title:
Fully Supersymmetric Hierarchies From A Energy Dependent Super Hill
  Operator

abstract:
  A super Hill operator with energy dependent potentials is proposed and the
associated integrable hierarchy is constructed explicitly. It is shown that in
the general case, the resulted hierarchy is multi-Hamiltonian system. The Miura
type transformations and modified hierarchies are also presented.


title:
On an integrable discretization of the modified Korteweg-de Vries
  equation

abstract:
  We find time discretizations for the two ''second flows'' of the
Ablowitz-Ladik hierachy. These discretizations are described by local equations
of motion, as opposed to the previously known ones, due to Taha and Ablowitz.
Certain superpositions of our maps allow a one-field reduction and serve
therefore as valid space-time discretizations of the modified Korteweg-de Vries
equation. We expect the performance of these discretizations to be much better
then that of the Taha-Ablowitz scheme. The way of finding interpolating
Hamiltonians for our maps is also indicated, as well as the solution of an
initial value problem in terms of matrix factorizations.


title:
Some eigenstates for a model associated with solutions of tetrahedron
  equation. II. A bit of algebraization

abstract:
  This paper adds two observations to the work solv-int/9701016 where some
eigenstates for a model based on tetrahedron equation have been constructed.
The first observation is that there exists a more "algebraic" construction of
one-particle states, resembling the 1+1-dimensional algebraic Bethe ansatz. The
second observation is that the strings introduced in solv-int/9701016 are
symmetries of a transfer matrix, rather than just eigenstates.


title:
Asymptotics for Solution to the Cauchy Problem for Volterra Lattice with
  Step-Like Initial Values

abstract:
  The connection between modulated Riemann surface of genus one and solution to
Volterra lattice that tends to constants at infinity is studied. The main term
of asymptotics for large time of solution to the mentioned Cauchy problem is
written out.


title: Quasi-BiHamiltonian Systems and Separability

abstract:
  Two quasi--biHamiltonian systems with three and four degrees of freedom are
presented. These systems are shown to be separable in terms of Nijenhuis
coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with
an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis
coordinates) and its separability is proved.


title: An Integral Operator Solution to the Matrix Toda Equations

abstract:
  In previous work the author found solutions to the Toda equations that were
expressed in terms of determinants of integral operators. Here it is observed
that a simple variant yields solutions to the matrix Toda equations. As an
application another derivation is given of a differential equation of Sato,
Miwa and Jimbo for a particular Fredholm determinant.

categories: solv-int funct-an hep-th math.FA nlin.SI


title:
Symbolic Computation of Conserved Densities for Systems of Nonlinear
  Evolution Equations

abstract:
  A new algorithm for the symbolic computation of polynomial conserved
densities for systems of nonlinear evolution equations is presented. The
algorithm is implemented in Mathematica. The program condens.m automatically
carries out the lengthy symbolic computations for the construction of conserved
densities. The code is tested on several well-known partial differential
equations from soliton theory. For systems with parameters, condens.m can be
used to determine the conditions on these parameters so that a sequence of
conserved densities might exist. The existence of a large number of
conservation laws is a predictor for integrability of the system.


title: Poisson brackets with divergence terms in field theories: two examples

abstract:
  In field theories one often works with the functionals which are integrals of
some densities. These densities are defined up to divergence terms (boundary
terms). A Poisson bracket of two functionals is also a functional, i.e., an
integral of a density. Suppose the divergence term in the density of the
Poisson bracket be fixed so that it becomes a bilinear form of densities of two
functionals. Then the left-hand side of the Jacobi identity written in terms of
densities is not necessarily zero but a divergence of a trilinear form. The
question is: what can be said about this trilinear form, what kind of a higher
Jacobi identity (involving four fields) it enjoys? Two examples whose origin is
the theory of integrable systems are given.


title:
An analog of the variational derivative and constructive necessary
  integrability condition for hyperbolic equation

abstract:
  An algorithm is constructed which allows to express conserved flows of
hyperbolic equations in terms of corresponding conserved densities and to
eliminate these flows from conservation laws of hyperbolic equations. The
application of this algorithm to canonical conservation laws gives constructive
necessary integrability conditions of hyperbolic equations in terms of the
generalized Laplace invariants of these equations.


title:
On the point transformations for the second order differential
  equations. I

abstract:
  Point transformations for the ordinary differential equations of the form
$y''=P(x,y)+3 Q(x,y) y'+3 R(x,y) (y')^2+S(x,y) (y')^3$ are considered. Some
classical results are resumed. Solution for the equivalence problem for the
equations of general position is described.


title:
A collection of integrable systems of the Toda type in continuous and
  discrete time, with 2x2 Lax representations

abstract:
  A fairly complete list of Toda-like integrable lattice systems, both in the
continuous and discrete time, is given. For each system the Newtonian,
Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax
representation and r-matrix structure. The material is given in the "no
comment" style, in particular, all proofs are omitted.


title: Non-autonomous H\'{e}non-Heiles Systems

abstract:
  Scaling similarity solutions of three integrable PDEs, namely the
Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered.
It is shown that the resulting ODEs may be written as non-autonomous
Hamiltonian equations, which are time-dependent generalizations of the
well-known integrable H\'{e}non-Heiles systems. The (time-dependent)
Hamiltonians are given by logarithmic derivatives of the tau-functions
(inherited from the original PDEs). The ODEs for the similarity solutions also
have inherited B\"{a}cklund transformations, which may be used to generate
sequences of rational solutions as well as other special solutions related to
the first Painlev\'{e} transcendent.


title:
Modified Korteweg-de Vries Hierachies in Multiple-Times Variables and
  the Solutions of Modified Boussinesq Equations

abstract:
  We study solitary-wave and kink-wave solutions of a modified Boussinesq
equation through a multiple-time reductive perturbation method. We use
appropriated modified Korteweg-de Vries hierarchies to eliminate secular
producing terms in each order of the perturbative scheme. We show that the
multiple-time variables needed to obtain a regular perturbative series are
completely determined by the associated linear theory in the case of a
solitary-wave solution, but requires the knowledge of each order of the
perturbative series in the case of a kink-wave solution. These appropriate
multiple-time variables allow us to show that the solitary-wave as well as the
kink-wave solutions of the modified Botussinesq equation are actually
respectively a solitary-wave and a kink-wave satisfying all the equations of
suitable modified Korteweg-de Vries hierarchies.


title: Raman Solitons and Raman spikes

abstract:
  Stimulated Raman scattering of a laser pump pulse seeded by a Stokes pulse
generically leaves a two-level medium initially at rest in an excited state
constituted of static solitons and radiation. The soliton birth manifests as
sudden very large variations of the phase of the output pump pulse. This is
proved by building the IST solution of SRS on the semi-line, which shows
moreover that initial Stokes phase flips induce Raman spikes in the pump output
also for short pulse experiments.


title:
Inelastic Collision and Switching of Coupled Bright Solitons in Optical
  Fibers

abstract:
  By constructing the general six-parameter bright two-soliton solution of the
integrable coupled nonlinear Schrodinger equation (Manakov model) using the
Hirota method, we find that the solitons exhibit certain novel inelastic
collision properties, which have not been observed in any other (1+1)
dimensional soliton system so far. In particular, we identify the exciting
possibility of switching solitons between modes by changing the phase. However,
the standard elastic collision property of solitons is regained with specific
choices of parameters.


title: Linearizability of the Perturbed Burgers Equation

abstract:
  We show in this letter that the perturbed Burgers equation $u_t = 2uu_x +
u_{xx} + \epsilon ( 3 \alpha_1 u^2 u_x + 3\alpha_2 uu_{xx} + 3\alpha_3 u_x^2 +
\alpha_4 u_{xxx} )$ is equivalent, through a near-identity transformation and
up to order \epsilon, to a linearizable equation if the condition $3\alpha_1 -
3\alpha_3 - 3/2 \alpha_2 + 3/2 \alpha_4 = 0$ is satisfied. In the case this
condition is not fulfilled, a normal form for the equation under consideration
is given. Then, to illustrate our results, we make a linearizability analysis
of the equations governing the dynamics of a one-dimensional gas.


title:
Some eigenstates for a model associated with solutions of tetrahedron
  equation. III. Tetrahedral Zamolodchikov algebras and perturbed strings

abstract:
  This paper continues the series begun with works solv-int/9701016 and
solv-int/9702004. Here we show how to construct eigenstates for a model based
on tetrahedron equation using the tetrahedral Zamolodchikov algebras. This
yields, in particular, new eigenstates for the model on infinite lattice --
`perturbed', or `broken', strings.


title: Non-standard Construction of Hamiltonian Structures

abstract:
  Examples of the construction of Hamiltonian structures for dynamical systems
in field theory (including one reputedly non-Hamiltonian problem) without using
Lagrangians, are presented. The recently developed method used requires the
knowledge of one constant of the motion of the system under consideration and
one solution of the symmetry equation.


title:
Lagrangian Description, Symplectic Structure, and Invariants of 3D Fluid
  Flow

abstract:
  Three dimensional unsteady flow of fluids in the Lagrangian description is
considered as an autonomous dynamical system in four dimensions. The condition
for the existence of a symplectic structure on the extended space is the frozen
field equations of the Eulerian description of motion. Integral invariants of
symplectic flow are related to conservation laws of the dynamical equation. A
scheme generating infinite families of symmetries and invariants is presented.
For the Euler equations these invariants are shown to have a geometric origin
in the description of flow as geodesic motion; they are also interpreted in
connection with the particle relabelling symmetry.


title: WDVV and DZM

abstract:
  We show how the WDVV equations and the DZM system can be characterized via a
background family of functions.


title: A survey of Hirota's difference equations

abstract:
  A review of selected topics in Hirota's bilinear difference equation (HBDE)
is given. This famous 3-dimensional difference equation is known to provide a
canonical integrable discretization for most important types of soliton
equations. Similarly to the continuous theory, HBDE is a member of an infinite
hierarchy. The central point of our exposition is a discrete version of the
zero curvature condition explicitly written in the form of discrete
Zakharov-Shabat equations for M-operators realized as difference or
pseudo-difference operators. A unified approach to various types of M-operators
and zero curvature representations is suggested. Different reductions of HBDE
to 2-dimensional equations are considered. Among them discrete counterparts of
the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical
examples are discussed in detail.


title:
Extended matrix Gelfand-Dickey hierarchies: reduction to classical Lie
  algebras

abstract:
  The Drinfeld-Sokolov reduction method has been used to associate with $gl_n$
extensions of the matrix r-KdV system. Reductions of these systems to the fixed
point sets of involutive Poisson maps, implementing reduction of $gl_n$ to
classical Lie algebras of type $B, C, D$, are here presented. Modifications
corresponding, in the first place to factorisation of the Lax operator, and
then to Wakimoto realisations of the current algebra components of the
factorisation, are also described.


title: Convergent Normal Forms of Symmetric Dynamical Systems

abstract:
  It is shown that the presence of Lie-point-symmetries of (non-Hamiltonian)
dynamical systems can ensure the convergence of the coordinate transformations
which take the dynamical sytem (or vector field) into Poincar\'e-Dulac normal
form.


title:
Spectral Curves and Whitham Equations in Isomonodromic Problems of
  Schlesinger Type

abstract:
  It has been known since the beginning of this century that isomonodromic
problems --- typically the Painlev\'e transcendents --- in a suitable
asymptotic region look like a kind of ``modulation'' of isospectral problem.
This connection between isomonodromic and isospectral problems is reconsidered
here in the light of recent studies related to the Seiberg-Witten solutions of
$N = 2$ supersymmetric gauge theories. A general machinary is illustrated in a
typical isomonodromic problem, namely the Schlesinger equation, which is
reformulated to include a small parameter $\epsilon$. In the small-$\epsilon$
limit, solutions of this isomonodromic problem are expected to behave as a
slowly modulated finite-gap solution of an isospectral problem. The modulation
is caused by slow deformations of the spectral curve of the finite-gap
solution. A modulation equation of this slow dynamics is derived by a heuristic
method. An inverse period map of Seiberg-Witten type turns out to give general
solutions of this modulation equation. This construction of general solution
also reveals the existence of deformations of Seiberg-Witten type on the same
moduli space of spectral curves. A prepotential is also constructed in the same
way as the prepotential of the Seiberg-Witten theory.


title:
A (2+1) dimensional integrable spin model: Geometrical and gauge
  equivalent counterpart, solitons and localized coherent structures

abstract:
  A non-isospectral (2+1) dimensional integrable spin equation is investigated.
It is shown that its geometrical and gauge equivalent counterparts is the (2+1)
dimensional nonlinear Schr\"odinger equation introduced by Zakharov and studied
recently by Strachan. Using a Hirota bilinearised form, line and curved soliton
solutions are obtained. Using certain freedom (arbitrariness) in the solutions
of the bilinearised equation, exponentially localized dromion-like solutions
for the potential is found. Also, breaking soliton solutions (for the spin
variables) of the shock wave type and algebraically localized nature are
constructed.


title: Complex Analysis of a Piece of Toda Lattice

abstract:
  We study a small piece of two dimensional Toda lattice as a complex dynamical
system. In particular the Julia set, which appears when the piece is deformed,
is shown analytically how it disappears as the system approaches to the
integrable limit.


title:
The Correspondence between Discrete Surface and Difference Geometry of
  the KP-hierarchy

abstract:
  The correspondence between two geometrical descriptions of the KP-hierarchy,
one by discrete surface and another by difference analogue of differential
geometry, is given.


title: Dual Resonance Model Solves the Yang-Baxter Equation

abstract:
  The duality of dual resonance models is shown to imply that the four point
string correlation function solves the Yang-Baxter equation. A reduction of
transfer matrices to $A_l$ symmetry is described by a restriction of the KP
$\tau$ function to Toda molecules.


title:
Singularity analysis towards nonintegrability of nonhomogeneous
  nonlinear lattices

abstract:
  We show non-integrability of the nonlinear lattice of Fermi-Pasta-Ulam type
via the singularity analysis(Picard-Vessiot theory) of normal variational
equations of Lam\'e type.


title: Krichever Maps, Faa' di Bruno Polynomials, and Cohomology in KP Theory

abstract:
  We study the geometrical meaning of the Faa' di Bruno polynomials in the
context of KP theory. They provide a basis in a subspace W of the universal
Grassmannian associated to the KP hierarchy. When W comes from geometrical data
via the Krichever map, the Faa' di Bruno recursion relation turns out to be the
cocycle condition for (the Welters hypercohomology group describing) the
deformations of the dynamical line bundle on the spectral curve together with
the meromorphic sections which give rise to the Krichever map. Starting from
this, one sees that the whole KP hierarchy has a similar cohomological meaning.


title:
Non-commutative and commutative integrability of generic Toda flows in
  simple Lie algebras

abstract:
  In this paper we prove the complete integrability of Toda flows on generic
coadjoint orbits in simple Lie algebras.


title:
Huygens' Principle in Minkowski Spaces and Soliton Solutions of the
  Korteweg-de Vries Equation

abstract:
  A new class of linear second order hyperbolic partial differential operators
satisfying Huygens' principle in Minkowski spaces is presented. The
construction reveals a direct connection between Huygens' principle and the
theory of solitary wave solutions of the Korteweg-de Vries equation.


title:
Some eigenstates for a model associated with solutions of tetrahedron
  equation. IV. String-particle marriage

abstract:
  This paper continues the series begun with works solv-int/9701016,
solv-int/9702004 and solv-int/9703010. Here we construct more sophisticated
strings, combining ideas from those papers and some considerations involving
solutions of tetrahedron equation due to Sergeev, Mangazeev and Stroganov.


title:
Trilinear representation and the Moutard transformation for the
  Tzitzeica equation

abstract:
  In the paper we present a trilinear form and a Darboux-type transformation to
an equation considered by Tzitzeica in 1910. This equation equivalent to the
Bullough-Dodd-Jiber-Shabat equation. Soliton solutions are constructed by
dressing the trivial solution.


title:
Fusion rules for Quantum Transfer Matrices as a Dynamical System on
  Grassmann Manifolds

abstract:
  We show that the set of transfer matrices of an arbitrary fusion type for an
integrable quantum model obey these bilinear functional relations, which are
identified with an integrable dynamical system on a Grassmann manifold (higher
Hirota equation). The bilinear relations were previously known for a particular
class of transfer matrices corresponding to rectangular Young diagrams. We
extend this result for general Young diagrams. A general solution of the
bilinear equations is presented.


title:
Computation of conserved densities for systems of nonlinear
  differential-difference equations

abstract:
  A new method for the computation of conserved densities of nonlinear
differential-difference equations is applied to Toda lattices and
discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations.
The algorithm, which can be implemented in computer algebra languages such as
Mathematica, can be used as an indicator of integrability.


title:
Hamiltonian structure and coset construction of the supersymmetric
  extensions of N=2 KdV hierarchy

abstract:
  A manifestly N=2 supersymmetric coset formalism is applied to analyse the
"fermionic" extensions of N=2 $a=4$ and $a=-2$ KdV hierarchies. Both these
hierarchies can be obtained from a manifest N=2 coset construction. This coset
is defined as the quotient of some local but non-linear superalgebra by a
$\hat{U(1)}$ subalgebra. Three superextensions of N=2 KdV hierarchy are
proposed, among which one seems to be entirely new.


title: Rational solutions to d-PIV

abstract:
  We study the rational solutions of the discrete version of Painleve's fourth
equation d-PIV. The solutions are generated by applying Schlesinger
transformations on the seed solutions -2z and -1/z. After studying the
structure of these solutions we are able to write them in a determinantal form
that includes an interesting parameter shift that vanishes in the continuous
limit.


title: On equation of geodesic deviation and its solutions

abstract:
  Equations of geodesic deviation for the 3-dimensional and 4-dimensional
Riemann spaces are discussed. Availability of wide classes of exact solutions
of such equations, due to recent results for the matrix Schr\"odinger equation,
is demonstrated. Particular classes of exact solutions for the geodesic
deviation equation as well as for the Raychaudhuri and generalized Raychaudhuri
equation are presented. Solutions of geodesic deviation equation for the
Schwarzshild and Kasner metrics are found.


title: On superintegrable systems closed to geodesic motion

abstract:
  In this work we consider superintegrable systems in the classical $r$-matrix
method. By using other authomorphisms of the loop algebras we construct new
superintegrable systems with rational potentials from geodesic motion on
$R^{2n}$.


title:
Some eigenstates for a model associated with solutions of tetrahedron
  equation. V. Two cases of string superposition

abstract:
  In paper IV (solv-int/9704013) we have considered a string living in the
infinite lattice that was, in a sense, generated by a "particle". Here we show
how to construct multi-string eigenstates generated by several particles. It
turns out that, at least in some cases, this allows us to bypass the
difficulties of constructing multi-particle states. We also present and discuss
the "dispersion relations" for our particles-strings.


title:
Investigation of dynamical systems using tools of the theory of
  invariants and projective geometry

abstract:
  The investigation of nonlinear dynamical systems of the type
$\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to
some ordinary differential equations of the second order in the form
$y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main
backbone of this investigation was provided by the theory of invariants
developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century
and the projective geometry of E. Cartan. In our work two, in some sense
supplementary, systems are considered: the Lorenz system $\dot{x}=\sigma (y-x),
\dot{y}=rx-y-zx,\dot{z}=xy-bz $ and the R\"o\ss ler system
$\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+xz-cz.$. The invarinats for the ordinary
differential equations, which correspond to the systems mentioned abouve, are
evaluated. The connection of values of the invariants with characteristics of
dynamical systems is established.


title: Constructing Integrable Third Order Systems:The Gambier Approach

abstract:
  We present a systematic construction of integrable third order systems based
on the coupling of an integrable second order equation and a Riccati equation.
This approach is the extension of the Gambier method that led to the equation
that bears his name. Our study is carried through for both continuous and
discrete systems. In both cases the investigation is based on the study of the
singularities of the system (the Painlev\'e method for ODE's and the
singularity confinement method for mappings).


title: On Adomian's Decomposition Method for Solving Differential Equations

abstract:
  We show that with a few modifications the Adomian's method for solving second
order differential equations can be used to obtain the known results of the
special functions of mathematical physics. The modifications are necessary in
order to take correctly into account the behaviour of the solutions in the
neighborhood of the singular points.


title:
Analytic-bilinear approach to integrable hierarchies. II. Multicomponent
  KP and 2D Toda lattice hierarchies

abstract:
  Analytic-bilinear approach for construction and study of integrable
hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice
hierarchies are considered. This approach allows to represent generalized
hierarchies of integrable equations in a condensed form of finite functional
equations. Generalized hierarchy incorporates basic hierarchy, modified
hierarchy, singularity manifold equation hierarchy and corresponding linear
problems. Different levels of generalized hierarchy are connected via
invariants of Combescure symmetry transformation. Resolution of functional
equations also leads to the $\tau $-function and addition formulae to it.


title: Integrability of Riccati equations and the stationary KdV equations

abstract:
  Using the S.Lie's infinitesimal approach we establish the connection between
integrability of a one-parameter family of the Riccati equations and the
stationary KdV hierarchy.


title:
Stationary mKdV hierarchy and integrability of the Dirac equations by
  quadratures

abstract:
  Using the Lie's infinitesimal method we establish that the Dirac equation in
one variable is integrable by quadratures if the potential V(x) is a solution
of one of the equations of the stationary mKdV hierarchy.


title:
Vectorial Darboux Transformations for the Kadomtsev-Petviashvili
  Hierarchy

abstract:
  We consider the vectorial approach to the binary Darboux transformations for
the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We
obtain explicit formulae for the Darboux transformed potentials in terms of
Grammian type determinants. We also study the $n$-th Gel'fand-Dickey hierarchy
introducing spectral operators and obtaining similar results. We reduce the
above mentioned results to the Kadomtsev-Petviashvili I and II real forms,
obtaining corresponding vectorial Darboux transformations. In particular for
the Kadomtsev-Petviashvili I hierarchy we get the line soliton, the lump
solution and the Johnson-Thompson lump, and the corresponding determinant
formulae for the non-linear superposition of several of them. For
Kadomtsev-Petviashvili II apart from the line solitons we get singular rational
solutions with its singularity set describing the motion of strings in the
plane. We also consider the I and II real forms for the Gel'fand-Dickey
hierarchies obtaining the vectorial Darboux transformation in both cases.


title:
Dispersionless Hierarchies, Hamilton-Jacobi Theory and Twistor
  Correspondences

abstract:
  The dispersionless KP and Toda hierarchies possess an underlying twistorial
structure. A twistorial approach is partly implemented by the method of
Riemann-Hilbert problem. This is however still short of clarifying geometric
ingredients of twistor theory, such as twistor lines and twistor surfaces. A
more geometric approach can be developed in a Hamilton-Jacobi formalism of
Gibbons and Kodama. AMS Subject Classifiation (1991): 35Q20, 58F07,70H99


title:
Time-Dependent Symmetries of Variable-Coefficient Evolution Equations
  and Graded Lie Algebras

abstract:
  Polynomial-in-time dependent symmetries are analysed for polynomial-in-time
dependent evolution equations. Graded Lie algebras, especially Virasoro
algebras, are used to construct nonlinear variable-coefficient evolution
equations, both in 1+1 dimensions and in 2+1 dimensions, which possess
higher-degree polynomial-in-time dependent symmetries. The theory also provides
a kind of new realisation of graded Lie algebras. Some illustrative examples
are given.


title:
Graded Symmetry Algebras of Time-Dependent Evolution Equations and
  Application to the Modified KP equations

abstract:
  By starting from known graded Lie algebras, including Virasoro algebras, new
kinds of time-dependent evolution equations are found possessing graded
symmetry algebras. The modified KP equations are taken as an illustrative
example: new modified KP equations with $m$ arbitrary time-dependent
coefficients are obtained possessing symmetries involving $m$ arbitrary
functions of time. A particular graded symmetry algebra for the modified KP
equations is derived in this connection homomorphic to the Virasoro algebras.


title: Dual Isomonodromic Problems and Whitham Equations

abstract:
  The author's recent results on an asymptotic description of the Schlesinger
equation are generalized to the JMMS equation. As in the case of the
Schlesinger equation, the JMMS equation is reformulated to include a small
parameter $\epsilon$. By the method of multiscale analysis, the isomonodromic
problem is approximated by slow modulations of an isospectral problem. A
modulation equation of this slow dynamics is proposed, and shown to possess a
number of properties similar to the Seiberg- Witten solutions of low energy
supersymmetric gauge theories.


title:
Laplace transformations of hydrodynamic type systems in Riemann
  invariants: periodic sequences

abstract:
  The conserved densities of hydrodynamic type system in Riemann invariants
satisfy a system of linear second order partial differential equations. For
linear systems of this type Darboux introduced Laplace transformations,
generalising the classical transformations in the scalar case. It is
demonstrated that Laplace transformations can be pulled back to the
transformations of the corresponding hydrodynamic type systems. We discuss
periodic Laplace sequences of with the emphasize on the simplest nontrivial
case of period 2. For 3-component systems in Riemann invariants a complete
discription of closed quadruples is proposed. They turn to be related to a
special quadratic reduction of the (2+1)-dimensional 3-wave system which can be
reduced to a triple of pairwize commuting Monge-Ampere equations. In terms of
the Lame and rotation coefficients Laplace transformations have a natural
interpretation as the symmetries of the Dirac operator, associated with the
(2+1)-dimensional n-wave system. The 2-component Laplace transformations can be
interpreted also as the symmetries of the (2+1)-dimensional integrable
equations of Davey-Stewartson type. Laplace transformations of hydrodynamic
type systems originate from a canonical geometric correspondence between
systems of conservation laws and line congruences in projective space.


title: A Characterization of All Elliptic Solutions of the AKNS Hierarchy

abstract:
  An explicit characterization of all elliptic algebro-geometric solutions of
the AKNS hierarchy is presented. Our approach is based on (an extension of) a
classical theorem of Picard, which guarantees the existence of solutions which
are elliptic of the second kind for n-th order ordinary differential equations
with elliptic coefficients associated with a common period lattice. As
by-products we offer a detailed Floquet analysis of Dirac-type differential
expressions with periodic coefficients, specifically emphasizing
algebro-geometric coefficients, and a constructive reduction of singular
hyperelliptic curves and their Baker-Akhiezer functions to the nonsingular
case.


title:
Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and
  Kac-van Moerbeke Hierarchies

abstract:
  Combining algebro-geometric methods and factorization techniques for finite
difference expressions we provide a complete and self-contained treatment of
all real-valued quasi-periodic finite-gap solutions of both the Toda and
Kac-van Moerbeke hierarchies. In order to obtain our principal new result, the
algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we
employ particular commutation methods in connection with Miura-type
transformations which enable us to transfer whole classes of solutions (such as
finite-gap solutions) from the Toda hierarchy to its modified counterpart, the
Kac-van Moerbeke hierarchy, and vice versa.


title: R-Matrices and Generalized Inverses

abstract:
  Four results are given that address the existence, ambiguities and
construction of a classical R-matrix given a Lax pair. They enable the uniform
construction of R-matrices in terms of any generalized inverse of $ad L$. For
generic $L$ a generalized inverse (and indeed the Moore-Penrose inverse) is
explicitly constructed. The R-matrices are in general momentum dependent and
dynamical. The construction applies equally to Lax matrices with spectral
parameter.


title: Integrable Fredholm Operators and Dual Isomonodromic Deformations

abstract:
  The Fredholm determinants of a special class of integral operators K
supported on the union of m curve segments in the complex plane are shown to be
the tau-functions of an isomonodromic family of meromorphic covariant
derivative operators D_l. These have regular singular points at the 2m
endpoints of the curve segments and a singular point of Poincare index 1 at
infinity. The rank r of the vector bundle over the Riemann sphere on which they
act equals the number of distinct terms in the exponential sums entering in the
numerator of the integral kernels. The deformation equations may be viewed as
nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M,
whose Poisson quotient, under a parametric family of Hamiltonian group actions,
is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with
respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem
method is used to identify the auxiliary space M with the data defining the
integral kernel of the resolvent operator at the endpoints of the curve
segments. A second associated isomonodromic family of covariant derivative
operators D_z is derived, having rank n=2m, and r finite regular singular
points at the values of the exponents defining the kernel of K. This family is
similarly embedded into the algebra Lgl_R(n) through a dual parametric family
of Poisson quotients of M. The operators D_z are shown to be analogously
associated to the integral operator obtained from K through a Fourier-Laplace
transform.

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


title:
On the point transformations for the equation $y''= P + 3Qy' + 3R{y'}^2
  + S{y'}^3$

abstract:
  For the equations $y''=P(x,y) + 3Q(x,y)y' + 3R(x,y){y'}^2 + S(x,y){y'}^3$ the
problem of equivalence is considered. Some classical results are resumed in
order to prepare the background for the study of special subclass of such
equations, which arises in the theory of dynamical systems admitting the normal
shift.


title: An analytic description of the vector constrained KP hierarchy

abstract:
  In this paper we give a geometric description in terms of the Grassmann
manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the
vector $k$-constrained KP hierarchy. We also show in a geometric way that these
hierarchies are equivalent to Krichever's general rational reductions of the KP
hierarchy.


title: A Lax Description for Polytropic Gas Dynamics

abstract:
  We give a Lax description for the system of polytropic gas equations. The
special structure of the Lax function naturally leads to the two infinite sets
of conserved charges associated with this system. We obtain closed form
expressions for the conserved charges as well as the generating functions for
them. We show how the study of these generating functions can naturally lead to
the recursion relation between the conserved quantities as well as the higher
order Hamiltonian structures.


title: The geometry of spinors and the multicomponent BKP and DKP hierarchies

abstract:
  We develop a formalism of multicomponent BKP hierarchies using elementary
geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence
all their reductions like KdV, NLS, AKNS or DS) are reductions of the
multicomponent BKP.


title:
On the Liouville transformation and exactly-solvable Schrodinger
  equations

abstract:
  The present article discusses the connection between exactly-solvable
Schrodinger equations and the Liouville transformation. This transformation
yields a large class of exactly-solvable potentials, including the
exactly-solvable potentials introduced by Natanzon. As well, this class is
shown to contain two new families of exactly solvable potentials.


title:
An Extension of the KdV Hierarchy Arising from a Representation of a
  Toroidal Lie Algebra

abstract:
  In this article we show how to construct hierarchies of partial differential
equations from the vertex operator representations of toroidal Lie algebras. In
the smallest example - rank 2 toroidal cover of $sl_2$ - we obtain an extension
of the KdV hierarchy. We use the action of the corresponding
infinite-dimensional group to construct solutions for these non-linear PDEs.


title: The spectrum of random matrices and integrable systems

abstract:
  What is the connection of random matrices with integrable systems? Is this
connection really useful? Introducing apprpriate times in the distribution of
the ensemble of matrices, one shows that the corresponding distribution of the
eigenvalues satisfies the KP-equation, the 1-Toda lattice or the 2-Toda
lattice, depending on the original distribution. The probability distribution
also satisfies Virasoro type constraints, which contain a time-part and a
boundary-part. These equations taken together lead to a system of PDE's for the
distribution of the spectrum in terms of the boundary of the set, under
consideration.


title:
Matrix Integrals, Toda symmetries, Virasoro constraints, and orthogonal
  polynomials

abstract:
  The relationship is made between matrix integrals, Toda master-symmetries,
Virasoro constraints and orthogonal polynomials.


title: The Distribution of the Largest Eigenvalue in the Gaussian Ensembles

abstract:
  The focus of this survey paper is on the distribution function for the
largest eigenvalue in the finite N Gaussian ensembles (GOE,GUE,GSE) in the edge
scaling limit of N->infinity. These limiting distribution functions are
expressible in terms of a particular Painleve II function. Comparisons are made
with finite N simulations as well as a discussion of the universality of these
distribution functions.


title: Trace Formula for a System of Particles with Elliptic Potential

abstract:
  We consider classical particles on the line with the Weierstrass $\wp$
function as potential. This system parameterizes special solutions of the KP
equation. We derive the trace formula which relates the Hamiltonian of the
particle system to the residues of some Abelian differential (meromorphic
one-form) on the spectral curve. Such formula is important for the construction
action-angle variables and study invariant Gibbs' states.


title:
Statistical Mechanics of Elastica on Plane as a Model of Supercoiled
  DNA-Origin of the MKdV hierarchy-

abstract:
  In this article, I have investigated statistical mechanics of a non-stretched
elastica in two dimensional space using path integral method. In the
calculation, the MKdV hierarchy naturally appeared as the equations including
the temperature fluctuation.I have classified the moduli of the closed elastica
in heat bath and summed the Boltzmann weight with the thermalfluctuation over
the moduli. Due to the bilinearity of the energy functional,I have obtained its
exact partition function.By investigation of the system,I conjectured that an
expectation value at a critical point of this system obeys the Painlev\'e
equation of the first kind and its related equations extended by the KdV
hierarchy.Furthermore I also commented onthe relation between the MKdV
hierarchy and BRS transformationin this system.


title: Nonlinear Quasiclassics and the Painlev\'e Equations

abstract:
  Problem of asymptotic description for global solutions to the six Painleve
equations was investigated. Elliptic anzatzes and appropriate modulation
equations were written out.


title: Solvability of the G_2 Integrable System

abstract:
  It is shown that the 3-body trigonometric G_2 integrable system is
exactly-solvable. If the configuration space is parametrized by certain
symmetric functions of the coordinates then, for arbitrary values of the
coupling constants, the Hamiltonian can be expressed as a quadratic polynomial
in the generators of some Lie algebra of differential operators in a
finite-dimensional representation. Four infinite families of eigenstates,
represented by polynomials, and the corresponding eigenvalues are described
explicitly.


title: Quantum Coupled Nonlinear Schr\"odinger System with Different Masses

abstract:
  In this letter, I have considered one-dimensional quantum system with
different masses $m$ and $M$, which does not appear integrable in general.
However I have found an exact two-body wave function and due to the extension
of the integrable system to more general system, it was concluded that the
rapidity or quasi-momentum in the integrable system should be regarded as a
modification of velocity rather than that of momentum. I have also considered
the three-body wave function and argued its integrable condition.


title:
On Density of State of Quantized Willmore Surface-A Way to Quantized
  Extrinsic String in R^3

abstract:
  Recently I quantized an elastica with Bernoulli-Euler functional in
two-dimensional space using the modified KdV hierarchy. In this article, I will
quantize a Willmore surface, or equivalently a surface with the Polyakov
extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation.
In other words, I show that the density of state of the partition function for
the quantized Willmore surface is expressed by volume of a subspace of the
moduli of the MNV equation.


title: Functional representation of the Ablowitz-Ladik hierarchy

abstract:
  The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the
inverse scattering approach. After establishing the structure of solutions of
the auxiliary linear problems, the ALH, which has been originally introduced as
an infinite system of difference-differential equations is presented as a
finite system of difference-functional equations. The representation obtained,
when rewritten in terms of Hirota's bilinear formalism, is used to demonstrate
relations between the ALH and some other integrable systems, the
Kadomtsev-Petviashvili hierarchy in particular.


title:
An Alternative Approach to Algebro-Geometric Solutions of the AKNS
  Hierarchy

abstract:
  We develop an alternative systematic approach to the AKNS hierarchy based on
elementary algebraic methods. In particular, we recursively construct Lax pairs
for the entire AKNS hierarchy by introducing a fundamental polynomial formalism
and establish the basic algebro-geometric setting including associated
Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas,
Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and
theta function representations for algebro-geometric solutions.


title:
A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its
  algebro-geometric solutions

abstract:
  We derive a zero-curvature formalism for a combined sine-Gordon (sG) and
modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV
hierarchy. In complete analogy to other completely integrable hierarchies of
soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV
hierarchy is recursively constructed by means of a fundamental polynomial
formalism involving a spectral parameter. We further illustrate our approach by
developing the basic algebro-geometric setting for the sGmKdV hierarchy,
including Baker-Akhiezer functions, trace formulas, Dubrovin-type equations,
and theta function representations for its algebro-geometric solutions.
Although we mainly focus on sG-type equations, our formalism also yields the
sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type
equations combined with the mKdV hierarchy.


title: Bifurcation diagram of a one-parameter family of dispersive waves

abstract:
  The Korteweg de Vries (KdV) equation with small dispersion is a model for the
formation and propagation of dispersive shock waves in one dimension.
Dispersive shock waves in KdV are characterized by the appearance of zones of
rapid modulated oscillations in the solution of the Cauchy problem with smooth
initial data. The modulation in time and space of the amplitudes, the
frequencies and the wave-numbers of these oscillations and their interactions
is approximately described by the $g$-phase Whitham equations. We study the
initial value problem for the Whitham equations for a one parameter family of
monotone decreasing initial data. We obtain the bifurcation diagram of the
number $g$ of interacting oscillatory zones.


title:
The Constrained MKP Hierarchy and the Generalized Kupershmidt-Wilson
  Theorem

abstract:
  The constrained Modified KP hierarchy is considered from the viewpoint of
modification. It is shown that its second Poisson bracket, which has a rather
complicated form, is associated to a vastly simpler bracket via Miura-type map.
The similar results are established for a natural reduction of MKP.


title:
Algebra of non-local charges in the O(N) WZNW model at and beyond
  criticality

abstract:
  We derive the classical algebra of the non-local conserved charges in the
O(N) WZNW model and analyze its dependence on the coupling constant of the
Wess-Zumino term. As in the non-linear sigma model, we find cubic deformations
of the O(N) affine algebra. The surprising result is that the cubic algebra of
the WZNW non-local charges does not obey the Jacobi identity, thus opposing our
expectations from the known Yangian symmetry of this model.


title:
The constrained modified KP hierarchy and the generalized Miura
  transformations

abstract:
  In this letter, we consider the second Hamiltonian structure of the
constrained modified KP hierarchy. After mapping the Lax operator to a pure
differential operator the second structure becomes the sum of the second and
the third Gelfand-Dickey brackets defined by this differential operator. We
simplify this Hamiltonian structure by factorizing the Lax operator into linear
terms.


title: B\"acklund transformation for the Krichever-Novikov equation

abstract:
  The B\"acklund transformation and its nonlinear superposition principle are
presented for the Krichever-Novikov equation $u_t= u_{xxx} - {3/(2u_x)}
(u^2_{xx} - r(u)) + cu_x, r^{(5)}=0$.


title:
Continuum limit of nonlinear discrete systems with long range
  interaction potentials

abstract:
  One dimensional nonlinear lattices with harmonic long range interaction
potentials (LRIP) having an inverse power kernel type, are studied. For the
nearest neighbour nonlinear interaction we consider the anharmonic potential of
the Fermi-Pasta-Ulam problem and the \phi^3+\phi^4 potential as well. The
continuum limit is obtained following the method used by Ishimori and several
Boussinesq and KdV type equations with supplementary Hilbert transform terms
are found. These nonlocal terms are introduced by the LRIP. For the
\phi^3+\phi^4 nearest neighbour interactions the continuum approximation turns
out to admit exact bilinearization in Hirota formalism. Exact rational
nonsingular solutions are found. The integrability of these nonlocal equations
and the connection with perturbed KdV are also discussed.


title: Bihamiltonian Reductions and W_n Algebras

abstract:
  We discuss the geometry of the Marsden-Ratiu reduction theorem for a
bihamiltonian manifold. We consider the case of the manifolds associated with
the Gel'fand-Dickey theory, i.e., loop algebras over sl(n+1). We provide an
explicit identification, tailored on the MR reduction, of the
Adler-Gel'fand-Dickey brackets with the Poisson brackets on the MR-reduced
bihamiltonian manifold N. Such an identification relies on a suitable immersion
of the space of sections of the cotangent bundle of N into the algebra of
pseudo differential operators connected to geometrical features of the theory
of (classical) W_n algebras.


title: Two-dimensional soliton cellular automaton of deautonomized Toda-type

abstract:
  A deautonomized version of the two-dimensional Toda lattice equation is
presented. Its ultra-discrete analogue and soliton solutions are also
discussed.


title: Nonstandard Drinfeld-Sokolov reduction

abstract:
  Subject to some conditions, the input data for the Drinfeld-Sokolov
construction of KdV type hierarchies is a quadruplet $(\A,\Lambda, d_1, d_0)$,
where the $d_i$ are $\Z$-gradations of a loop algebra $\A$ and $\Lambda\in \A$
is a semisimple element of nonzero $d_1$-grade. A new sufficient condition on
the quadruplet under which the construction works is proposed and examples are
presented. The proposal relies on splitting the $d_1$-grade zero part of $\A$
into a vector space direct sum of two subalgebras. This permits one to
interpret certain Gelfand-Dickey type systems associated with a nonstandard
splitting of the algebra of pseudo-differential operators in the
Drinfeld-Sokolov framework.


title:
Three Graded Modified Classical Yang-Baxter Equations and Integrable
  Systems

abstract:
  The $6 = 3\times 2$ huge Lie algebra $\Xi$ of all local and non local
differential operators on a circle is applied to the standard
Adler-Kostant-Symes (AKS) R-bracket sckeme. It is shown in particular that
there exist three additional Lie structures, associated to three graded
modified classical Yang-Baxter(GMCYB) equations. As we know from the standard
case, these structures can be used to classify in a more consitent way a wide
class of integrable systems. Other algebraic properties are also presented.


title: Criterion and Regions of Stability for Quasi-Equidistant Soliton Trains

abstract:
  Using the complex Toda chain (CTC) as a model for the propagation of the
N-soliton pulse trains of the nonlinear Schrodinger (NLS) equation, we predict
the asymptotic behavior of these trains. The following asymptotic regimes are
stable: (i)~asymptotically free propagation of all N solitons; (ii)~bound state
regime where the N solitons may move quasi-equidistantly (QED); and
(iii)~various different combinations of (i) and (ii). For N=2 and 3 we
determine analytically the set of initial soliton parameters corresponding to
each of these regimes. We find excellent agreement between the solutions of CTC
and NLS for all regimes and propose realistic choices for the sets of
amplitudes, for which the solitons propagate QED to very large run lengths.
This is of importance for optical fiber communication.


title: On the Integrable Hierarchies Associated With N=2 Super $W_n$ Algebra

abstract:
  A new Lax operator is proposed from the viewpoint of constructing the
integrable hierarchies related with N=2 super $W_n$ algebra. It is shown that
the Poisson algebra associated to the second Hamiltonian structure for the
resulted hierarchy contains the N=2 super Virasoro algebra as a proper
subalgebra. The simplest cases are discussed in detail. In particular, it is
proved that the supersymmetric two-boson hierarchy is one of N=2 supersymmetric
KdV hierarchies. Also, a Lax operator is supplied for one of N=2 supersymmetric
Boussinesq hierarchies.


title: Introduction to the Hirota bilinear method

abstract:
  We give an elementary introduction to Hirota's direct method of constructing
multisoliton solutions to integrable nonlinear evolution equations. We discuss
in detail how this works for equations in the Korteweg-de Vries class. We also
show how Hirota's method can be used to search for new integrable evolution
equations by testing for the existence of 3- and 4-soliton solutions, and list
the results that have been obtained this way for the KdV, mKdV/sG and nlS
classes of equations.


title: Generating function of correlators in the sl_2 Gaudin model

abstract:
  For the sl_2 Gaudin model (degenerated quantum integrable XXX spin chain) an
exponential generating function of correlators is calculated explicitely. The
calculation relies on the Gauss decomposition for the SL_2 loop group. From the
generating function a new explicit expression for the correlators is derived
from which the determinant formulas for the norms of Bethe eigenfunctions due
to Richardson and Gaudin are obtained.


title:
Nonlocal extended conformal algebras associated with multi-constraint KP
  hierarchy and their free-field realizations

abstract:
  We study the conformal properties of the multi-constraint KP hierarchy and
its nonstandard partner by covariantizing their corresponding Lax operators.
The associated second Hamiltonian structures turn out to be nonlocal extension
of $W_n$ algebra by some integer or half-integer spin fields depending on the
order of the Lax operators. In particular, we show that the complicated second
Hamiltonian structure of the nonstandard multi-constraint KP hierarchy can be
simplified by factorizing its Lax operator to multiplication form. We then
diagonalize this simplified Poisson matrix and obtain the free-field
realizations of its associated nonlocal algebras.


title: d=2, N=2 Superconformally Covariant Operators and Super W-Algebras

abstract:
  We construct and classify superconformally covariant differential operators
defined on N=2 super Riemann surfaces. By contrast to the N=1 theory, these
operators give rise to partial rather than ordinary differential equations
which leads to novel features for their matrix representation. The latter is
applied to the derivation of N=2 super W-algebras in terms of N=2 superfields.


title: Pure quantum integrability

abstract:
  The correspondence between the integrability of classical mechanical systems
and their quantum counterparts is not a 1-1, although some close
correspondencies exist. If a classical mechanical system is integrable with
invariants that are polynomial in momenta one can construct a corresponding
commuting set of differential operators. Here we discuss some 2- or
3-dimensional purely quantum integrable systems (the 1-dimensional counterpart
is the Lame equation). That is, we have an integrable potential whose amplitude
is not free but rather proportional to $\hbar^2$, and in the classical limit
the potential vanishes. Furthermore it turns out that some of these systems
actually have N+1 commuting differential operators, connected by a nontrivial
algebraic relation. Some of them have been discussed recently by A.P. Veselov
et. al.} from the point of view of Baker-Akheizer functions.


title: Separation of variables via integral transformations

abstract:
  For a system of linear partial differential equations (LPDEs) we introduce an
operator equation for auxiliary operators. These operators are used to
construct a kernel of an integral transformation leading the LPDE to the
separation of variables (SoV). The auxiliary operators are found for various
types of the SoV including conventional SoV in the scalar second order LPDE and
the SoV by the functional Bethe anzatz. The operators are shown to relate to
separable variables. This approach is similar to the position-momentum
transformation to action angle coordinates in the classical mechanics. General
statements are illustrated by some examples.


title: Field Theory for Coherent Optical Pulse Propagation

abstract:
  We introduce a new notion of "matrix potential" to nonlinear optical systems.
In terms of a matrix potential $g$, we present a gauge field theoretic
formulation of the Maxwell-Bloch equation that provides a semiclassical
description of the propagation of optical pulses through resonant multi-level
media. We show that the Bloch part of the equation can solved identically
through $g$ and the remaining Maxwell equation becomes a second order
differential equation with reduced set of variables due to the gauge invariance
of the system. Our formulation clarifies the (nonabelian) symmetry structure of
the Maxwell-Bloch equations for various multi-level media in association with
symmetric spaces $G/H$. In particular, we associate nondegenerate two-level
system for self-induced transparency with $G/H=SU(2)/U(1)$ and three-level $\L
$- or V-systems with $G/H = SU(3)/U(2)$. We give a detailed analysis for the
two-level case in the matrix potential formalism, and address various new
properties of the system including soliton numbers, effective potential energy,
gauge and discrete symmetries, modified pulse area, conserved topological and
nontopological charges. The nontopological charge measures the amount of
self-detuning of each pulse. Its conservation law leads to a new type of pulse
stability analysis which explains nicely earlier numerical results.


title: Matched Pulse Propagation in a Three-Level System

abstract:
  The B\"{a}cklund transformation for the three-level Maxwell-Bloch equation is
presented in the matrix potential formalism. By applying the B\"{a}cklund
transformation to a constant electric field background, we obtain a general
solution for matched pulses (a pair of solitary waves) which can emit or absorb
a light velocity solitary pulse but otherwise propagate with their shapes
invariant. In the special case, this solution describes a steady state pulse
without emission or absorption, and becomes the matched pulse solution recently
obtained by Hioe and Grobe. A nonlinear superposition rule is derived from the
B\"{a}cklund transformation and used for the explicit construction of two
solitons as well as nonabelian breathers. Various new features of these
solutions are addressed. In particular, we analyze in detail the scattering of
"invertons", a specific pair of different wavelength solitons one of which
moving with the velocity of light. Unlike the usual case of soliton scattering,
the broader inverton changes its sign through the scattering. Surprisingly, the
light velocity inverton receives time advance through the scattering thereby
moving faster than light, which however does not violate causality.


title:
Solitons from Dressing in an Algebraic Approach to the Constrained KP
  Hierarchy

abstract:
  The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras
leads to a variety of 1+1 soliton equations. By varying the rank of the
underlying $sl (n)$ algebra as well as its gradation in the affine setting, one
encompasses the set of the soliton equations of the constrained KP hierarchy.
The soliton solutions are then obtained as elements of the orbits of the
dressing transformations constructed in terms of representations of the vertex
operators of the affine $sl (n)$ algebras realized in the unconventional
gradations. Such soliton solutions exhibit non-trivial dependence on the KdV
(odd) time flows and KP (odd and even) time flows which distinguishes them from
the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of
the constrained KP hierarchy.


title:
Integrable discretizations for lattice systems: local equations of
  motion and their Hamiltonian properties

abstract:
  We develop the approach to the problem of integrable discretization based on
the notion of $r$--matrix hierarchies. One of its basic features is the
coincidence of Lax matrices of discretized systems with the Lax matrices of the
underlying continuous time systems. A common feature of the discretizations
obtained in this approach is non--locality. We demonstrate how to overcome this
drawback. Namely, we introduce the notion of localizing changes of variables
and construct such changes of variables for a large number of examples,
including the Toda and the relativistic Toda lattices, the Volterra lattice and
its integrable perturbation, the second flows of the Toda and of the Volterra
hierarchies, the modified Volterra lattice, the Belov-Chaltikian lattice, the
Bogoyavlensky lattices, the Bruschi-Ragnisco lattice. We also introduce a novel
class of constrained lattice KP systems, discretize all of them, and find the
corresponding localizing change of variables. Pulling back the differential
equations of motion under the localizing changes of variables, we find also
(sometimes novel) integrable one-parameter perturbations of integrable lattice
systems. Poisson properties of the localizing changes of variables are also
studied: they produce interesting one-parameter deformations of the known
Poisson algebras.


title:
Solutions of the functional tetrahedron equation connected with the
  local Yang -- Baxter equation for the ferro-electric

abstract:
  Local (or modified) Yang -- Baxter equation (LYBE) gives the functional map
from the parameters of the weights in the left hand side to the parameters of
the correspondent weights in the right hand side of LYBE. Such maps solve the
functional tetrahedron equation. In this paper all the maps associated with
LYBE of the ferro-electric type with single parameter in each weight matrix are
classified.


title:
Discrete Levy Transformations and Casorati Determinant Solutions of
  Quadrilateral Lattices

abstract:
  Sequences of discrete Levy and adjoint Levy transformations for the
multidimensional quadrilateral lattices are studied. After a suitable number of
iterations we show how all the relevant geometrical features of the transformed
quadrilateral lattice can be expressed in terms of multi-Casorati determinants.
As an example we dress the Cartesian lattice.


title: Nonlinear Physics: Integrability, Chaos and Beyond

abstract:
  Integrability and chaos are two of the main concepts associated with
nonlinear physical systems which have revolutionized our understanding of them.
Highly stable exponentially localized solitons are often associated with many
of the important integrable nonlinear systems while motions which are
sensitively dependent on initial conditions are associated with chaotic
systems. Besides dramatically raising our perception of many natural phenomena,
these concepts are opening up new vistas of applications and unfolding
technologies: Optical soliton based information technology, magnetoelectronics,
controlling and synchronization of chaos and secure communications, to name a
few. These developments have raised further new interesting questions and
potentialities. We present a particular view of some of the challenging
problems and payoffs ahead in the next few decades by tracing the early
historical events, summarizing the revolutionary era of 1950-70 when many
important new ideas including solitons and chaos were realized and reviewing
the current status. Important open problems both at the basic and applied
levels are discussed.


title:
Motion of Curves and Surfaces and Nonlinear Evolution Equations in (2+1)
  Dimensions

abstract:
  It is shown that a class of important integrable nonlinear evolution
equations in (2+1) dimensions can be associated with the motion of space curves
endowed with an extra spatial variable or equivalently, moving surfaces.
Geometrical invariants then define topological conserved quantities. Underlying
evolution equations are shown to be associated with a triad of linear
equations. Our examples include Ishimori equation and Myrzakulov equations
which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov
-Strachan (2+1) dimensional nonlinear Schr\"odinger equations respectively.


title: The Camassa-Holm Equation: A Loop Group Approach

abstract:
  A map is presented that associates with each element of a loop group a
solution of an equation related by a simple change of coordinates to the
Camassa-Holm (CH) Equation. Certain simple automorphisms of the loop group give
rise to Backlund transformations of the equation. These are used to find
2-soliton solutions of the CH equation, as well as some novel singular
solutions.


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

abstract:
  Rational solutions for the Painlev\'e IV equation are investigated by Hirota
bilinear formalism. It is shown that the solutions in one hierarchy are
expressed by 3-reduced Schur functions, and those in another two hierarchies by
Casorati determinant of the Hermite polynomials, or by special case of the
Schur polynomials.


title:
The Whitham Equations for Optical Communications: Mathematical Theory of
  NRZ

abstract:
  We present a model of optical communication system for high-bit-rate data
transmission in the nonreturn-to-zero (NRZ) format over transoceanic distance.
The system operates in a small group velocity dispersion regime, and the model
equation is given by the Whitham equations describing the slow modulation of
multi-phase wavetrains of the (defocusing) nonlinear Schr\"odinger (NLS)
equation. The model equation is of hyperbolic type, and certain initial NRZ
pulse with phase modulation develops a shock. We then show how one can obtain a
global solution by choosing an appropriate Riemann surface on which the Whitham
equation is defined. The present analysis may be interpreted as an alternative
to the method of inverse scattering transformation for the NLS solitons. We
also discuss wavelength-division-multiplexing (WDM) in the NRZ format by using
the Whitham equation for a coupled NLS equation, and show that there exists a
hydro-dynamic-type instability between channels.


title:
On a two dimensional system associated with the complex of the solutions
  of the Tetrahedron equation

abstract:
  A sort of two dimensional linear auxiliary problem for the complex of 3D $R$
-- operators associated with the Zamolodchikov -- Bazhanov -- Baxter
statistical model is proposed. This problem resembles the problem of the local
Yang -- Baxter equation but does not coincide with it. The formulation of the
auxiliary problem admits a notion of a ``fusion'', and usual local Yang --
Baxter equation appears among other results of this ``fusion''.


title: Susy Hierarchies and Affine Algebras

abstract:
  We review some basic features of the Lie-algebraic classification of
W-algebras and related integrable hierarchies in 1+1 dimensions, pointing out
the role of affine Lie algebras. We emphasize that the supersymmetric
extensions of the above construction possibly lead, though some questions are
still opened, to the classification of supersymmetric hierarchies based on
``generic'' supersymmetric affine Lie algebras. Here the word generic is used
to make clear that well-known procedures, as those introduced by Inami and
Kanno, are too restricted and do not lead to the full spectrum of
supersymmetric integrable hierarchies one can construct. A particular attention
is devoted to the large-N supersymmetric extensions (here N=4). The attention
paid by large-N theories being due to the fact that they arise as dimensional
reduction of N=1 models, and moreover that they realize an ``unification'' of
known hierarchies.


title: Correlation functions for a strongly correlated boson system

abstract:
  The correlation functions for a strongly correlated exactly solvable
one-dimensional boson system on a finite chain as well as in the thermodynamic
limit are calculated explicitly. This system which we call the phase model is
the strong coupling limit of the integrable q-boson hopping model. The results
are presented as determinants.

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


title: Shock waves in one-dimensional Heisenberg ferromagnets

abstract:
  We use SU(2) coherent state path integral formulation with the stationary
phase approximation to investigate, both analytically and numerically, the
existence of shock waves in the one- dimensional Heisenberg ferromagnets with
anisotropic exchange interaction. As a result we show the existence of shock
waves of two types,"bright" and "dark", which can be interpreted as moving
magnetic domains.


title:
Two-body Elliptic Model in proper variables: Lie-algebraic forms and
  their discretizations

abstract:
  Two Lie algebraic forms of the 2-body Elliptic Calogero model are presented.
Translation-invariant and dilatation-invariant discretizations of the model are
obtained.


title:
Binary Darboux-Backlund Transformations for the Manin-Radul Super KdV
  Hierarchy

abstract:
  We construct the supersymmetric extensions of the Darboux-Backlund
transformations (DBTs) for the Manin-Radul super KdV hierarchy using the
super-pseudo-differential operators. The elementary DBTs are triggered by the
gauge operators constructed from the wave functions and adjoint wave functions
of the hierarchy. Iterating these elementary DBTs, we obtain not only Wronskian
type but also binary type superdeterminant representations of the solutions.


title: AKS scheme for face and Calogero-Moser-Sutherland type models

abstract:
  We give the construction of quantum Lax equations for IRF models and
difference versions of Calogero-Moser-Sutherland models introduced by
Ruijsenaars. We solve the equations using factorization properties of the
underlying face Hopf algebras/elliptic quantum groups. This construction is in
the spirit of the Adler-Kostant-Symes method and generalizes our previous work
to the case of face Hopf algebras/elliptic quantum groups with dynamical
R-matrices.


title: Equations of the reaction-diffusion type with a loop algebra structure

abstract:
  A system of equations of the reaction-diffusion type is studied in the
framework of both the direct and the inverse prolongation structure. We find
that this system allows an incomplete prolongation Lie algebra, which is used
to find the spectral problem and a whole class of nonlinear field equations
containing the original ones as a special case.


title: On classical string configurations

abstract:
  Equations which define classical configurations of strings in $R^3$ are
presented in a simple form. General properties as well as particular classes of
solutions of these equations are considered.


title: A new explicit expression for the Korteweg-De Vries hierarchy

abstract:
  We derive an improved fully explicit expression for the right-hand sides of
the matrix KdV hierarchy using the relation to the heat kernel of the
one-dimensional Schr\"odinger operator. Our method of "matrix elements"
produces, moreover, an explicit expression for the powers of a
Schr\"odinger-like differential operator of any order.


title: On Fifth Order KdV-Type Equation

abstract:
  The dynamics of the highly nonlinear fifth order $KdV$-type equation is
discussed in the framework of the Lagrangian and Hamiltonian formalisms. The
symmetries of the Lagrangian produce three commuting conserved quantities that
are found to be recursively related to one-another for a certain specific value
of the power of nonlinearity. The above cited recursion relations are obeyed
with a second Poisson bracket which sheds light on the integrability properties
of the above nonlinear equation. It is shown that a Miura-type transformation
can be made to obtain the fifth order $mKdV$-type equation from the fifth order
$KdV$-type equation. The spatial dependence of the fields involved is, however,
not physically interesting from the point of view of the solitonic solutions.
As a consequence, it seems that the fifth order $KdV$- and $mKdV$-type
equations are completely independent nonlinear evolution equations in their own
right.


title: A Conjectured R-Matrix

abstract:
  A new spectral parameter independent R-matrix (that depends on all of the
dynamical variables) is proposed for the elliptic Calogero-Moser models.
Necessary and sufficient conditions for this R-matrix to exist reduce to an
equality between determinants of matrices involving elliptic functions. The
needed identity appears new and is still unproven in full generality: we
present it as a conjecture.


title:
Hamiltonian Dynamics, Classical R-matrices and Isomonodromic
  Deformations

abstract:
  The Hamiltonian approach to the theory of dual isomonodromic deformations is
developed within the framework of rational classical R-matrix structures on
loop algebras. Particular solutions to the isomonodromic deformation equations
appearing in the computation of correlation functions in integrable quantum
field theory models are constructed through the Riemann-Hilbert problem method.
The corresponding $\tau$-functions are shown to be given by the Fredholm
determinant of a special class of integral operators.


title: Asymptotic dynamics of short-waves in nonlinear dispersive models

abstract:
  The multiple-scale perturbation theory, well known for long-waves, is
extended to the study of the far-field behaviour of short-waves, commonly
called ripples. It is proved that the Benjamin-Bona-Mahony- Peregrine equation
can propagates short-waves. This result contradict the Benjamin hypothesis that
short-waves tends not to propagate in this model and close a part of the old
controversy between Korteweg-de Vries and Benjamin-Bona-Mahony-Peregrine
equations. We shown that a nonlinear (quadratic) Klein-Gordon type equation
substitutes in a short-wave analysis the ubiquitous Korteweg-de Vries equation
of long-wave approach. Moreover the kink solutions of phi-4 and sine-Gordon
equations are understood as an all orders asymptotic behaviour of short-waves.
It is proved that the antikink solution of phi-4 model which was never obtained
perturbatively can be obtained by perturbation expansion in the wave-number k
in the short-wave limit.


title: Vectorial Ribaucour Transformations for the Lame Equations

abstract:
  The vectorial extension of the Ribaucour transformation for the Lame
equations of orthogonal conjugates nets in multidimensions is given. We show
that the composition of two vectorial Ribaucour transformations with
appropriate transformation data is again a vectorial Ribaucour transformation,
from which it follows the permutability of the vectorial Ribaucour
transformations. Finally, as an example we apply the vectorial Ribaucour
transformation to the Cartesian background.


title: Hidden quantum R-matrix in discrete time classical Heisenberg magnet

abstract:
  We construct local M-operators for an integrable discrete time version of the
classical Heisenberg magnet by convolution of the twisted quantum trigonometric
4$\times$4 R-matrix with certain vectors in its "quantum" space. Components of
the vectors are identified with $\tau$-functions of the model. Hirota's
bilinear formalism is extensively used. The construction generalizes the known
representation of M-operators in continuous time models in terms of Lax
operators and classical $r$-matrix.


title:
Bispectral Operators, Dual Isomonodromic Deformations and the
  Riemann-Hilbert Dressing Method

abstract:
  A comparison is made between bispectral systems and dual isomonodromic
deformation equations. A number of examples are given, showing how bispectral
systems may be embedded into isomonodromic ones. Sufficiency conditions are
given for the construction of rational solutions of isomonodromic deformation
equations through the Riemann-Hilbert problem dressing method, and these are
shown, in certain cases, to reduce to bispectral systems.


title:
Jack polynomials, generalized binomial coefficients and polynomial
  solutions of the generalized Laplace's equation

abstract:
  We discuss the symmetric homogeneous polynomial solutions of the generalized
Laplace's equation which arises in the context of the Calogero-Sutherland model
on a line. The solutions are expressed as linear combinations of Jack
polynomials and the constraints on the coefficients of expansion are derived.
These constraints involve generalized binomial coefficients defined through
Jack polynomials. Generalized binomial coefficients for partitions of $k$ upto
$k=6$ are tabulated.


title: Unifying structures in quantum integrable systems

abstract:
  Basic concepts of quantum integrable systems (QIS) are presented stressing on
the unifying structures underlying such diverse models. Variety of ultralocal
and nonultralocal models is shown to be described by a few basic relations
defining novel algebraic entries. Such properties can generate and classify
integrable models systematically and also help to solve exactly their
eigenvalue problem in an almost model-independent way. The unifying thread
stretches also beyond the QIS to establish its deep connections with
statistical models, conformal field theory etc. as well as with abstract
mathematical objects like quantum group, braided or quadratic algebra


title: D-modules and Darboux transformations

abstract:
  A method of G. Wilson for generating commutative algebras of ordinary
differential operators is extended to higher dimensions. Our construction,
based on the theory of D-modules, leads to a new class of examples of
commutative rings of partial differential operators with rational spectral
varieties. As an application, we briefly discuss their link to the bispectral
problem and to the theory of lacunas.


title: The Painlev\'e approach to nonlinear ordinary differential equations

abstract:
  The ``Painlev\'e analysis'' is quite often perceived as a collection of
tricks reserved to experts. The aim of this course is to demonstrate the
contrary and to unveil the simplicity and the beauty of a subject which is in
fact the theory of the (explicit) integration of nonlinear differential
equations.
  To achieve our goal, we will not start the exposition with a more or less
precise ``Painlev\'e test''. On the contrary, we will finish with it, after a
gradual introduction to the rich world of singularities of nonlinear
differential equations, so as to remove any cooking recipe.
  The emphasis is put on embedding each method of the test into the well known
theorem of perturbations of Poincar\'e. A summary can be found at the beginning
of each chapter.


title: The Discrete Painlev\'e I Hierarchy

abstract:
  The discrete Painlev\'e I equation (dP$\rm_I$) is an integrable difference
equation which has the classical first Painlev\'e equation (P$\rm_I$) as a
continuum limit. dP$\rm_I$ is believed to be integrable because it is the
discrete isomonodromy condition for an associated (single-valued) linear
problem. In this paper, we derive higher-order difference equations as
isomonodromy conditions that are associated to the same linear deformation
problem. These form a hierarchy that may be compared to hierarchies of
integrable ordinary differential equations (ODEs). We strengthen this
comparison by continuum limit calculations that lead to equations in the
P$\rm_I$ hierarchy. We propose that our difference equations are discrete
versions of higher-order Painlev\'e equations.


title:
The Second Painlev\'e Equation in the Large-Parameter Limit I: Local
  Asymptotic Analysis

abstract:
  In this paper, we find all possible asymptotic behaviours of the solutions of
the second Painlev\'e equation $y''=2y^3+xy +\alpha$ as the parameter
$\alpha\to\infty$ in the local region $x\ll\alpha^{2/3}$. We prove that these
are asymptotic behaviours by finding explicit error bounds. Moreover, we show
that they are connected and complete in the sense that they correspond to all
possible values of initial data given at a point in the local region.


title:
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a
  Review and Extensions of Tests for the Painlev\'e Property

abstract:
  The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.


title:
Complete Nondiagonal Reflection Matrices of RSOS/SOS and Hard Hexagon
  Models

abstract:
  In this paper we compute the most general nondiagonal reflection matrices of
the RSOS/SOS models and hard hexagon model using the boundary Yang-Baxter
equations. We find new one-parameter family of reflection matrices for the RSOS
model in addition to the previous result without any parameter. We also find
three classes of reflection matrices for the SOS model, which has one or two
parameters. For the hard hexagon model which can be mapped to RSOS(5) model by
folding four RSOS heights into two, the solutions can be obtained similarly
with a main difference in the boundary unitarity conditions. Due to this, the
reflection matrices can have two free parameters. We show that these extra
terms can be identified with the `decorated' solutions. We also generalize the
hard hexagon model by `folding' the RSOS heights of the general RSOS(p) model
and show that they satisfy the integrability conditions such as the Yang-
Baxter and boundary Yang-Baxter equations. These models can be solved using the
results for the RSOS models.


title: Self-duality of the SL_2 Hitchin integrable system at genus two

abstract:
  We revisit the Hitchin integrable system whose phase space is the bundle
cotangent to the moduli space $N$ of holomorphic $SL_2$-bundles over a smooth
complex curve of genus two. $N$ may be identified with the 3-dimensional
projective space of theta functions of the second order, We prove that the
Hitchin system on $T^*N$ possesses a remarkable symmetry: it is invariant under
the interchange of positions and momenta. This property allows to complete the
work of van Geemen-Previato which, basing on the classical results on geometry
of the Kummer quartic surfaces, specified the explicit form of the Hamiltonians
of the Hitchin system. The resulting integrable system resembles the classic
Neumann systems which are also self-dual. Its quantization produces a commuting
family of differential operators of the second order acting on homogeneous
polynomials in four complex variables. As recently shown by van Geemen-de Jong,
these operators realize the Knizhnik-Zamolodchikov-Bernard-Hitchin connection
for group SU(2) and genus 2 curves.

categories: solv-int alg-geom hep-th math.AG nlin.SI


title: The sAKNS Hierarchy

abstract:
  We study, systematically, the properties of the supersymmetric AKNS (sAKNS)
hierarchy. In particular, we discuss the Lax representation in terms of a
bosonic Lax operator and some special features of the equations and construct
the bosonic local charges as well as the fermionic nonlocal charges associated
with the system starting from the Lax operator. We obtain the Hamiltonian
structures of the system and check the Jacobi identity through the method of
prolongation. We also show that this hierarchy of equations can equivalently be
described in terms of a fermionic Lax operator. We obtain the zero curvature
formulation as well as the conserved charges of the system starting from this
fermionic Lax operator which suggests a connection between the two. Finally,
starting from the fermionic description of the system, we construct the soliton
solutions for this system of equations through Darboux-Backlund transformations
and describe some open problems.


title:
Surfaces of Revolution via the Schroedinger Equation : Construction,
  Integrable Dynamics and Visualization

abstract:
  Surfaces of revolution in three-dimensional Euclidean space are considered.
Several new examples of surfaces of revolution associated with well-known
solvable cases of the Schoedinger equation (infinite well, harmonic oscillator,
Coulomb potential, Bargmann potential, etc.) are analyzed and visualized. The
properties of such surfaces are discussed. Two types of deformations
(evolutions), namely 1) preserving the Gaussian curvature and 2) via the
dynamics of the Korteweg-de-Vries equation are discussed.


title:
Equal-time temperature correlators of the one-dimensional Heisenberg XY
  chain

abstract:
  Representations as determinants of $M\times M$ dimensional matrices are
obtained for equal-time temperature correlators of the anisotropic Heisenberg
XY chain. These representations are simple deformations of the answers for the
isotropic XX0 chain. In the thermodynamic limit, the correlators are expressed
in terms of the Fredholm determinants of linear integral operators.


title:
Integrable Models and the Higher Dimensional Representations of Graded
  Lie Algebras

abstract:
  We construct a zero curvature formulation, in superspace, for the sTB-B
hierarchy which naturally reduces to the zero curvature condition in terms of
components, thus solving one of the puzzling features of this model. This
analysis, further, suggests a systematic method of constructing higher
dimensional representations for the zero curvature condition starting with the
fundamental representation. We illustrate this with the examples of the sTB
hierarchy and the sKdV hierarchy. This would be particularly useful in
constructing explicit higher dimensional representations of graded Lie
algebras.


title: Darboux Transformations for SUSY Integrable Systems

abstract:
  Several types of Darboux transformations for supersymmetric integrable
systems such as the Manin-Radul KdV, Mathieu KdV and SUSY sine-Gordon equations
are considered. We also present solutions such as supersolitons and superkinks.


title:
Additional symmetries of the Zakharov-Shabat hierarchy, String equation
  and Isomonodromy

abstract:
  Isomonodromic deformations are nothing but symmetries of the Zakharov-Shabat
(isospectral) hierarchy, both the basic ones (belonging to the hierarchy) and
additional, restricted to the submanifold of solutions to the string equation.


title: On the Integrability of the One-Dimensional Open XYZ Spin Chain

abstract:
  The Lax pair for the one-dimensional open XYZ spin chain is constructed, this
shows that the system is completely integrable .


title:
A Note on the Gauge Equivalence between the Manin-Radul and
  Laberge-Mathieu Super KdV Hierarchies

abstract:
  The gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV
hierarchies is revisited. Apart from the Inami-Kanno transformation, we show
that there is another gauge transformation which also possess the canonical
property. We explore the relationship of these two gauge transformations from
the Kupershmidt-Wilson theorem viewpoint and, as a by-product, obtain the
Darboux-Backlund transformation for the Manin-Radul super KdV hierarchy. The
geometrical intepretation of these transformations is also briefly discussed.


title:
Comment on ``Equal-time temperature correlators of the one-dimensional
  Heisenberg XY chain'', preprint solv-int/9710028

abstract:
  In the comment we give references to our papers where the problem was solved
for more general case of time-dependent finite temperature correlators.


title: A new class of completely integrable quantum spin chains

abstract:
  A large (infinitely-dimensional) class of completely integrable (possibly
non-autonomous) spin chains is discovered associated to an infinite-dimensional
Lie Algebra of infinite rank. The complete set of integrals of motion is
constructed explicitly, as well as their eigenstates and spectra. As an example
we outline kicked Ising model: Ising chain periodically kicked with transversal
magnetic field.


title: The nondynamical r-matrix structure of the elliptic Calogero-Moser model

abstract:
  In this paper, we construct a new Lax operator for the elliptic
Calogero-Moser model with N=2. The nondynamical r-matrix structure of this Lax
operator is also studied .
  The relation between our Lax operator and the Lax operator given by Krichever
is also obtained.


title: The N=2 supersymmetric matrix GNLS hierarchies

abstract:
  We construct the matrix generalization of the N=2 supersymmetric GNLS
hierarchies. This is done by exhibiting the corresponding matrix super Lax
operators in terms of N=2 superfields in two different superfield bases. We
present the second Hamiltonian structure and discrete symmetries. We then
extend our discussion by conjecturing the Lax operators of different reductions
of the N=2 supersymmetric matrix KP hierarchy and discuss the simplest
examples.


title: Few remarks on Baecklund transformations for many-body systems

abstract:
  Using the n-particle periodic Toda lattice and the relativistic
generalization due to Ruijsenaars of the elliptic Calogero-Moser system as
examples, we revise the basic properties of the Baecklund transformations
(BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's
quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a
conjugated variable mu for the parameter lambda of the BT B_lambda such that mu
belongs to the spectrum of the Lax operator L(lambda). As a consequence, the
generating function of the composition of n BT's gives rise also to another
canonical transformation separating variables for the model. For the Toda
lattice the dual BT parametrized by mu is introduced.


title: Vertex Operators and Solitons of Constrained KP Hierarchies

abstract:
  We construct the vertex operator representation for the Affine Kac-Moody
$SL(M+K+1)$ algebra, which is relevant for the construction of the soliton
solutions of the constrained KP hierarchies. The oscillators involved in the
vertex operator construction are provided by the Heisenberg subalgebras of
$SL(M+K+1)$ realized in the unconventional gradations. The well-known limiting
cases are the homogeneous Heisenberg subalgebra of $SL(M+1)$ and the principal
Heisenberg subalgebra of ${\hat{sl}}(K+1)$. The explicit example of $M=K=1$ is
discussed in detail and the corresponding soliton solutions and tau-functions
are given.


title: Coset approach to the N=2 supersymmetric matrix GNLS hierarchies

abstract:
  We discuss a large class of coset constructions of the N=2 sl(n|n-1) affine
superalgebra. We select admissible subalgebras, i.e. subalgebras that induce
linear chiral/antichiral constraints on the coset supercurrents. We show that
all the corresponding coset constructions lead to N=2 matrix GNLS hierarchies.
We develop an algorithm to compute the relative Hamiltonians and flows. We
spell out completely the case of the N=2 affine sl(3|2), which possesses four
admissible subalgebras. The non-local second Hamiltonian structure of the N=2
matrix GNLS hierarchies is obtained via Dirac procedure from the local N=2
sl(n|n-1) affine superalgebra. We observe that to any second Hamiltonian
structure with pure bosonic or pure fermionic superfield content there
correspond two different N=2 matrix GNLS hierarchies.


title:
Commuting Charges of the Quantum Korteweg-deVries and Boussinesq
  Theories from the Reduction of W(infinity) and W(1+infinity) Algebras

abstract:
  Integrability of the quantum Boussinesq equation for c=-2 is demonstrated by
giving a recursive algorithm for generating explicit expressions for the
infinite number of commuting charges based on a reduction of the W(infinity)
algebra. These charges exist for all spins $s \geq 2$. Likewise, reductions of
the W(infinity/2) and W((1+infinity)/2) algebras yield the commuting quantum
charges for the quantum KdV equation at c=-2 and c=1/2, respectively.


title: Singularity confinement and chaos in discrete systems

abstract:
  We present a number of second order maps, which pass the singularity
confinement test commonly used to identify integrable discrete systems, but
which nevertheless are non-integrable. As a more sensitive integrability test,
we propose the analysis of the complexity (``algebraic entropy'') of the map
using the growth of the degree of its iterates: integrability is associated
with polynomial growth while the generic growth is exponential for chaotic
systems.


title: Integrable Coupled KdV Systems

abstract:
  We give the conditions for a system of N- coupled Korteweg de Vries(KdV) type
of equations to be integrable. Recursion operators of each subclasses are also
given. All examples for N=2 are explicitly given.


title:
On the geometry of point-expansions for certain class of differential
  equations of the second order

abstract:
  Second order ordinary differential equations of the form $y'' = P(x,y)
  + 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered
and their point-expansions are constructed. Geometrical structures connected
with these expansions are described.


title: Extended N=2 supersymmetric matrix (1,s)-KdV hierarchies

abstract:
  We propose the Lax operators for N=2 supersymmetric matrix generalization of
the bosonic (1,s)-KdV hierarchies. The simplest examples - the N=2
supersymmetric a=4 KdV and a=5/2 Boussinesq hierarchies - are discussed in
detail.


title: The Stackel systems and algebraic curves

abstract:
  We show how the Abel-Jacobi map provides all the principal properties of an
ample family of integrable mechanical systems associated to hyperelliptic
curves. We prove that derivative of the Abel-Jacobi map is just the St\"{a}ckel
matrix, which determines $n$-orthogonal curvilinear coordinate systems in a
flat space. The Lax pairs, $r$-matrix algebras and explicit form of the flat
coordinates are constructed. An application of the Weierstrass reduction theory
allows to construct several flat coordinate systems on a common hyperelliptic
curve and to connect among themselves different integrable systems on a single
phase space.


title:
The Complex Toda Chains and the Simple Lie Algebras - Solutions and
  Large Time Asymptotics

abstract:
  The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends
related to the classical series of simple Lie algebras are classified. It is
shown that the CTC models have much richer variety of asymptotic regimes than
the real Toda chain (RTC). Besides asymptotically free propagation (the only
possible regime for the RTC), CTC allow bound state regimes, various
intermediate regimes when one (or several) group(s) of particles form bound
state(s), singular and degenerate solutions. These results can be used e.g., in
describing the soliton interactions of the nonlinear Schroedinger equation.
Explicit expressions for the solutions in terms of minimal sets of scattering
data are proposed for all classical series B_r - D_r.


title: The New Identity for the Scattering Matrx of Exactly Solvable Models

abstract:
  We discovered a simple quadratic equation, which relates scattering phases of
particles on Fermi surface. We consider one dimensional Bose gas and XXZ
Heisenberg spin chain.


title: Integrability in 3+1 Dimensions: Relaxing a Tetrahedron Relation

abstract:
  I propose a scheme of constructing classical integrable models in 3+1
discrete dimensions, based on a relaxed version of the problem of factorizing a
matrix into the product of four matrices of a special form.


title:
Dynamical Correlation Functions for an Impenetrable Bose gas with open
  boundary conditions

abstract:
  We study the time and temperature dependent correlation functions for an
impenetrable bose gas with open boundary conditions. We derive the Fredholm
determinant formulae for the correlation functions, by means of the Bethe
Ansatz. In the case of time independent ground state, our Fredholm determinant
formulae degenerate to the one which have been obtained by the help of fermions
[T. Kojima, J.Stat.Phys.Vol.88,713-(1997)]


title: The XXC Models

abstract:
  A class of recently introduced multi-states XX models is generalized to
include a deformation parameter. This corresponds to an additional
nearest-neighbor CC interaction in the defining quadratic hamiltonian. Complete
integrability of the one-dimensional models is shown in the context of the
quantum inverse scattering method. The new R-matrix is derived. The
diagonalization of the XXC models is carried out using the algebraic Bethe
Ansatz.


title: Canonical gauge equivalences of the sAKNS and sTB hierarchies

abstract:
  We study the gauge transformations between the supersymmetric AKNS (sAKNS)
and supersymmetric two-boson (sTB) hierarchies. The Hamiltonian nature of these
gauge transformations is investigated, which turns out to be canonical. We also
obtain the Darboux-Backlund transformations for the sAKNS hierarchy from these
gauge transformations.


title:
The averaging of Hamiltonian structures in discrete variant of Whitham
  method

abstract:
  Paper is devoted to the construction of averaging procedure of Hamiltonian
structures in discrete Whitham method. The procedure is analogous to
Dubrovin-Novikov procedure of averaging of local field-theoretical Poisson
brackets and gives the Poisson bracket of Hydrodynamic Type starting from
Poisson bracket for a discrete chain.


title: Boundary K-matrices and the Lax pair for 1D open XYZ spin-chain

abstract:
  We analysis the symmetries of the reflection equation for open $XYZ$ model
and find their solutions $K^{\pm}$ case by case. In the general open boundary
conditions, the Lax pair for open one-dimensional $XYZ$ spin-chain is given.


title: A New ``Dual'' Symmetry Structure of the KP Hierarchy

abstract:
  A new infinite set of commuting additional (``ghost'') symmetries is proposed
for the KP-type integrable hierarchy. These symmetries allow for a Lax
representation in which they are realized as standard isospectral flows. This
gives rise to a new double-KP hierarchy embedding ``ghost'' and original
KP-type Lax hierarchies connected to each other via a ``duality'' mapping
exchanging the isospectral and ``ghost'' ``times''. A new representation of 2D
Toda lattice hierarchy as a special Darboux-Backlund orbit of the double-KP
hierarchy is found and parametrized entirely in terms of (adjoint)
eigenfunctions of the original KP subsystem.


title: A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator

abstract:
  \We consider an inverse scattering problem for Schr\"odinger operators with
energy dependent potentials. The inverse problem is formulated as a
Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for
two distinct symmetry classes. As an application we prove global existence
theorems for the two distinct systems of partial differential equations
$u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted,
complementary classes of initial data.