of the Korteweg-de Vries equation are considered, and some
explicit solutions are presented. There are situations where solutions engender
the interesting property of being chiral, that is, of having velocity
determined in terms of the parameters that define the generalized equation,
with a definite sign.


title: Lax pairs for N=2,3 Supersymmetric KdV Equations and their Extensions

abstract:
  We present the Lax operator for the N=3 KdV hierarchy and consider its
extensions. We also construct a new infinite family of N=2 supersymmetric
hierarchies by exhibiting the corresponding super Lax operators. The new
realization of N=4 supersymmetry on the two general N=2 superfields, bosonic
spin 1 and fermionic spin 1/2, is discussed.


title:
Computation of Higher-order Symmetries for Nonlinear Evolution and
  Lattice Equations

abstract:
  A straightforward algorithm for the symbolic computation of higher-order
symmetries of nonlinear evolution equations and lattice equations is presented.
The scaling properties of the evolution or lattice equations are used to
determine the polynomial form of the higher-order symmetries. The coefficients
of the symmetry can be found by solving a linear system. The method applies to
polynomial systems of PDEs of first-order in time and arbitrary order in one
space variable. Likewise, lattices must be of first order in time but may
involve arbitrary shifts in the discretized space variable.
  The algorithm is implemented in Mathematica and can be used to test the
integrability of both nonlinear evolution equations and semi-discrete lattice
equations. With our Integrability Package, higher-order symmetries are obtained
for several well-known systems of evolution and lattice equations. For PDEs and
lattices with parameters, the code allows one to determine the conditions on
these parameters so that a sequence of higher-order symmetries exist. The
existence of a sequence of such symmetries is a predictor for integrability.


title: Hierarchy of Higher Dimensional Integrable System

abstract:
  Integrable equations in ($1 + 1$) dimensions have their own higher order
integrable equations, like the KdV, mKdV and NLS hierarchies etc. In this paper
we consider whether integrable equations in ($2 + 1$) dimensions have also the
analogous hierarchies to those in ($1 + 1$) dimensions. Explicitly is discussed
the Bogoyavlenskii-Schiff(BS) equation. For the BS hierarchy, there appears an
ambiguity in the Painlev\'e test. Nevertheless, it may be concluded that the BS
hierarchy is integrable.


title:
Surfaces, curves and the Lakshmanan equivalent counterparts of the some
  Myrzakulov equations


title: Chiral Solutions to Generalized Burgers and Burgers-Huxley Equations

abstract:
  We investigate generalizations of the Burgers and Burgers-Huxley equations.
The investigations we offer focus attention mainly on presenting explict
analytical solutions by means of relating these generalized equations to
relativistic 1+1 dimensional systems of scalar fields where topological
solutions are known to play a role. Emphasis is given on chiral solutions, that
is, on the possibility of finding solutions that travel with velocities
determined in terms of the parameters that identify the generalized equation,
with a definite sign.


title:
A systematic construction of completely integrable Hamiltonians from
  coalgebras

abstract:
  A universal algorithm to construct N-particle (classical and quantum)
completely integrable Hamiltonian systems from representations of coalgebras
with Casimir element is presented. In particular, this construction shows that
quantum deformations can be interpreted as generating structures for integrable
deformations of Hamiltonian systems with coalgebra symmetry. In order to
illustrate this general method, the $so(2,1)$ algebra and the oscillator
algebra $h_4$ are used to derive new classical integrable systems including a
generalization of Gaudin-Calogero systems and oscillator chains. Quantum
deformations are then used to obtain some explicit integrable deformations of
the previous long-range interacting systems and a (non-coboundary) deformation
of the $(1+1)$ Poincar\'e algebra is shown to provide a new
Ruijsenaars-Schneider-like Hamiltonian.


title:
Multi-parameter deformed and nonstandard $Y(gl_M)$ Yangian symmetry in
  integrable variants of Haldane-Shastry spin chain

abstract:
  By using `anyon like' representations of permutation algebra, which pick up
nontrivial phase factors while interchanging the spins of two lattice sites, we
construct some integrable variants of Haldane-Shastry (HS) spin chain. Lax
equations for these spin chains allow us to find out the related conserved
quantities. However, it turns out that such spin chains also possess a few
additional conserved quantities which are apparently not derivable from the Lax
equations. Identifying these additional conserved quantities, and the usual
ones related to Lax equations, with different modes of a monodromy matrix, it
is shown that the above mentioned HS like spin chains exhibit multi-parameter
deformed and `nonstandard' variants of $Y(gl_M)$ Yangian symmetry.


title:
A Dirac Sea and thermodynamic equilibrium for the quantized three-wave
  interaction

abstract:
  The classical version of the three wave interaction models the creation and
destruction of waves; the quantized version models the creation and destruction
of particles. The quantum three wave interaction is described and the Bethe
Ansatz for the eigenfunctions is given in closed form. The Bethe equations are
derived in a rigorous fashion and are shown to have a thermodynamic limit. The
Dirac sea of negative energy states is obtained as the infinite density limit.
Finite particle/hole excitations are determined and the asymptotic relation of
energy and momentum is obtained. The Yang-Yang functional for the relative free
energy of finite density excitations is constructed and is shown to be convex
and bounded below. The equations of thermal equilibrium are obtained.


title: Quadratic reductions of quadrilateral lattices

abstract:
  It is shown that quadratic constraints are compatible with the geometric
integrability scheme of the multidimensional quadrilateral lattice equation.
The corresponding Ribaucour reduction of the fundamental transformation of
quadrilateral lattices is found as well, and superposition of the Ribaucour
transformations is presented in the vectorial framework. Finally, the quadratic
reduction approach is illustrated on the example of multidimensional circular
lattices.


title: Eigenvector and eigenvalue problem for 3D bosonic model

abstract:
  In this paper we reformulate free field theory models defined on the
rectangular $D+1$ dimensional lattices as $D+1$ evolution models. This
evolution is in part a simple linear evolution on free (``creation'' and
``annihilation'') operators. Formal eigenvectors of this linear evolution can
be directly constructed, and them play the role of the ``physical'' creation
and annihilation operators. These operators being completed by a ``physical''
vacuum vector give the spectrum of the evolution operator, as well as the trace
of the evolution operator give a correct expression for the partition function.
As an example, Bazhanov -- Baxter's free bosonic model is considered.


title: Supersymmetric Drinfeld-Sokolov reduction

abstract:
  The Drinfeld-Sokolov construction of integrable hierarchies, as well as its
generalizations, may be extended to the case of loop superalgebras. A
sufficient condition on the algebraic data for the resulting hierarchy to be
invariant under supersymmetry transformation is given. The method used is a
construction of the hierarchies in superspace, where supersymmetry is manifest.
Several examples are discussed.


title: 3D symplectic map

abstract:
  Quantum 3D R-matrix in the classical (i.e. functional) limit gives a
symplectic map of dynamical variables. The corresponding 3D evolution model is
considered. An auxiliary problem for it is a system of linear equations playing
the role of the monodromy matrix in 2D models. A generating function for the
integrals of motion is constructed as a determinant of the auxiliary system.


title:
The nondynamical r-matrix structure of the elliptic
  Ruijsenaars-Schneider model with N=2

abstract:
  We demonstrate that in a certain gauge the elliptic Ruijsenaars-Shneider
model with N=2 admits a nondynamical r-matrix structure and the corresponding
classical r-matrix is the same as that of its non-relativistic counterpart
(Calogero-Moser model) in the same gauge.The relation between our
(classical)Lax operator and the Lax operator given by Ruijsenaars is also
obtained.


title: What is the relativistic Volterra lattice?

abstract:
  We develop a systematic procedure of finding integrable ''relativistic''
(regular one-parameter) deformations for integrable lattice systems. Our
procedure is based on the integrable time discretizations and consists of three
steps. First, for a given system one finds a local discretization living in the
same hierarchy. Second, one considers this discretization as a particular
Cauchy problem for a certain 2-dimensional lattice equation, and then looks for
another meaningful Cauchy problems, which can be, in turn, interpreted as new
discrete time systems. Third, one has to identify integrable hierarchies to
which these new discrete time systems belong. These novel hierarchies are
called then ''relativistic'', the small time step $h$ playing the role of
inverse speed of light. We apply this procedure to the Toda lattice (and
recover the well-known relativistic Toda lattice), as well as to the Volterra
lattice and a certain Bogoyavlensky lattice, for which the ''relativistic''
deformations were not known previously.


title:
Solitons, Surfaces, Curves, and the Spin Description of Nonlinear
  Evolution Equations

abstract:
  The briefly review on the common spin description of the nonlinear evolution
equations.


title: On gauge-equivalent formulations of N=4 SKdV hierarchy

abstract:
  We point out that the N=4 supersymmetric KdV hierarchy, when written through
the prepotentials of the bosonic chiral and antichiral N=2 supercurrents,
exhibits a freedom related to the possibility to choose different gauges for
the prepotentials. In particular, this implies that the Lax operator for the
N=4 SKdV system and the associated realization of N=4 supersymmetry obtained in
solv-int/9802003 are reduced to the previously known ones. We give the
prepotential form of the `small' N=4 superconformal algebra, the second
hamiltonian structure algebra of the N=4 SKdV hierarchy, for two choices of
gauge.


title: Motion of Curves on Two Dimensional Surfaces and Soliton Equations

abstract:
  A connection is established between the soliton equations and curves moving
in a three dimensional space $V_{3}$. The sign of the self-interacting terms of
the soliton equations are related to the signature of $V_{3}$. It is shown that
there corresponds a moving curve to each soliton equations.


title: Extension of Hereditary Symmetry Operators

abstract:
  Two models of candidates for hereditary symmetry operators are proposed and
thus many nonlinear systems of evolution equations possessing infinitely many
commutative symmetries may be generated. Some concrete structures of hereditary
symmetry operators are carefully analyzed on the base of the resulting general
conditions and several corresponding nonlinear systems are explicitly given out
as illustrative examples.


title:
Finsler-Geometrical Approach to the Studying of Nonlinear Dynamical
  Systems

abstract:
  A two dimensional Finsler space associated with the differential equation
$y''=Y_3 y'^3+Y_2 y'^2+Y_1 y'+Y_0$ is characterized by a tensor equation and
called the Douglas space. An application to the Lorenz nonlinear dynamical
equation is discussed from the standpoint of Finsler geometry.


title:
On the Law of Transformation of Affine Connection and its Integration.
  Part 1. Generalization of the Lame equations

abstract:
  The law of transformation of affine connection for n-dimensional manifolds as
the system of nonlinear equations on local coordinates of manifold is
considered. The extension of the Darboux-Lame system of equations to the spaces
of constant negative curvature is demonstrated. Geodesic deviation equation as
well as the equations of geodesics are presented in the form of the matrix
Darboux-Lame system of equations.


title:
Algorithmic Integrability Tests for Nonlinear Differential and Lattice
  Equations

abstract:
  Three symbolic algorithms for testing the integrability of polynomial systems
of partial differential and differential-difference equations are presented.
The first algorithm is the well-known Painlev\'e test, which is applicable to
polynomial systems of ordinary and partial differential equations. The second
and third algorithms allow one to explicitly compute polynomial conserved
densities and higher-order symmetries of nonlinear evolution and lattice
equations.
  The first algorithm is implemented in the symbolic syntax of both Macsyma and
Mathematica. The second and third algorithms are available in Mathematica. The
codes can be used for computer-aided integrability testing of nonlinear
differential and lattice equations as they occur in various branches of the
sciences and engineering. Applied to systems with parameters, the codes can
determine the conditions on the parameters so that the systems pass the
Painlev\'e test, or admit a sequence of conserved densities or higher-order
symmetries.


title: The Painlev\'e Integrability Test

abstract:
  The Painlev\'e test is a widely applied and quite successful technique to
investigate the integrability of nonlinear ODEs and PDEs by analyzing the
singularity structure of the solutions. The test is named after the French
mathematician Paul Painlev\'e ....


title: Miura Transformation between two Non-Linear Equations in 2+1 dimensions

abstract:
  A Dispersive Wave Equation in 2+1 dimensions (2LDW) widely discussed by
different authors is shown to be nothing but the modified version of the
Generalized Dispersive Wave Equation (GLDW). Using Singularity Analysis and
techniques based upon the Painleve Property leading to the Double Singular
Manifold Expansion we shall find the Miura Transformation which converts the
2LDW Equation into the GLDW Equation. Through this Miura Transformation we
shall also present the Lax pair of the 2LDW Equation as well as some
interesting reductions to several already known integrable systems in 1+1
dimensions.


title:
To the Gel'fand-Tsetlin realization of irreducible representations of
  classical semisimple algebras

abstract:
  It is shown that the Gel'fand-Tsetlin realization of irreducible
representations of the $A_n$ algebra is directly connected with a linear
exactly integrable system in the n-dimensional space. General solution for this
system is explicitly given.


title: A Class of Coupled KdV systems and Their Bi-Hamiltonian Formulations

abstract:
  A Hamiltonian pair with arbitrary constants is proposed and thus a sort of
hereditary operators is resulted. All the corresponding systems of evolution
equations possess local bi-Hamiltonian formulation and a special choice of the
systems leads to the KdV hierarchy. Illustrative examples are given.


title: The solution of the N=(0|2) superconformal f-Toda lattice

abstract:
  The general solution of the two-dimensional integrable generalization of the
f-Toda chain with fixed ends is explicitly presented in terms of matrix
elements of various fundamental representations of the SL(n|n-1) supergroup.
The dominant role of the representation theory of graded Lie algebras in the
problem of constructing integrable mappings and lattices is demonstrated.


title:
On the simplest (2+1) dimensional integrable spin systems and their
  equivalent nonlinear Schr\"odinger equations

abstract:
  Using a moving space curve formalism, geometrical as well as gauge
equivalence between a (2+1) dimensional spin equation (M-I equation) and the
(2+1) dimensional nonlinear Schr\"odinger equation (NLSE) originally discovered
by Calogero, discussed then by Zakharov and recently rederived by Strachan,
have been estabilished. A compatible set of three linear equations are obtained
and integrals of motion are discussed. Through stereographic projection, the
M-I equation has been bilinearized and different types of solutions such as
line and curved solitons, breaking solitons, induced dromions, and domain wall
type solutions are presented. Breaking soliton solutions of (2+1) dimensional
NLSE have also been reported. Generalizations of the above spin equation are
discussed.


title: N-Soliton Solutions to a New (2 + 1) Dimensional Integrable Equation

abstract:
  We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation,
$\phi_{xt} + \phi_{xxxz}/4 + \phi_x \phi_{xz} + \phi_{xx} \phi_z/2 +
\partial_x^{-1} \phi_{zzz}/4 = 0$. This equation is obtained by unifying two
directional generalization of the KdV equation, composing the closed ring with
the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura
transformation which yields the same ring in the corresponding modified
equations.


title:
Towards second order Lax pairs to discrete Painlev\'e equations of first
  degree

abstract:
  We investigate the question of finding discrete Lax pairs for the six
discrete Painlev\'e equations (Pn). The choice we make is to discretize the
pairs of Garnier, once converted to matricial form.


title: Rules of discretization for Painlev\'e equations

abstract:
  The discrete Painlev\'e property is precisely defined, and basic
discretization rules to preserve it are stated. The discrete Painlev\'e test is
enriched with a new method which perturbs the continuum limit and generates
infinitely many no-log conditions. A general, direct method is provided to
search for discrete Lax pairs.


title:
Charged Free Fermions, Vertex Operators and Classical Theory of
  Conjugate Nets

abstract:
  We show that the quantum field theoretical formulation of the $\tau$-function
theory has a geometrical interpretation within the classical transformation
theory of conjugate nets. In particular, we prove that i) the partial charge
transformations preserving the neutral sector are Laplace transformations, ii)
the basic vertex operators are Levy and adjoint Levy transformations and iii)
the diagonal soliton vertex operators generate fundamental transformations. We
also show that the bilinear identity for the multicomponent
Kadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, a
bilinear identity for the multidimensional quadrilateral lattice equations.


title: Integrable discretizations of the Euler top

abstract:
  Discretizations of the Euler top sharing the integrals of motion with the
continuous time system are studied. Those of them which are also Poisson with
respect to the invariant Poisson bracket of the Euler top are characterized.
For all these Poisson discretizations a solution in terms of elliptic functions
is found, allowing a direct comparison with the continuous time case. We
demonstrate that the Veselov--Moser discretization also belongs to our family,
and apply our methods to this particular example.


title:
All generalized SU(2) chiral models have spectral dependent Lax
  formulation

abstract:
  The equations that define the Lax pairs for generalized principal chiral
models can be solved for any nondegenerate bilinear form on $su(2)$. The
solution is dependent on one free variable that can serve as the spectral
parameter.


title: On lump instability of Davey--Stewartson II equation

abstract:
  We show that lumps (solitons) of the Davey--Stewartson II equation fail under
small perturbations of initial data.


title:
The system of three vortexes of two dimensional ideal hydrodinamics as a
  new example of the (integrable) Nambu- Poisson mechanics

abstract:
  A Nambu-Poisson formulation of the system of three ordinary differential
equations describing dynamics of three vortexes of the ideal two-dimensional
hydrodynamics is given. The system is integrated by quadratures.


title: Painlev\'e analysis for nonlinear partial differential equations

abstract:
  The Painlev\'e analysis introduced by Weiss, Tabor and Carnevale (WTC) in
1983 for nonlinear partial differential equations (PDE's) is an extension of
the method initiated by Painlev\'e and Gambier at the beginning of this century
for the classification of algebraic nonlinear differential equations (ODE's)
without movable critical points. In these lectures we explain the WTC method in
its invariant version introduced by Conte in 1989 and its application to
solitonic equations in order to find algorithmically their associated
B\"acklund transformation. A lot of remarkable properties are shared by these
so-called ``integrable'' equations but they are generically no more valid for
equations modelising physical phenomema. Belonging to this second class, some
equations called ``partially integrable'' sometimes keep remnants of
integrability. In that case, the singularity analysis may also be useful for
building closed form analytic solutions, which necessarily % Conte agree with
the singularity structure of the equations. We display the privileged role
played by the Riccati equation and systems of Riccati equations which are
linearisable, as well as the importance of the Weierstrass elliptic function,
for building solitary waves or more elaborate solutions.


title:
Correlation Functions, Cluster Functions and Spacing Distributions for
  Random Matrices

abstract:
  The usual formulas for the correlation functions in orthogonal and symplectic
matrix models express them as quaternion determinants. From this representation
one can deduce formulas for spacing probabilities in terms of Fredholm
determinants of matrix-valued kernels. The derivations of the various formulas
are somewhat involved. In this article we present a direct approach which leads
immediately to scalar kernels for unitary ensembles and matrix kernels for the
orthogonal and symplectic ensembles, and the representations of the correlation
functions, cluster functions and spacing distributions in terms of them.


title:
On the relation between orthogonal, symplectic and unitary matrix
  ensembles

abstract:
  For the unitary ensembles of $N\times N$ Hermitian matrices associated with a
weight function $w$ there is a kernel, expressible in terms of the polynomials
orthogonal with respect to the weight function, which plays an important role.
For the orthogonal and symplectic ensembles of Hermitian matrices there are
$2\times2$ matrix kernels, usually constructed using skew-orthogonal
polynomials, which play an analogous role. These matrix kernels are determined
by their upper left-hand entries. We derive formulas expressing these entries
in terms of the scalar kernel for the corresponding unitary ensembles. We also
show that whenever $w'/w$ is a rational function the entries are equal to the
scalar kernel plus some extra terms whose number equals the order of $w'/w$.
General formulas are obtained for these extra terms. We do not use
skew-orthogonal polynomials in the derivations.


title: Modular Solutions to Equations of Generalized Halphen Type

abstract:
  Solutions to a class of differential systems that generalize the Halphen
system are determined in terms of automorphic functions whose groups are
commensurable with the modular group. These functions all uniformize Riemann
surfaces of genus zero and have $q$--series with integral coefficients.
Rational maps relating these functions are derived, implying subgroup relations
between their automorphism groups, as well as symmetrization maps relating the
associated differential systems.

categories: solv-int hep-th math-ph math.MP math.QA nlin.SI


title: Novel integrable spin-particle models from gauge theories on a cylinder

abstract:
  We find and solve a large class of integrable dynamical systems which
includes Calogero-Sutherland models and various novel generalizations thereof.
In general they describe $N$ interacting particles moving on a circle and
coupled to an arbitrary number, $m$, of $su(N)$ spin degrees of freedom with
interactions which depend on arbitrary real parameters $x_j$, $j=1,2,...,m$. We
derive these models from SU(N) Yang-Mills gauge theory coupled to non-dynamic
matter and on spacetime which is a cylinder. This relation to gauge theories is
used to prove integrability, to construct conservation laws, and solve these
models.


title:
On the exact solutions of the Bianchi IX cosmological model in the
  proper time

abstract:
  It has recently been argued that there might exist a four-parameter analytic
solution to the Bianchi IX cosmological model, which would extend the
three-parameter solution of Belinskii et al. to one more arbitrary constant. We
perform the perturbative Painlev\'e test in the proper time variable, and
confirm the possible existence of such an extension.


title: The Davey Stewartson system and the B\"{a}cklund Transformations

abstract:
  We consider the (coupled) Davey-Stewartson (DS) system and its B\"{a}cklund
transformations (BT). Relations among the DS system, the double
Kadomtsev-Petviashvili (KP) system and the Ablowitz-Ladik hierarchy (ALH) are
established. The DS hierarchy and the double KP system are equivalent. The ALH
is the BT of the DS system in a certain reduction. {From} the BT of coupled DS
system we can obtain new coupled derivative nonlinear Schr\"{o}dinger
equations.


title: Determinant formula for the six-vertex model with reflecting end

abstract:
  Using the Quantum Inverse Scattering Method for the XXZ model with open
boundary conditions, we obtained the determinant formula for the six vertex
model with reflecting end.


title: The Gambier Mapping, Revisited

abstract:
  We examine critically the Gambier equation and show that it is the generic
linearisable equation containing, as reductions, all the second-order equations
which are integrable through linearisation. We then introduce the general
discrete form of this equation, the Gambier mapping, and present conditions for
its integrability. Finally, we obtain the reductions of the Gambier mapping,
identify their integrable forms and compute their continuous limits.


title: Again, Linearizable Mappings

abstract:
  We examine a family of 3-point mappings that include mappings solvable
through linearization. The different origins of mappings of this type are
examined: projective equations and Gambier systems. The integrable cases are
obtained through the application of the singularity confinement criterion and
are explicitly integrated.


title:
Higher Order Asymptotics of the Modified Non-Linear Schr\"{o}dinger
  Equation

abstract:
  Using the matrix Riemann-Hilbert factorisation approach for non-linear
evolution systems which take the form of Lax-pair isospectral deformations, the
higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the
solution to the Cauchy problem for the modified non-linear Schr\"{o}dinger
equation, $i \partial_{t} u + {1/2} \partial_{x}^{2} u + | u |^{2} u + i s
\partial_{x} (| u |^{2} u) = 0$, $s \in \Bbb R_{> 0}$, which is a model for
non-linear pulse propagation in optical fibres in the subpicosecond time scale,
are obtained: also derived are analogous results for two gauge-equivalent
non-linear evolution equations; in particular, the derivative non-linear
Schr\"{o}dinger equation, $i \partial_{t} q + \partial_{x}^{2} q - i
\partial_{x}(| q |^{2} q) = 0$.


title:
The Gel'fand-Tsetlin Selection Rules and Representations of Quantum
  Algebras

abstract:
  The problem of construction of irreducible representations of quantum $A^q_n$
algebras is solved at the level of explicit integration of the linear
(inhomogeneous) system in finite differences in the n-dimensional space. The
general solution of this system is given explicitly and particular ones, which
correspond to the irreducible representations are selected.


title:
Knizhnik-Zamolodchikov-Bernard equations connected with the eight-vertex
  model

abstract:
  Using quasiclassical limit of Baxter's 8 - vertex R - matrix, an elliptic
generalization of the Knizhnik-Zamolodchikov equation is constructed. Via
Off-Shell Bethe ansatz an integrable representation for this equation is
obtained. It is shown that there exists a gauge transformation connecting this
equation with Knizhnik-Zamolodchikov-Bernard equation for SU(2)-WZNW model on
torus.

categories: solv-int cond-mat.stat-mech hep-th nlin.SI


title:
Elliptic solutions to difference non-linear equations and nested Bethe
  ansatz equations

abstract:
  We outline an approach to a theory of various generalizations of the elliptic
Calogero-Moser (CM) and Ruijsenaars-Shneider (RS) systems based on a special
inverse problem for linear operators with elliptic coefficients. Hamiltonian
theory of such systems is developed with the help of the universal symplectic
structure proposed by D.H. Phong and the author. Canonically conjugated
action-angle variables for spin generalizations of the elliptic CM and RS
systems are found.


title: Travelling Wave Solutions in Nonlinear Diffusive and Dispersive Media

abstract:
  We investigate the presence of soliton solutions in some classes of nonlinear
partial differential equations, namely generalized Korteweg-de Vries-Burgers,
Korteveg-de Vries-Huxley, and Korteveg-de Vries-Burgers-Huxley equations, which
combine effects of diffusion, dispersion, and nonlinearity. We emphasize the
chiral behavior of the travelling solutions, whose velocities are determined by
the parameters that define the equation. For some appropriate choices, we show
that these equations can be mapped onto equations of motion of relativistic 1+1
dimensional phi^{4} and phi^{6} field theories of real scalar fields. We also
study systems of two coupled nonlinear equations of the types mentioned.


title: On The Stability of the Compacton Solutions

abstract:
  The stability of the recently discovered compacton solutions is studied by
means of both linear stability analysis as well as Lyapunov stability criteria.
  From the results obtained it follows that, unlike solitons, all the allowed
compacton solutions are stable, as the stability condition is satisfied for
arbitrary values of the nonlinearity parameter. The results are shown to be
true even for the higher order nonlinear dispersion equations for compactons.
Some new conservation laws for the higher order nonlinear dispersion equations
are also presented.

categories: solv-int cond-mat hep-th nlin.SI quant-ph


title: A nonlinear indentity for the scattering phase of integrable models

abstract:
  A nonlinear identity for the scattering phase of quantum integrable models is
proved.


title: Analytical Study of the Julia Set of a Coupled Generalized Logistic Map

abstract:
  A coupled system of two generalized logistic maps is studied. In particular
influence of the coupling to the behaviour of the Julia set in two dimensional
complex space is analyzed both analytically and numerically. It is proved
analytically that the Julia set disappears from the complex plane uniformly as
a parameter interpolates from the chaotic phase to the integrable phase, if the
coupling strength satisfies a certain condition.


title:
On the Miura and Backlund transformations associated with the
  supersymmetric Gelfand-Dickey bracket

abstract:
  The supersymmetric version of the Miura and B\"acklund transformations
associated with the supersymmetric Gelfand-Dickey bracket are investigated from
the point of view of the Kupershmidt-Wilson theorem.


title: Hidden Algebra of Three-Body Integrable Systems

abstract:
  It is shown that all 3-body quantal integrable systems that emerge in the
Hamiltonian reduction method possess the same hidden algebraic structure. All
of them are given by a second degree polynomial in generators of an
infinite-dimensional Lie algebra of differential operators. It leads to new
families of the orthogonal polynomials in two variables.

categories: solv-int cond-mat.stat-mech hep-th math-ph math.MP math.RT math.SP nlin.SI


title: Long range integrable oscillator chains from quantum algebras

abstract:
  Completely integrable Hamiltonians defining classical mechanical systems of
$N$ coupled oscillators are obtained from Poisson realizations of
Heisenberg--Weyl, harmonic oscillator and $sl(2,\R)$ coalgebras. Various
completely integrable deformations of such systems are constructed by
considering quantum deformations of these algebras. Explicit expressions for
all the deformed Hamiltonians and constants of motion are given, and the
long-range nature of the interactions is shown to be linked to the underlying
coalgebra structure. The relationship between oscillator systems induced from
the $sl(2,\R)$ coalgebra and angular momentum chains is presented, and a
non-standard integrable deformation of the hyperbolic Gaudin system is
obtained.


title:
Reduced Vectorial Ribaucour Transformation for the Darboux-Egoroff
  Equations

abstract:
  The vectorial fundamental transformation for the Darboux equations is reduced
to the symmetric case. This is combined with the orthogonal reduction of Lame
type to obtain those vectorial Ribaucour transformations which preserve the
Egoroff reduction. We also show that a permutability property holds for all
these transformations. Finally, as an example, we apply these transformations
to the Cartesian background.

categories: solv-int math-ph math.DG math.MP nlin.SI


title: On Grassmannian Description of the Constrained KP Hierarchy

abstract:
  This note develops an explicit construction of the constrained KP hierarchy
within the Sato Grassmannian framework. Useful relations are established
between the kernel elements of the underlying ordinary differential operator
and the eigenfunctions of the associated KP hierarchy as well as between the
related bilinear concomitant and the squared eigenfunction potential.


title:
Integrable Extensions of N=2 Supersymmetric KdV Hierarchy Associated
  with the Nonuniqueness of the Roots of the Lax operator

abstract:
  We preesent a new supersymmetric integrable extensions of the a=4,N=2 KdV
hierarchy. The root of the supersymmetric Lax operator of the KdV equation is
generalized, by including additional fields. This generalized root generate new
hierarchy of integrable equations, for which we investigate the hamiltonian
structure. In special case our system describes the interaction of the KdV
equation with the two MKdV equations.


title:
Exact Kink Solitons in the Presence of Diffusion, Dispersion, and
  Polynomial Nonlinearity

abstract:
  We describe exact kink soliton solutions to nonlinear partial differential
equations in the generic form u_{t} + P(u) u_{x} + \nu u_{xx} + \delta u_{xxx}
= A(u), with polynomial functions P(u) and A(u) of u=u(x,t), whose generality
allows the identification with a number of relevant equations in physics. We
emphasize the study of chirality of the solutions, and its relation with
diffusion, dispersion, and nonlinear effects, as well as its dependence on the
parity of the polynomials $P(u)$ and $A(u)$ with respect to the discrete
symmetry $u\to-u$. We analyze two types of kink soliton solutions, which are
also solutions to 1+1 dimensional phi^{4} and phi^{6} field theories.


title: Multiplicity A_m Models

abstract:
  Models generalizing the su(2) XX spin-chain were recently introduced. These
XXC models also have an underlying su(2) structure. Their construction method
is shown to generalize to the chains based on the fundamental representations
of the A_m Lie algebras. Integrability of the new models is shown in the
context of the quantum inverse scattering method. Their R-matrix is found and
shown to yield a representation of the Hecke algebra. The diagonalization of
the transfer matrices is carried out using the algebraic Bethe Ansatz. I
comment on eventual generalizations and possible links to reaction-diffusion
processes.


title:
From Ramond Fermions to Lame Equations for Orthogonal Curvilinear
  Coordinates

abstract:
  We show how Ramond free neutral Fermi fields lead to a $\tau$-function theory
of BKP type which describes iso-orthogonal deformations of systems of ortogonal
curvilinear coordinates. We also provide a vertex operator representation for
the classical Ribaucour transformation.

categories: solv-int hep-th math-ph math.DG math.MP nlin.SI


title:
Connection formulae for degenerated asymptotic solutions of the fourth
  Painleve equation

abstract:
  All possible 1-parametric classical and transcendent degenerated solutions of
the fourth Painleve equation with the corresponding connection formulae of the
asymptotic parameters are described.


title: On integrability of a (2+1)-dimensional perturbed Kdv equation

abstract:
  A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma
and B. Fuchssteiner, is proven to pass the Painlev\'e test for integrability
well, and its 4$\times $4 Lax pair with two spectral parameters is found. The
results show that the Painlev\'e classification of coupled KdV equations by A.
Karasu should be revised.

categories: solv-int math-ph math.AP math.MP nlin.SI


title: Quantization of cohomology in semi-simple Lie algebras

abstract:
  The space of realizations of a finite-dimensional Lie algebra by first order
differential operators is naturally isomorphic to H^1 with coefficients in the
module of functions. The condition that a realization admits a
finite-dimensional invariant subspace of functions seems to act as a kind of
quantization condition on this H^1. It was known that this quantization of
cohomology holds for all realizations on 2-dimensional homogeneous spaces, but
the extent to which quantization of cohomology is true in general was an open
question. The present article presents the first known counter-examples to
quantization of cohomology; it is shown that quantization can fail even if the
Lie algebra is semi-simple, and even if the homogeneous space in question is
compact. A explanation for the quantization phenomenon is given in the case of
semi-simple Lie algebras. It is shown that the set of classes in H^1 that admit
finite-dimensional invariant subspaces is a semigroup that lies inside a
finitely-generated abelian group. In order for this abelian group be a discrete
subset of H^1, i.e. in order for quantization to take place, some extra
conditions on the isotropy subalgebra are required. Two different instances of
such necessary conditions are presented.


title:
Singular sector of the KP hierarchy, $\bar{\partial}$-operators of
  non-zero index and associated integrable systems

abstract:
  Integrable hierarchies associated with the singular sector of the KP
hierarchy, or equivalently, with $\dbar$-operators of non-zero index are
studied. They arise as the restriction of the standard KP hierarchy to
submanifols of finite codimension in the space of independent variables. For
higher $\dbar$-index these hierarchies represent themselves families of
multidimensional equations with multidimensional constraints. The
$\dbar$-dressing method is used to construct these hierarchies. Hidden KdV,
Boussinesq and hidden Gelfand-Dikii hierarchies are considered too.


title:
Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP
  Hierarchy

abstract:
  We use ideas of the geometry of bihamiltonian manifolds, developed by
Gel'fand and Zakharevich, to study the KP equations. In this approach they have
the form of local conservation laws, and can be traded for a system of ordinary
differential equations of Riccati type, which we call the Central System. We
show that the latter can be linearized by means of a Darboux covering, and we
use this procedure as an alternative technique to construct rational solutions
of the KP equations.


title:
Imprimitively generated Lie-algebraic Hamiltonians and separation of
  variables

abstract:
  Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose
domain is a subset of the Euclidean plane admits a separation of variables. A
proof of this conjecture is given in those cases where the generating
Lie-algebra acts imprimitively. The general form of the conjecture is false. A
counter-example is given based on the trigonometric Olshanetsky-Perelomov
potential corresponding to the A_2 root system.


title: Pfaffian form of the Grammian determinant solutions of the BKP hierarchy

abstract:
  The Grammian determinant type solutions of the KP hierarchy, obtained through
the vectorial binary Darboux transformation, are reduced, imposing suitable
differential constraint on the transformation data, to Pfaffian solutions of
the BKP hierarchy.


title:
Pfaffian Solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY
  sine-Gordon Equations

abstract:
  We reduce the vectorial binary Darboux transformation for the Manin-Radul
supersymmetric KdV system in such a way that it preserves the
Manin-Radul-Mathieu supersymmetric KdV equation reduction. Expressions in terms
of bosonic Pfaffians are provided for transformed solutions and wave functions.
We also consider the implications of these results for the supersymmetric
sine-Gordon equation.


title: Initial boundary value problem on a half-line for the MKdV equation

abstract:
  Initial boundary value problem on a half-line for the Modified KdV equation
is considered with the boundary conditions equal to zero at the origin and
initial condition chosen arbitrary decreasing rapidly enough and this problem
is plunged into the scheme of the inverse scattering method. Here the inverse
scattering problem is reduced to the Riemann problem on a system of rays on the
complex plane.


title:
Phonon Scattering by Breathers in the Discrete Nonlinear Schroedinger
  Equation

abstract:
  Linear theory for phonon scattering by discrete breathers in the discrete
nonlinear Schroedinger equation using the transfer matrix approach is
presented. Transmission and reflection coefficients are obtained as a function
of the wave vector of the input phonon. The occurrence of a nonzero
transmission, which in fact becomes perfect for a symmetric breather, is shown
to be connected with localized eigenmodes thresholds. In the weak-coupling
limit, perfect reflection are shown to exist, which requires two scattering
channels. A necessary condition for a system to have a perfect reflection is
also considered in a general context.

categories: solv-int cond-mat.dis-nn nlin.SI quant-ph


title:
M\"obius invariant integrable lattice equations associated with KP and
  2DTL hierarchies

abstract:
  Integrable lattice equations arising in the context of singular manifold
equations for scalar, multicomponent KP hierarchies and 2D Toda lattice
hierarchy are considered. These equation generate the corresponding continuous
hierarchy of singular manifold equations, its B\"acklund transformations and
different forms of superposition principles. They possess rather special form
of compatibility representation. The distinctive feature of these equations is
invariance under the action of M\"obius transformation. Geometric
interpretation of these discrete equations is given.


title: Geometric B\"acklund--Darboux transformations for the KP hierarchy

abstract:
  We shown that, if you have two planes in the Segal-Wilson Grassmannian that
have an intersection of finite codimension, then the corresponding solutions of
the KP hierarchy are linked by B\"acklund-Darboux transformations (BDT). The
pseudodifferential operator that performs this transformation is shown to be
built up in a geometric way from elementary BDT's and is given here in a closed
form. The geometric description of elementary DBT's requires that one has a
geometric interpretation of the dual wavefunctions involved. This is done here
with the help of a suitable algebraic characterization of the wavefunction. The
BDT's also induce transformations of the tau-function associated to a plane in
the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric
characterization of the BDT'ss that preserves these subsystems of the KP
hierarchy. This generalizes the classical Darboux-transformations. we also
determine an explicit expression for the squared eigenfunction potentials. Next
a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy.
It is shown that infinite flags in the Grassmannian yield solutions of the
latter hierarchy. these flags can be constructed by means of BDT's, starting
from some plane. Other applications of these BDT's are a geometric way to
characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy
and the construction of a vast collection of orthogonal polynomials, playing a
role in matrix models.


title: DNA Transcription Mechanism with a Moving Enzyme

abstract:
  Previous numerical investigations of an one-dimensional DNA model with an
extended modified coupling constant by transcripting enzyme are integrated to
longer time and demonstrated explicitly the trapping of breathers by DNA chains
with realistic parameters obtained from experiments. Furthermore, collective
coordinate method is used to explain a previously observed numerical evidence
that breathers placed far from defects are difficult to trap, and the motional
effect of RNA-polymerase is investigated.


title:
Extending Hamiltonian Operators to Get Bi-Hamiltonian Coupled KdV
  Systems

abstract:
  An analysis of extension of Hamiltonian operators from lower order to higher
order of matrix paves a way for constructing Hamiltonian pairs which may result
in hereditary operators. Based on a specific choice of Hamiltonian operators of
lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a
consequence of bi-Hamiltonian structure, they all possess infinitely many
symmetries and infinitely many conserved densities.


title: Schlesinger Transformations for Linearisable Equations

abstract:
  We introduce the Schlesinger transformations of the Gambier equation. The
latter can be written, in both the continuous and discrete cases, as a system
of two coupled Riccati equations in cascade involving an integer parameter n.
In the continuous case the parameter appears explicitly in the equation while
in the discrete case it corresponds to the number of steps for singularity
confinement. Two Schlesinger transformations are obtained relating the
solutions for some value $n$ to that corresponding to either n+1 or n+2.


title:
Degenerate Frobenius manifolds and the bi-Hamiltonian structure of
  rational Lax equations

abstract:
  The bi-Hamiltonian structure of certain multi-component integrable systems,
generalizations of the dispersionless Toda hierarchy, is studies for systems
derived from a rational Lax function. One consequence of having a rational
rather than a polynomial Lax function is that the corresponding bi-Hamiltonian
structures are degenerate, i.e. the metric which defines the Hamiltonian
structure has vanishing determinant. Frobenius manifolds provide a natural
setting in which to study the bi-Hamiltonian structure of certain classes of
hydrodynamic systems. Some ideas on how this structure may be extanded to
include degenerate bi-Hamiltonian structures, such as those given in the first
part of the paper, are given.


title: 'Universality' of the Ablowitz-Ladik hierarchy

abstract:
  The aim of this paper is to summarize some recently obtained relations
between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It
has been shown that solutions of finite subsystems of the ALH can be used to
derive a wide range of solutions for, e.g., the 2D Toda lattice, nonlinear
Schr\"odinger, Davey-Stewartson, Kadomtsev-Petviashvili (KP) and some other
equations. Similar approach has been used to construct new integrable models:
O(3,1) and multi-field sigma models. Such 'universality' of the ALH becomes
more transparent in the framework of the Hirota's bilinear method. The ALH,
which is usually considered as an infinite set of differential-difference
equations, has been presented as a finite system of functional-difference
equations, which can be viewed as a generalization of the famous bilinear
identities for the KP tau-functions.


title: Dubrovin equations and integrable systems on hyperelliptic curves

abstract:
  We introduce the most general version of Dubrovin-type equations for divisors
on a hyperelliptic curve of arbitrary genus, and provide a new argument for
linearizing the corresponding completely integrable flows. Detailed
applications to completely integrable systems, including the KdV, AKNS, Toda,
and the combined sine-Gordon and mKdV hierarchies, are made. These
investigations uncover a new principle for 1+1-dimensional integrable soliton
equations in the sense that the Dubrovin equations, combined with appropriate
trace formulas, encode all hierarchies of soliton equations associated with
hyperelliptic curves. In other words, completely integable hierarchies of
soliton equations determine Dubrovin equations and associated trace formulas
and, vice versa, Dubrovin-type equations combined with trace formulas permit
the construction of hierarchies of soliton equations.


title: On the integrability of nonlinear partial differential equations

abstract:
  We investigate the integrability of Nonlinear Partial Differential Equations
(NPDEs). The concepts are developed by firstly discussing the integrability of
the KdV equation. We proceed by generalizing the ideas introduced for the KdV
equation to other NPDEs. The method is based upon a linearization principle
which can be applied on nonlinearities which have a polynomial form. We
illustrate the potential of the method by finding solutions of the (coupled)
nonlinear Schr\"{o}dinger equation and the Manakov equation which play an
important role in optical fiber communication. Finally, it is shown that the
method can also be generalized to higher-dimensions.


title: Separation of Variables in the Elliptic Gaudin Model

abstract:
  For the elliptic Gaudin model (a degenerate case of XYZ integrable spin
chain) a separation of variables is constructed in the classical case. The
corresponding separated coordinates are obtained as the poles of a suitably
normalized Baker-Akhiezer function. The classical results are generalized to
the quantum case where the kernel of separating integral operator is
constructed. The simplest one-degree-of-freedom case is studied in detail.


title:
Non-classical symmetries and the singular manifold method: A further two
  examples

abstract:
  This paper discusses two equations with the conditional Painleve property.
The usefulness of the singular manifold method as a tool for determining the
non-classical symmetries that reduce the equations to ordinary differential
equations with the Painleve property is confirmed once more


title: An Approach to Master Symmetries of Lattice Equations

abstract:
  An approach to master symmetries of lattice equations is proposed by the use
of discrete zero curvature equation. Its key is to generate non-isospectral
flows from the discrete spectral problem associated with a given lattice
equation. A Volterra-type lattice hierarchy and the Toda lattice hierarchy are
analyzed as two illustrative examples.


title:
$A_n^{(1)}$ Toda Solitons: a Relation between Dressing transformations
  and Vertex Operators

abstract:
  Affine Toda equations based on simple Lie algebras arise by imposing zero
curvature condition on a Lax connection which belongs to the corresponding loop
Lie algebra in the principal gradation. In the particular case of $A_n^{(1)}$
  Toda models, we exploit the symmetry of the underlying linear problem to
calculate the dressing group element which generates arbitrary $N$-soliton
solution from the vacuum. Starting from this result we recover the vertex
operator representation of the soliton tau functions.


title: Polynomial rings of the chiral $SU(N)_{2}$ models

abstract:
  Via explicit diagonalization of the chiral $SU(N)_{2}$ fusion matrices, we
discuss the possibility of representing the fusion ring of the chiral SU(N)
models, at level K=2, by a polynomial ring in a single variable when $N$ is odd
and by a polynomial ring in two variables when $N$ is even.


title: From One-Component KP Hierarchy to Two-Component KP Hierarchy and Back

abstract:
  We show that the system of the standard one-component KP hierarchy endowed
with a special infinite set of abelian additional symmetries, generated by
squared eigenfunction potentials, is equivalent to the two-component KP
hierarchy.


title:
Berezinian Construction of Super-Solitons in Supersymmetric Constrained
  KP Hierarchies

abstract:
  We consider a broad class of consistently reduced Manin-Radul supersymmetric
KP hierarchies (MR-SKP) which are supersymmetric analogs of the ordinary
bosonic constrained KP models. Compatibility of these reductions with the MR
fermionic isospectral flows is achieved via appropriate modification of the
latter preserving their (anti-)commutation algebra. Unlike the general
unconstrained MR-SKP case, Darboux-Backlund transformations do preserve the
fermionic isospectral flows of the reduced MR-SKP hierarchies. This allows for
a systematic derivation of explicit Berezinian solutions for the
super-tau-functions (super-solitons) for these models.


title:
Relationship Between the Energy Eigenstates of Calogero-Sutherland
  Models With Oscillator and Coulomb-like Potentials

abstract:
  We establish a simple algebraic relationship between the energy eigenstates
of the rational Calogero-Sutherland model with harmonic oscillator and
Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in
both of these models which plays a crucial role in such an identification.
Further, we show that our analysis is in fact valid for any many-particle
system in arbitrary dimensions whose potential term (apart from the oscillator
or the Coulomb-like potential) is a homogeneous function of coordinates of
degree -2. The explicit coordinate transformation which maps the Coulomb-like
problem to the oscillator one has also been determined in some specific cases.


title:
Darboux transformations for twisted so(p,q) system and local isometric
  immersion of space forms

abstract:
  For the n-dimensional integrable system with a twisted so(p,q) reduction,
Darboux transformations given by Darboux matrices of degree 2 are constructed
explicitly. These Darboux transformations are applied to the local isometric
immersion of space forms with flat normal bundle and linearly independent
curvature normals to give the explicit expression of the position vector. Some
examples are given from the trivial solutions and standard imbedding T^n\to
R^{2n}.


title: Equations of Geodesic Deviation and the Inverse Scattering Transform

abstract:
  Solutions of equations of geodesic deviation in three- and four- dimensional
spaces obtained by the inverse scattering transform are considered. It is shown
that in the case of three-dimensional space solutions of geodesic deviation
equations are reduced to solutions of the well-known Zakharov-Shabat problem.
In four- dimensional space system of geodesic deviation equations is associated
with $3\times 3$ matrix Schr\"{o}dinger equation, and dependence on parameters
defined by the nonlinear equations of three-wave interaction.


title: The construction of Frobenius manifolds from KP tau-functions

abstract:
  Frobenius manifolds (solutions of WDVV equations) in canonical coordinates
are determined by the system of Darboux-Egoroff equations. This system of
partial differential equations appears as a specific subset of the
$n$-component KP hierarchy. KP representation theory and the related Sato
infinite Grassmannian are used to construct solutions of this Darboux-Egoroff
system and the related Frobenius manifolds. Finally we show that for these
solutions Dubrovin's isomonodromy tau-function can be expressed in the KP
tau-function.


title: Finite gap integration of a discrete Euler top

abstract:
  In [1] new discretizations of the Euler top have been found. They can be
discribed with a Lax pair with a spectral parameter on an elliptic curve. This
is used in this paper to perform a finite gap integration.


title:
Binary Nonlinearization of AKNS Spectral Problem under Higher-Order
  Symmetry Constraints

abstract:
  Binary nonlinearization of AKNS spectral problem is extended to the cases of
higher-order symmetry constraints. The Hamiltonian structures, Lax
representations, $r$-matrices and integrals of motion in involution are
explicitly proposed for the resulting constrained systems in the cases of the
first four orders. The obtained integrals of motion are proved to be
functionally independent and thus the constrained systems are completely
integrable in the Liouville sense.


title: Localized solitons of hyperbolic su(N) AKNS system

abstract:
  Using the nonlinear constraint and Darboux transformation methods, the
(m_1,...,m_N) localized solitons of the hyperbolic su(N) AKNS system are
constructed. Here "hyperbolic su(N)" means that the first part of the Lax pair
is F_y=JF_x+U(x,y,t)F where J is constant real diagonal and U^*=-U. When
different solitons move in different velocities, each component U_{ij} of the
solution U has at most m_i m_j peaks as t tends to infinity. This corresponds
to the (M,N) solitons for the DSI equation. When all the solitons move in the
same velocity, U_{ij} still has at most m_i m_j peaks if the phase differences
are large enough.


title: On The KMS Condition for the critical Ising model

abstract:
  Using the KMS condition and exchange algebras we discuss the monodromy and
modular properties of two-point KMS states of the critical Ising model.


title: On fusion algebra of chiral $SU(N)_{k}$ models

abstract:
  We discuss some algebraic setting of chiral $SU(N)_{k}$ models in terms of
the statistical dimensions of their fields. In particular, the conformal
dimensions and the central charge of the chiral $SU(N)_{k}$ models are
calculated from their braid matrices. Futhermore, at level K=2, we present the
characteristic polynomials of their fusion matrices in a factored form.


title:
Inversion of the linearized Korteweg-deVries equation at the
  multi-soliton solutions

abstract:
  Uniform estimates for the decay structure of the $n$-soliton solution of the
Korteweg-deVries equation are obtained. The KdV equation, linearized at the
$n$-soliton solution is investigated in a class $\WW$ consisting of sums of
travelling waves plus an exponentially decaying residual term. An analog of the
kernel of the time-independent equation is proposed, leading to solvability
conditions on the inhomogeneous term. Estimates on the inversion of the
linearized KdV equation at the $n$-soliton are obtained.


title: On the Solution of a Painlev\'e III Equation

abstract:
  In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the
solution of a Painlev\'e equation in terms of Fredholm determinants of integral
operators. This equation is $\psi''(t)+t^{-1}\psi'(t)=(1/2) \sinh 2\psi+2\alpha
t^{-1} \sinh\psi$, a special case of the Painlev\'e III equation. The proof in
the cited paper is complicated, and the purpose of this note is to give a more
straightforward one. First we give an equivalent formulation of the solution in
terms of the kernel ${e^{-t (x+x^{-1})/2}\over x+y}\Big|{x-1\over
x+1}\Big|^{2\alpha}$. There are already in the literature relatively simple
proofs of the fact that when $\alpha=0$ Fredholm determinants of this kernel
give solutions to the equation. We extend this result here to general $\alpha$.


title: Legendre transformations on the triangular lattice

abstract:
  The main purpose of the paper is to demonstrate that condition of invariance
with respect to the Legendre transformations allows effectively isolate the
class of integrable difference equations on the triangular lattice, which can
be considered as discrete analogues of relativistic Toda type lattices. Some of
obtained equations are new, up to the author knowledge. As an example, one of
them is studied in more details, in particular, its higher continuous
symmetries and zero curvature representation are found.


title:
On Some One-Parameter Families of Three-Body Problems in One Dimension:
  Exchange Operator Formalism in Polar Coordinates and Scattering Properties

abstract:
  We apply the exchange operator formalism in polar coordinates to a
one-parameter family of three-body problems in one dimension and prove the
integrability of the model both with and without the oscillator potential. We
also present exact scattering solution of a new family of three-body problems
in one dimension.


title:
On Explicit Parametrisation of Spectral Curves for Moser-Calogero
  Particles and its Applications

abstract:
  The system of $N$ classical particles on the line with the Weierstrass $\wp$
function as potential is known to be completely integrable. Recently D'Hoker
and Phong found a beautiful parameterization by the polynomial of degree $N$ of
the space of Riemann surfaces associated with this system. In the trigonometric
limit of the elliptic potential these Riemann surfaces degenerate into rational
curves. The D'Hoker-Phong polynomial in the limit describes the intersection
points of the rational curves. We found an explicit determinant representation
of the polynomial in the trigonometric case. We consider applications of this
result to the theory of Toeplitz determinants and to geometry of the spectral
curves. We also prove our earlier conjecture on the asymptotic behavior of the
ratio of two symplectic volumes when the number of particles tends to infinity.


title: Hypercomplex Integrable Systems

abstract:
  In this paper we study hypercomplex manifolds in four dimensions. Rather than
using an approach based on differential forms, we develop a dual approach using
vector fields. The condition on these vector fields may then be interpreted as
Lax equations, exhibiting the integrability properties of such manifolds. A
number of different field equations for such hypercomplex manifolds are
derived, one of which is in Cauchy-Kovaleskaya form which enables a formal
general solution to be given. Various other properties of the field equations
and their solutions are studied, such as their symmetry properties and the
associated hierarchy of conservation laws.


title:
Bethe ansatz solution of a closed spin 1 XXZ Heisenberg chain with
  quantum algebra symmetry

abstract:
  A quantum algebra invariant integrable closed spin 1 chain is introduced and
analysed in detail. The Bethe ansatz equations as well as the energy
eigenvalues of the model are obtained. The highest weight property of the Bethe
vectors with respect to U_q(sl(2)) is proved.


title:
A Bethe ansatz solution for the closed $U_{q}[sl(2)]$ Temperley-Lieb
  quantum spin chains

abstract:
  We solve the spectrum pf the closed Temperley-Lieb quantum spin chains using
the coordinate Bethe ansatz. These Hamiltonians are invariante under the
quantum group $U_{q}[sl(2)]$


title: Osp(1|2) Off-shell Bethe Ansatz Equations

abstract:
  The semiclassical limit of the algebraic quantum inverse scattering method is
used to solve the theory of the Gaudin model. Via Off-shell Bethe ansatz
equations of an integrable representation of the graded osp(1|2) vertex model
we find the spectrum of N-1 independent Hamiltonians of Gaudin. Integral
representations of the N-point correlators are presented as solutions of the
Knizhnik-Zamolodchikov equation. These results are extended for highest
representations of the osp(1|2) Gaudin algebra.


title: Algebro-Geometric Solutions of the Boussinesq Hierarchy

abstract:
  We continue a recently developed systematic approach to the Bousinesq (Bsq)
hierarchy and its algebro-geometric solutions. Our formalism includes a
recursive construction of Lax pairs and establishes associated
Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations
for analogs of Dirichlet and Neumann divisors. The principal aim of this paper
is a detailed theta function representation of all algebro-geometric
quasi-periodic solutions and related quantities of the Bsq hierarchy.


title:
Elliptic Algebro-Geometric Solutions of the KdV and AKNS Hierarchies -
  An Analytic Approach

abstract:
  We provide an overview of elliptic algebro-geometric solutions of the KdV and
AKNS hierarchies, with special emphasis on Floquet theoretic and spectral
theoretic methods. Our treatment includes an effective characterization of all
stationary elliptic KdV and AKNS solutions based on a theory developed by
Hermite and Picard.


title: From 2D Integrable Systems to Self-Dual Gravity

abstract:
  We explain how to construct solutions to the self-dual Einstein vacuum
equations from solutions of various two-dimensional integrable systems by
exploiting the fact that the Lax formulations of both systems can be embedded
in that of the self-dual Yang--Mills equations. We illustrate this by
constructing explicit self-dual vacuum metrics on $\R^2\times \Sigma$, where
$\Sigma$ is a homogeneous space for a real subgroup of $SL(2, \C)$ associated
with the two-dimensional system.


title: A comparison of two discrete mKdV equations

abstract:
  We consider here two discrete versions of the modified KdV equation. In one
case, some solitary wave solutions, B\"acklund transformations and integrals of
motion are known. In the other one, only solitary wave solutions were given,
and we supply the corresponding results for this equation. We also derive the
integrability of the second equation and give a transformation which links the
two models.


title:
Linear r-Matrix Algebra for a Hierarchy of One-Dimensional Particle
  Systems Separable in Parabolic Coordinates

abstract:
  We consider a hierarchy of many-particle systems on the line with polynomial
potentials separable in parabolic coordinates. The first non-trivial member of
this hierarchy is a generalization of an integrable case of the H\'enon-Heiles
system. We give a Lax representation in terms of $2\times 2$ matrices for the
whole hierarchy and construct the associated linear r-matrix algebra with the
r-matrix dependent on the dynamical variables. A Yang-Baxter equation of
dynamical type is proposed. Classical integration in a particular case is
carried out and quantization of the system is discussed with the help of
separation variables. This paper was published in the rary issues: Sfb 288
Preprint No. 110, Berlin and Nonlinear Mathematical Physics, {\bf 1(3)},
275-294 (1994)


title:
Algebraic Structure of Discrete Zero Curvature Equations and Master
  Symmetries of Discrete Evolution Equations

abstract:
  An algebraic structure related to discrete zero curvature equations is
established. It is used to give an approach for generating master symmetries of
first degree for systems of discrete evolution equations and an answer to why
there exist such master symmetries. The key of the theory is to generate
nonisospectral flows $(\lambda_t=\lambda ^l, l\ge0)$ from the discrete spectral
problem associated with a given system of discrete evolution equations. Three
examples are given.


title:
Symmetries and exact solutions of some integrable Haldane-Shastry like
  spin chains

abstract:
  By using a class of `anyon like' representations of permutation algebra,
which pick up nontrivial phase factors while interchanging the spins of two
lattice sites, we construct some integrable variants of $SU(M)$ Haldane-Shastry
(HS) spin chain. Lax pairs and conserved quantities for these spin chains are
also found and it is established that these models exhibit multi-parameter
deformed or nonstandard variants of $Y(gl_M)$ Yangian symmetry. Moreover, by
projecting the eigenstates of Dunkl operators in a suitable way, we derive a
class of exact eigenfunctions for such HS like spin chain and subsequently
conjecture that these exact eigenfunctions would lead to the highest weight
states associated with a multi-parameter deformed or nonstandard variant of
$Y(gl_M)$ Yangian algebra. By using this conjecture, and acting descendent
operator on the highest weight states associated with a nonstandard $Y(gl_2)$
Yangian algebra, we are able to find out the complete set of eigenvalues and
eigenfunctions for the related HS like spin-${1\over 2}$ chain. It turns out
that some additional energy levels, which are forbidden due to a selection rule
in the case of SU(2) HS model, interestingly appear in the spectrum of above
mentioned HS like spin chain having nonstandard $Y(gl_2)$ Yangian symmetry.


title: The Generalised mKdV Equations for Level -3 of $\hat{sl}_2$

abstract:
  A certain generalisation of the hierarchy of mKdV equations (modified KdV),
which forms an integrable system, is studied here. This generalisation is based
on a Lax operator associated to the equations, with principal components of
degrees between -3 and 0. The results are the following ones: 1) an isomorphism
between the space of jets of the system and a quotient of ${Sl}_2({\CC}((t)))$;
2) the fact that the monodromy matrixes of the Lax operators have, morover,
Poisson brackets given by the trigonometric r-matrix; 3) a definition of the
action of screening operators on the densities; 4) an identification of the
intersection of the kernel with the integrals of motion.


title: The complex geometry of Lagrange top

abstract:
  We prove that the heavy symmetric top (Lagrange, 1788) linearizes on a
two-dimensional non-compact algebraic group -- the generalized Jacobian of an
elliptic curve with two points identified. This leads to a transparent
description of its complex and real invariant level sets. We also deduce, by
making use of a Baker-Akhiezer function, simple explicit formulae for the
general solution of Lagrange top.


title: Lax Pairs for Integrable Lattice Systems

abstract:
  This paper studies the structure of Lax pairs associated with integrable
lattice systems (where space is a one-dimensional lattice, and time is
continuous). It describes a procedure for generating examples of such systems,
and emphasizes the features that are needed to obtain equations which are local
on the spatial lattice.


title: A Bilinear Approach to Discrete Miura Transformations

abstract:
  We present a systematic approach to the construction of Miura transformations
for discrete Painlev\'e equations. Our method is based on the bilinear
formalism and we start with the expression of the nonlinear discrete equation
in terms of $\tau$-functions. Elimination of $\tau$-functions from the
resulting system leads to another nonlinear equation, which is a ``modified''
version of the original equation. The procedure therefore yields Miura
transformations. In this letter, we illustrate this approach by reproducing
previously known Miura transformations and constructing new ones.


title:
Generalized Jacobians of spectral curves and completely integrable
  systems

abstract:
  Consider an ordinary differential equation which has a Lax pair
representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a
fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such
an equation can be considered as a completely integrable complex Hamiltonian
system. We show that the generic complex invariant manifold {A(x): det(A(x)-y
I)= P(x,y)} of this Lax pair is an affine part of a non-compact commutative
algebraic group---the generalized Jacobian of the spectral curve {(x,y):
P(x,y)=0} with its points at "infinity" identified. Moreover, for suitable
B(x), the Hamiltonian vector field defined by the Lax pairon the generalized
Jacobian is translation--invariant. We provide two examples in which the above
result applies.


title: Integrable open boundary conditions for XXC models

abstract:
  The XXC models are multistate generalizations of the well known spin 1/2 XXZ
model. These integrable models share a common underlying su(2) structure. We
derive integrable open boundary conditions for the hierarchy of conserved
quantities of the XXC models . Due to lack of crossing unitarity of the
R-matrix, we develop specific methods to prove integrability. The symmetry of
the spectrum is determined.

categories: solv-int cond-mat.str-el math-ph math.MP nlin.SI


title: Tau-function for discrete sine-Gordon equation and quantum R-matrix

abstract:
  We prove that the tau-function of the integrable discrete sine-Gordon model
apart from the "standard" bilinar identities obeys a number of "non-standard"
ones. They can be combined into a bivector 3-dimensional difference equation
which is shown to contain Hirota's difference analogue of the sine-Gordon
equation and both auxiliary linear problems for it. We observe that this
equation is most naturally written in terms of the quantum R-matrix for the XXZ
spin chain and looks then like a relation of the "vertex-face correspondence"
type.


title: Zeros of the Jimbo, Miwa, Ueno tau function

abstract:
  We introduce a family of local deformations for meromorphic connections on
the Riemann sphere in the neighborhood of a higher rank (simple) singularity.
Following a scheme introduced by Malgrange we use these local models to prove
that the zeros of the tau function introduced by Jimbo, Miwa and Ueno occur
precisely at those points in the deformation space at which a certain
Birkhoff-Riemann- Hilbert problem fails to have a solution.


title: Long range interaction corrections on the quantum vibronic soliton

abstract:
  Self-localized modes in a quantum vibronic system, with long range
interaction of Kac-Baker type and interacting nonlinearly with an acoustical
phonon bath, is studied. One works in the coherent state approximation.
Following a procedure of Sarker and Krumhansl, the problem is reduced to a
nearest neighbours one. In the continuum limit the localized state satisfy a
mKdV equation. An approximate expression for its frequency is found.


title: Classical Integrable 2-dim Models Inspired by SUSY Quantum Mechanics

abstract:
  A class of integrable 2-dim classical systems with integrals of motion of
fourth order in momenta is obtained from the quantum analogues with the help of
deformed SUSY algebra. With similar technique a new class of potentials
connected with Lax method is found which provides the integrability of
corresponding 2-dim hamiltonian systems. In addition, some integrable 2-dim
systems with potentials expressed in elliptic functions are explored.

categories: solv-int hep-th math-ph math.MP nlin.SI quant-ph


title:
Quelques applications de l'Ansatz de Bethe (Some applications of the
  Bethe Ansatz)

abstract:
  The Bethe Ansatz is a method that is used in quantum integrable models in
order to solve them explicitly. This method is explained here in a general
framework, which applies to 1D quantum spin chains, 2D statistical lattice
models (vertex models) and relativistic field theories with 1 space dimension
and 1 time dimension. The connection with quantum groups is expounded. Several
applications are then presented. Finite size corrections are calculated via two
methods: The Non-Linear Integral Equations, which are applied to the study of
the states of the affine Toda model with imaginary coupling, and their
interpolation between the high energy (ultra-violet) and low energy (infra-red)
regions; and the Thermodynamic Bethe Ansatz Equations, along with the
associated Fusion Equations, which are used to determine the thermodynamic
properties of the generalized multi-channel Kondo model. The latter is then
studied in more detail, still using the Bethe Ansatz and quantum groups, so as
to characterize the spectrum of the low energy excitations.


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

abstract:
  The paper is withdrawn due to an error in Section 3.2. The remaining of the
results are included in the preprint solv-int/9605004.


title:
Fermionic flows and tau function of the N=(1|1) superconformal Toda
  lattice hierarchy

abstract:
  An infinite class of fermionic flows of the N=(1|1) superconformal Toda
lattice hierarchy is constructed and their algebraic structure is studied. We
completely solve the semi-infinite N=(1|1) Toda lattice and chain hierarchies
and derive their tau functions, which may be relevant for building
supersymmetric matrix models. Their bosonic limit is also discussed.


title:
Pseudo-orthogonal groups and integrable dynamical systems in two
  dimensions

abstract:
  Integrable systems in low dimensions, constructed through the symmetry
reduction method, are studied using phase portrait and variable separation
techniques. In particular, invariant quantities and explicit periodic solutions
are determined. Widely applied models in Physics are shown to appear as
particular cases of the method.


title: Lax pair tensors in arbitrary dimensions

abstract:
  A recipe is presented for obtaining Lax tensors for any n-dimensional
Hamiltonian system admitting a Lax representation of dimension n. Our approach
is to use the Jacobi geometry and coupling-constant metamorphosis to obtain a
geometric Lax formulation. We also exploit the results to construct integrable
spacetimes, satisfying the weak energy condition.