title: Generating Quadrilateral and Circular Lattices in KP Theory

abstract:
  The bilinear equations of the $N$-component KP and BKP hierarchies and a
corresponding extended Miwa transformation allow us to generate quadrilateral
and circular lattices from conjugate and orthogonal nets, respectively. The
main geometrical objects are expressed in terms of Baker functions.


title: Stochastic Soliton Lattices

abstract:
  We introduce a new concept, Stochastic Soliton Lattice, as a random process
generated by a finite-gap potential of the Shroedinger operator. We study the
basic properties of this stochastic process and consider its KdV evolution


title: On Discretizations of the Vector Nonlinear Schrodinger Equation

abstract:
  Two discretizations of the vector nonlinear Schrodinger (NLS) equation are
studied. One of these discretizations, referred to as the symmetric system, is
a natural vector extension of the scalar integrable discrete NLS equation. The
other discretization, referred to as the asymmetric system, has an associated
linear scattering pair. General formulae for soliton solutions of the
asymmetric system are presented. Formulae for a constrained class of solutions
of the symmetric system may be obtained. Numerical studies support the
hypothesis that the symmetric system has general soliton solutions.


title:
The Lax operators $\cal L$ of the Benney type equations bound with the
  circle

abstract:
  The Lax operators of the Benney type equations are studied on the circle. The
vectors fields of the Lax operators are showed to commute with each other


title:
Asymptotics of the Fredholm determinant associated with the correlation
  functions of the quantum Nonlinear Schrodinger equation

abstract:
  The correlation functions of the quantum nonlinear Schrodinger equation can
be presented in terms of a Fredholm determinant. The explicit expression for
this determinant is found for the large time and long distance.


title:
Algebraic Exact Solvability of trigonometric-type Hamiltonians
  associated to root systems

abstract:
  In this article, we study and settle several structural questions concerning
the exact solvability of the Olshanetsky-Perelomov quantum Hamiltonians
corresponding to an arbitrary root system. We show that these operators can be
written as linear combinations of certain basic operators admitting infinite
flags of invariant subspaces, namely the Laplacian and the logarithmic gradient
of invariant factors of the Weyl denominator. The coefficients of the
constituent linear combination become the coupling constants of the final
model. We also demonstr ate the $L^2$ completeness of the eigenfunctions
obtained by this procedure, and describe a straight-forward recursive procedure
based on the Freudenthal multiplicity formula for constructing the
eigenfunctions explicitly.

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


title:
Discrete time Lagrangian mechanics on Lie groups, with an application to
  the Lagrange top

abstract:
  We develop the theory of discrete time Lagrangian mechanics on Lie groups,
originated in the work of Veselov and Moser, and the theory of Lagrangian
reduction in the discrete time setting. The results thus obtained are applied
to the investigation of an integrable time discretization of a famous
integrable system of classical mechanics, -- the Lagrange top. We recall the
derivation of the Euler--Poinsot equations of motion both in the frame moving
with the body and in the rest frame (the latter ones being less widely known).
We find a discrete time Lagrange function turning into the known continuous
time Lagrangian in the continuous limit, and elaborate both descriptions of the
resulting discrete time system, namely in the body frame and in the rest frame.
This system naturally inherits Poisson properties of the continuous time
system, the integrals of motion being deformed. The discrete time Lax
representations are also found. Kirchhoff's kinetic analogy between elastic
curves and motions of the Lagrange top is also generalised to the discrete
context.


title: Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation

abstract:
  The purpose of this paper is to present a class of particular solutions of a
C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry
reduction. Using the subgroups of similitude group reduced ordinary
differential equations of second order and their solutions by a singularity
analysis are classified. In particular, it has been shown that whenever they
have the Painlev\'e property, they can be transformed to standard forms by
Moebius transformations of dependent variable and arbitrary smooth
transformations of independent variable whose solutions, depending on the
values of parameters, are expressible in terms of either elementary functions
or Jacobi elliptic functions.


title: On Integrability and Chaos in Discrete Systems

abstract:
  The scalar nonlinear Schrodinger (NLS) equation and a suitable discretization
are well known integrable systems which exhibit the phenomena of ``effective''
chaos. Vector generalizations of both the continuous and discrete system are
discussed. Some attention is directed upon the issue of the integrability of a
discrete version of the vector NLS equation.


title: Hirota bilinear forms with 2-toroidal symmetry

abstract:
  In this note, we compute Hirota bilinear forms arising from both homogeneous
and principal realization of vertex representations of 2-toroidal Lie algebras
of type $A_l, D_l, E_l$.


title:
Bethe ansatz solution of the anisotropic correlated electron model
  associated with the Temperley-Lieb algebra

abstract:
  A recently proposed strongly correlated electron system associated with the
Temperley-Lieb algebra is solved by means of the coordinate Bethe ansatz for
periodic and closed boundary conditions.


title: Quantum 2+1 evolution model

abstract:
  A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.


title:
On Darboux-B\"acklund Transformations for the Q-Deformed Korteweg-de
  Vries Hierarchy

abstract:
  We study Darboux-B\"acklund transformations (DBTs) for the $q$-deformed
Korteweg-de Vries hierarchy by using the $q$-deformed pseudodifferential
operators. We identify the elementary DBTs which are triggered by the gauge
operators constructed from the (adjoint) wave functions of the hierarchy.
Iterating these elementary DBTs we obtain not only $q$-deformed Wronskian-type
but also binary-type representations of the tau-function to the hierarchy.


title: A unified treatment of cubic invariants at fixed and arbitrary energy

abstract:
  Cubic invariants for two-dimensional Hamiltonian systems are investigated
using the Jacobi geometrization procedure. This approach allows for a unified
treatment of invariants at both fixed and arbitrary energy. In the geometric
picture the invariant generally corresponds to a third rank Killing tensor,
whose existence at a fixed energy value forces the metric to satisfy a
nonlinear integrability condition expressed in terms of a Kahler potential.
Further conditions, leading to a system of equations which is overdetermined
except for singular cases, are added when the energy is arbitrary. As solutions
to these equations we obtain several new superintegrable cases in addition to
the previously known cases. We also discover a superintegrable case where the
cubic invariant is of a new type which can be represented by an energy
dependent linear invariant. A complete list of all known systems which admit a
cubic invariant at arbitrary energy is given.


title: Canonical variables for multiphase solutions of the KP equation

abstract:
  The KP equation has a large family of quasiperiodic multiphase solutions.
These solutions can be expressed in terms of Riemann-theta functions. In this
paper, a finite-dimensional canonical Hamiltonian system depending on a finite
number of parameters is given for the description of each such solution. The
Hamiltonian systems are completely integrable in the sense of Liouville. In
effect, this provides a solution of the initial-value problem for the
theta-function solutions. Some consequences of this approach are discussed.


title:
On a class of dynamical systems both quasi-bi-Hamiltonian and
  bi-Hamiltonian

abstract:
  It is shown that a class of dynamical systems (encompassing the one recently
considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both
quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the
separability of these systems; the second one is obtained trough a non
canonical map whose form is directly suggested by the associated Nijenhuis
tensor.


title:
Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation
  variables

abstract:
  We discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi
separability of a few dynamical systems. They are shown to admit, in their
natural phase space, a quasi-bi-Hamiltonian formulation of Pfaffian type. This
property allows us to straightforwardly recover a set of separation variables
for the corresponding Hamilton-Jacobi equation.


title: Spectral Difference Equations Satisfied by KP Soliton Wavefunctions

abstract:
  The Baker-Akhiezer (wave) functions corresponding to soliton solutions of the
KP hierarchy are shown to satisfy eigenvalue equations for a commutative ring
of translational operators in the spectral parameter. In the rational limit,
these translational operators converge to the differential operators in the
spectral parameter previously discussed as part of the theory of
"bispectrality". Consequently, these translational operators can be seen as
demonstrating a form of bispectrality for the non-rational solitons as well.


title:
Q-deformed KP hierarchy: Its additional symmetries and infinitesimal
  B\"acklund transformations

abstract:
  We study the additional symmetries associated with the $q$-deformed
Kadomtsev-Petviashvili ($q$-KP) hierarchy. After identifying the resolvent
operator as the generator of the additional symmetries, the $q$-KP hierarchy
can be consistently reduced to the so-called $q$-deformed constrained KP
($q$-cKP) hierarchy. We then show that the additional symmetries acting on the
wave function can be viewed as infinitesimal B\"acklund transformations by
acting the vertex operator on the tau-function of the $q$-KP hierarchy. This
establishes the Adler-Shiota-van Moerbeke formula for the $q$-KP hierarchy.


title: Darboux Transformations and solutions for an equation in 2+1 dimensions

abstract:
  Painleve analysis and the singular manifold method are the tools used in this
paper to perform a complete study of an equation in 2+1 dimensions. This
procedure has allowed us to obtain the Lax pair, Darboux transformation and tau
functions in such a way that a plethora of different solutions with solitonic
behavior can be constructed iteratively


title: Two Integrable Systems Related to Hyperbolic Monopoles

abstract:
  Monopoles on hyperbolic 3-space were introduced by Atiyah in 1984. This
article describes two integrable systems which are closely related to
hyperbolic monopoles: a one-dimensional lattice equation (the Braam-Austin or
discrete Nahm equation), and a soliton system in (2+1)-dimensional
anti-deSitter space-time.


title: Integrable KdV Systems: Recursion Operators of Degree Four

abstract:
  The recursion operator and bi-Hamiltonian formulation of the Drinfeld-
Sokolov system are given


title:
B\"acklund transformations for the second Painlev\'e hierarchy: a
  modified truncation approach

abstract:
  The second Painlev\'e hierarchy is defined as the hierarchy of ordinary
differential equations obtained by similarity reduction from the modified
Korteweg-de Vries hierarchy. Its first member is the well-known second
Painlev\'e equation, P2.
  In this paper we use this hierarchy in order to illustrate our application of
the truncation procedure in Painlev\'e analysis to ordinary differential
equations. We extend these techniques in order to derive auto-B\"acklund
transformations for the second Painlev\'e hierarchy. We also derive a number of
other B\"acklund transformations, including a B\"acklund transformation onto a
hierarchy of P34 equations, and a little known B\"acklund transformation for P2
itself.
  We then use our results on B\"acklund transformations to obtain, for each
member of the P2 hierarchy, a sequence of special integrals.


title:
Nonlinear Evolution Equations Invariant Under Schroedinger Group in
  three-dimensional Space-time

abstract:
  A classification of all possible realizations of the Galilei,
Galilei-similitude and Schroedinger Lie algebras in three-dimensional
space-time in terms of vector fields under the action of the group of local
diffeomorphisms of the space $\R^3\times\C$ is presented. Using this result a
variety of general second order evolution equations invariant under the
corresponding groups are constructed and their physical significance are
discussed.


title:
The supersymmetric Camassa-Holm equation and geodesic flow on the
  superconformal group

abstract:
  We study a family of fermionic extensions of the Camassa-Holm equation.
Within this family we identify three interesting classes: (a) equations, which
are inherently hamiltonian, describing geodesic flow with respect to an H^1
metric on the group of superconformal transformations in two dimensions, (b)
equations which are hamiltonian with respect to a different hamiltonian
structure and (c) supersymmetric flow equations. Classes (a) and (b) have no
intersection, but the intersection of classes (a) and (c) gives a candidate for
a new supersymmetric integrable system. We demonstrate the Painlev\'e property
for some simple but nontrivial reductions of this system.

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


title: Duality between integrable Stackel systems

abstract:
  For the Stackel family of the integrable systems a non-canonical
transformation of the time variable is considered. This transformation may be
associated to the ambiguity of the Abel map on the corresponding hyperelliptic
curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax
representations and the dynamical r-matrix algebras are constructed. As an
examples the Henon-Heiles systems, integrable Holt potentials and the
integrable deformations of the Kepler problem are discussed in detail.


title: Integrable boundary conditions for nonlinear lattices

abstract:
  Integrable boundary conditions in 1+1 and 2+1 dimensions are discussed from
the higher symmetries point of view. Boundary conditions consistent with the
discrete Landau-Lifshitz model and infinite 2D Toda lattice are represented.


title: The Coupled Modified Korteweg-de Vries Equations

abstract:
  Generalization of the modified KdV equation to a multi-component system, that
is expressed by $(\partial u_i)/(\partial t) + 6 (\sum_{j,k=0}^{M-1} C_{jk} u_j
u_k) (\partial u_i)/(\partial x) + (\partial^3 u_{i})/(\partial x^3) = 0, i=0,
1, ..., M-1 $, is studied. We apply a new extended version of the inverse
scattering method to this system. It is shown that this system has an infinite
number of conservation laws and multi-soliton solutions. Further, the initial
value problem of the model is solved.


title: On Calogero wave functions

abstract:
  Two properties of Calogero wave functions for rational Calogero models are
studied: (i) the representation of the wave functions in terms of the
exponential of Lassalle operators, (ii) the $sL(2,\rr)$ structure of the
Calogero--Moser wave functions.


title: On the Calogero model with negative harmonic term

abstract:
  The Calogero model with negative harmonic term is shown to be equivalent to
the set of negative harmonic oscillators. Two time-independent canonical
transformations relating both models are constructed: one based on the recent
results concerning quantum Calogero model and one obtained from dynamical
$sL(2,\rr)$ algebra. The two-particle case is discussed in some detail.


title: Random matrices, Virasoro algebras, and noncommutative KP

abstract:
  What is the connection of random matrices with integrable systems? Is this
connection really useful? The answer to these questions leads to a new and
unifying approach to the theory of random matrices. Introducing an appropriate
time t-dependence in the probability distribution of the matrix ensemble, leads
to vertex operator expressions for the n-point correlation functions
(probabilities of n eigenvalues in infinitesimal intervals) and the
corresponding Fredholm determinants (probabilities of no eigenvalue in a Borel
subset E); the latter probability is a ratio of tau-functions for the
KP-equation, whose numerator satisfy partial differential equations, which
decouple into the sum of two parts: a Virasoro-like part depending on time only
and a Vect(S^1)-part depending on the boundary points A_i of E. Upon setting
t=0, and using the KP-hierarchy to eliminate t-derivatives, these PDE's lead to
a hierarchy of non-linear PDE's, purely in terms of the A_i. These PDE's are
nothing else but the KP hierarchy for which the t-partials, viewed as commuting
operators, are replaced by non-commuting operators in the endpoints A_i of the
E under consideration. When the boundary of E consists of one point and for the
known kernels, one recovers the Painleve equations, found in prior work on the
subject.


title: Perturbative methods for the Painlev\'e test

abstract:
  There exist many situations where an ordinary differential equation admits a
movable critical singularity which the test of Kowalevski and Gambier fails to
detect. Some possible reasons are: existence of negative Fuchs indices,
insufficient number of Fuchs indices, multiple family, absence of an algebraic
leading order. Mainly giving examples, we present the methods which answer all
these questions. They are all based on the theorem of perturbations of
Poincar\'e and computerizable.


title: Various truncations in Painlev\'e analysis of PDEs

abstract:
  The ``truncation procedure'' initiated by Weiss et al. is best understood as
a Darboux transformation. If it leads to the Lax pair of the PDE under study,
the B\"acklund transformation follows by an elimination, thus proving the
integrability. We present the state of the art of this powerful technique. The
easy situations were all handled by the WTC one-family truncation and its
homographically invariant version. An updated version of this method has been
recently developed, which is now able to handle the Kaup-Kupershmidt and
Tzitz\'eica equations. It incorporates a new feature, namely the distinction
between two entire functions usually mingled, which are shown to be linked by
formulae established by Gambier for his classification.


title: The symplectic structure of rational Lax pair systems

abstract:
  We consider dynamical systems associated to Lax pairs depending rationnally
on a spectral parameter. We show that we can express the symplectic form in
terms of algebro--geometric data provided that the symplectic structure on L is
of Kirillov type. In particular, in this case the dynamical system is
integrable.


title: Coverings and integrability of the Gauss-Mainardi-Codazzi equations

abstract:
  Using covering theory approach (zero-curvature representations with the gauge
group SL2), we insert the spectral parameter into the Gauss-Mainardi-Codazzi
equations in Tchebycheff and geodesic coordinates. For each choice, four
integrable systems are obtained.


title:
On Discrete Painleve Equations Associated with the Lattice KdV Systems
  and the Painleve VI Equation

abstract:
  A new integrable nonautonomous nonlinear ordinary difference equation is
presented which can be considered to be a discrete analogue of the Painleve V
equation. Its derivation is based on the similarity reduction on the
two-dimensional lattice of integrable partial difference equations of KdV type.
The new equation which is referred to as GDP (generalised discrete Painleve
equation) contains various ``discrete Painleve equations'' as subcases for
special values/limits of the parameters, some of which were already given
before in the literature. The general solution of the GDP can be expressed in
terms of Painleve VI (PVI) transcendents. In fact, continuous PVI emerges as
the equation obeyed by the solutions of the discrete equation in terms of the
lattice parameters rather than the lattice variables that label the lattice
sites. We show that the bilinear form of PVI is embedded naturally in the
lattice systems leading to the GDP. Further results include the establishment
of Baecklund and Schlesinger transformations for the GDP, the corresponding
isomonodromic deformation problem, and the self-duality of its bilinear scheme.


title: Optical Fiber Communications:Group of the Nonlinear Transformations

abstract:
  A new method for finding solutions of the nonlinear Shr\"{o}dinger equation
is proposed. Comutative multiplicative group of the nonlinear transformations,
which operate on stationary localized solutions, enables a consideration of
fractal subspaces in the solution space, stability and deterministic chaos. An
increase of the transmission rate at the optical fiber communications can be
based on new forms of localized stationary solutions, without significant
change of input power. The estimated transmission rate is 50 Gbit/s, for
certain available soliton transmission systems.


title: Solutions to the Optical Cascading Equations

abstract:
  Group theoretical methods are used to study the equations describing
\chi^{(2)}:\chi^{(2)} cascading. The equations are shown not to be integrable
by inverse scattering techniques. On the other hand, these equations do share
some of the nice properties of soliton equations. Large families of explicit
analytical solutions are obtained in terms of elliptic functions. In special
cases, these periodic solutions reduce to localized ones, i.e., solitary waves.
All previously known explicit solutions are recovered, and many additional ones
are obtained


title: Inhomogeneous Burgers Equation and the Feynman-Kac Path Integral

abstract:
  By linearizing the inhomogeneous Burgers equation through the Hopf-Cole
transformation, we formulate the solution of the initial value problem of the
corresponding linear heat type equation using the Feynman-Kac path integral
formalism. For illustration, we present the exact solution for the forcing term
of the form: $F(x,t)=\omega ^2x+f(t).$ We also present the initial value
problem solution for the case with a constant forcing term to compare with the
known result.


title: A Realization of Discrete Geometry by String Model

abstract:
  A realization of discrete conjugate net is presented by using correlation
functions of strings in a gauge covariant form.


title:
Finite-dimensional analogs of string s <-> t duality and pentagon
  equation

abstract:
  We put forward one of the forms of functional pentagon equation (FPE), known
from the theory of integrable models, as an algebraic explanation to the
phenomenon known in physics as s<->t duality. We present two simple geometrical
examples of FPE solutions, one of them yielding in a particular case the
well-known Veneziano expression for 4-particle amplitude. Finally, we interpret
our solutions of FPE in terms of relations in Lie groups.


title:
Hamiltonian Structures of Generalized Manin-Radul Super KdV and
  Constrained Super KP Hierarchies

abstract:
  A study of Hamiltonian structures associated with supersymmetric Lax
operators is presented. Following a constructive approach, the Hamiltonian
structures of Inami-Kanno super KdV hierarchy and constrained modified super KP
hierarchy are investigated from the reduced supersymmetric Gelfand-Dickey
brackets. By applying a gauge transformation on the Hamiltonian structures
associated with these two nonstandard super Lax hierarchies, we obtain the
Hamiltonian structures of generalized Manin-Radul super KdV and constrained
super KP hierarchies. We also work out a few examples and compare them with the
known results.


title: The periodic Lax operators $\cL$ of the equations of Benney type II

abstract: This text has been withdrawn by the author.


title: Symmetries of Discrete Dynamical Systems Involving Two Species

abstract:
  The Lie point symmetries of a coupled system of two nonlinear
differential-difference equations are investigated. It is shown that in special
cases the symmetry group can be infinite dimensional, in other cases up to 10
dimensional. The equations can describe the interaction of two long molecular
chains, each involving one type of atoms.


title: Functional representation of the Ablowitz-Ladik hierarchy. II

abstract:
  In this paper I continue studies of the functional representation of the
Ablowitz-Ladik hierarchy (ALH). Using formal series solutions of the
zero-curvature condition I rederive the functional equations for the
tau-functions of the ALH and obtain some new equations which provide more
straightforward description of the ALH and which were absent in the previous
paper. These results are used to establish relations between the ALH and the
discrete-time nonlinear Schrodinger equations, to deduce the superposition
formulae (Fay's identities) for the tau-functions of the hierarchy and to
obtain some new results related to the Lax representation of the ALH and its
conservation laws. Using the previously found connections between the ALH and
other integrable systems I derive functional equations which are equivalent to
the AKNS, derivative nonlinear Schrodinger and Davey-Stewartson hierarchies.


title: Spontaneous magnetization of the XXZ Heisenberg spin-1/2 chain

abstract:
  Determinant representations of form factors are used to represent the
spontaneous magnetization of the Heisenberg XXZ chain (Delta >1) on the finite
lattice as the ratio of two determinants. In the thermodynamic limit (the
lattice of infinite length), the Baxter formula is reproduced in the framework
of Algebraic Bethe Ansatz. It is shown that the finite size corrections to the
Baxter formula are exponentially small.


title: Extension of the bilinear formalism to supersymmetric KdV-type equations

abstract:
  Extending the gauge-invariance principle for \tau functions of the standard
bilinear formalism to the supersymmetric case, we define N=1 supersymmetric
Hirota operators. Using them, we bilinearize SUSY KdV-type equations (KdV,
Sawada-Kotera, Hirota-Satsuma). The solutions for multiple collisions of
super-solitons and extension to SUSY sine-Gordon are also discussed.


title: Hamiltonian structure of real Monge-Amp\`ere equations

abstract:
  The real homogeneous Monge-Amp\`{e}re equation in one space and one time
dimensions admits infinitely many Hamiltonian operators and is completely
integrable by Magri's theorem. This remarkable property holds in arbitrary
number of dimensions as well, so that among all integrable nonlinear evolution
equations the real homogeneous Monge-Amp\`{e}re equation is distinguished as
one that retains its character as an integrable system in multi-dimensions.
This property can be traced back to the appearance of arbitrary functions in
the Lagrangian formulation of the real homogeneous Monge-Amp\`ere equation
which is degenerate and requires use of Dirac's theory of constraints for its
Hamiltonian formulation. As in the case of most completely integrable systems
the constraints are second class and Dirac brackets directly yield the
Hamiltonian operators. The simplest Hamiltonian operator results in the
Kac-Moody algebra of vector fields and functions on the unit circle.


title: Discrete and Continuous Linearizable Equations

abstract:
  We study the projective systems in both continuous and discrete settings.
These systems are linearizable by construction and thus, obviously, integrable.
We show that in the continuous case it is possible to eliminate all variables
but one and reduce the system to a single differential equation. This equation
is of the form of those singled-out by Painlev\'e in his quest for integrable
forms. In the discrete case, we extend previous results of ours showing that,
again by elimination of variables, the general projective system can be written
as a mapping for a single variable. We show that this mapping is a member of
the family of multilinear systems (which is not integrable in general). The
continuous limit of multilinear mappings is also discussed.


title: The Cole-Hopf and Miura transformations revisited

abstract:
  An elementary yet remarkable similarity between the Cole-Hopf transformation
relating the Burgers and heat equation and Miura's transformation connecting
the KdV and mKdV equations is studied in detail.


title: The classical Boussinesq hierarchy revisited

abstract:
  We develop a systematic approach to the classical Boussinesq (cBsq) hierarchy
based on an elementary polynomial recursion formalism. Moreover, the gauge
equivalence between the cBsq and AKNS hierarchies is studied in detail and used
to provide an effortless derivation of algebro-geometric solutions and their
theta function representations of the cBsq hierarchy.


title:
On some soliton equations in 2+1 dimensions and their 1+1 and/or 2+0
  dimensional integrable reductions

abstract:
  Some soliton equation in 2+1 dimensions and their 1+1 and/or dimensional
integrable reductions are considered.


title:
Integral equations for the correlation functions of the quantum
  one-dimensional Bose gas

abstract:
  The large time and long distance behavior of the temperature correlation
functions of the quantum one-dimensional Bose gas is considered. We obtain
integral equations, which solutions describe the asymptotics. These equations
are closely related to the thermodynamic Bethe Ansatz equations. In the low
temperature limit the solutions of these equations are given in terms of
observables of the model.


title: The relation between the Toda hierarchy and the KdV hierarchy

abstract:
  Under three relations connecting the field variables of Toda flows and that
of KdV flows, we present three new sequences of combination of the equations in
the Toda hierarchy which have the KdV hierarchy as a continuous limit. The
relation between the Poisson structures of the KdV hierarchy and the Toda
hierarchy in continuous limit is also studied.


title: The Lax pairs for the Holt system

abstract:
  By using non-canonical transformation between the Holt system and the
Henon-Heiles system the Lax pairs for all the integrable cases of the Holt
system are constructed from the known Lax representations for the Henon-Heiles
system.


title:
Two- and Many-Dimensional Quasi-Exactly Solvable Models With An
  Inhomogeneous Magnetic Field

abstract:
  Let group generators having finite-dimensional representation be realized as
Hermitian linear differential operators without nhomogeneous terms as takes
place, for example, for the SO(n) group. Then orresponding group Hamiltonians
containing terms linear in generators (along with quadratic ones) give rise to
quasi-exactly solvable models with a magnetic field in a curved space. In
particular, in the two-dimensional case such models are generated by quantum
tops. In the three-dimensional one for the SO(4) Hamiltonian with an isotropic
quadratic part the manifold within which a quantum particle moves has the
geometry of the Einstein universe.

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


title:
The Camassa-Holm Equation: Conserved Quantities and the Initial Value
  Problem

abstract:
  Using a Miura-Gardner-Kruskal type construction, we show that the
Camassa-Holm equation has an infinite number of local conserved quantities. We
explore the implications of these conserved quantities for global
well-posedness.


title: Unified algebraic Bethe ansatz for two-dimensional lattice models

abstract:
  We develop a unified formulation of the quantum inverse scattering method for
lattice vertex models associated to the non-exceptional $A^{(2)}_{2r}$,
$A^{(2)}_{2r-1}$, $B^{(1)}_r$, $C^{(1)}_r$, $D^{(1)}_{r+1}$ and $D^{(2)}_{r+1}$
Lie algebras. We recast the Yang-Baxter algebra in terms of novel commutation
relations between creation, annihilation and diagonal fields. The solution of
the $D^{(2)}_{r+1}$ model is based on an interesting sixteen-vertex model which
is solvable without recourse to a Bethe ansatz.


title: Universality of the distribution functions of random matrix theory

abstract:
  This paper first surveys the connection of integrable systems of the Painleve
type to various distribution functions appearing in Wigner-Dyson random matrix
theory. A short discussion is then given of the appearance of these same
distributions in other areas of mathematics.


title: Airy Kernel and Painleve II

abstract:
  We prove that the distribution function of the largest eigenvalue in the
Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in
terms of Painlev\'e II. Our goal is to concentrate on this important example of
the connection between random matrix theory and integrable systems, and in so
doing to introduce the newcomer to the subject as a whole. We also give
sketches of the results for the limiting distribution of the largest eigenvalue
in the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble
(GSE). This work we did some years ago in a more general setting. These notes,
therefore, are not meant for experts in the field.


title: Coupled KdV equations of Hirota-Satsuma type

abstract:
  It is shown that the system of two coupled Korteweg-de Vries equations passes
the Painlev\'e test for integrability in nine distinct cases of its
coefficients. The integrability of eight cases is verified by direct
construction of Lax pairs, whereas for one case it remains unknown.

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


title: The hunting for the discrete Painlev\'e VI is over

abstract:
  We present the discrete, q-, form of the Painlev\'e VI equation written as a
three-point mapping and analyse the structure of its singularities. This
discrete equation goes over to P_{VI} at the continuous limit and degenerates
towards the discrete q-P_{V} through coalescence. It possesses special
solutions in terms of the q-hypergeometric function. It can bilinearised and,
under the appropriate assumptions, ultradiscretised. A new discrete form for
P_{V} is also obtained which is of difference type, in contrast with the
`standard' form of the discrete P_{V}. Finally, we present the `asymmetric'
form of q-P_{VI}$ as a system of two first-order mappings involving seven
arbitrary parameters.


title: Acoustic Scattering and the Extended Korteweg deVries hierarchy

abstract:
  The acoustic scattering operator on the real line is mapped to a
Schr\"odinger operator under the Liouville transformation. The potentials in
the image are characterized precisely in terms of their scattering data, and
the inverse transformation is obtained as a simple, linear quadrature. An
existence theorem for the associated Harry Dym flows is proved, using the
scattering method. The scattering problem associated with the Camassa-Holm
flows on the real line is solved explicitly for a special case, which is used
to reduce a general class of such problems to scattering problems on finite
intervals.


title: Darboux-type transformations and hyperelliptic curves

abstract:
  We systematically study Darboux-type transformations for the KdV and AKNS
hierarchies and provide a complete account of their effects on hyperelliptic
curves associated with algebro-geometric solutions of these hierarchies.


title: Exact solutions of the associated Camassa-Holm equation

abstract:
  Recently the associated Camassa-Holm (ACH) equation, related to the
Fuchssteiner-Fokas-Camassa-Holm equation by a hodograph transformation, was
introduced by Schiff, who derived B\"{a}cklund transformations by a loop group
technique and used these to obtain some simple soliton and rational solutions.
We show how the ACH equation is related to Schr\"{o}dinger operators and the
KdV hierarchy, and use this connection to obtain exact solutions (rational and
N-soliton solutions). More generally, we show that solutions of ACH on a
constant background can be obtained directly from the tau-functions of known
solutions of the KdV hierarchy on a zero background. We also present exact
solutions given by a particular case of the third Painlev\'{e} transcendent.


title:
Multi-soliton Solution of the Integrable Coupled Nonlinear Schrodinger
  Equation of Manakov Type

abstract:
  The general multi-soliton solution of the integrable coupled nonlinear
Schrodinger equation (NLS) of Manakov type is investigated by using
Zakharov-Shabat (ZS) scheme. We get the bright and dark multi-soliton solution
using inverse scattering method of ZS scheme. Elastic and inelastic collision
of N-solitons solution of the equation are also discussed.

categories: solv-int hep-th math-ph math.DS math.MP nlin.PS nlin.SI patt-sol


title: A note on the third family of N=2 supersymmetric KdV hierarchies

abstract:
  We propose a hamiltonian formulation of the $N=2$ supersymmetric KP type
hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic
hamiltonian structure which allows for several reductions of the KP type
hierarchy. In particular, the third family of $N=2$ KdV hierarchies is
recovered. We also give an easy construction of Wronskian solutions of the KP
and KdV type equations.


title:
Spontaneous polarization of the Kondo problem associated with the
  higher-spin six-vertex model

abstract:
  We study the multi-channel Kondo model associated with an integrable
higher-spin analogue of the anti-ferroelectric six-vertex model, which is
constructed by inserting spin 1/2 to spin 1 lines: $... C^3 \otimes C^3 \otimes
C^2 \otimes C^3 \otimes C^3 ... $. We formulate the problem in terms of
representation theory of quantum affine algebra $U_q(hat{sl}_2)$. We derive an
exact formula for the spontaneous staggered polarization for our model, which
corresponds to Baxter`s formula for the six-vertex model.


title:
On time-dependent symmetries and formal symmetries of evolution
  equations

abstract:
  We present the explicit formulae, describing the structure of symmetries and
formal symmetries of any scalar (1+1)-dimensional evolution equation. Using
these results, the formulae for the leading terms of commutators of two
symmetries and two formal symmetries are found. The generalization of these
results to the case of system of evolution equations is also discussed.


title:
Miura transformations for Toda--type integrable systems, with
  applications to the problem of integrable discretizations

abstract:
  We study lattice Miura transformations for the Toda and Volterra lattices,
relativistic Toda and Volterra lattices, and their modifications. In
particular, we give three successive modifications for the Toda lattice, two
for the Volterra lattice and for the relativistic Toda lattice, and one for the
relativistic Volterra lattice. We discuss Poisson properties of the Miura
transformations, their permutability properties, and their role as localizing
changes of variables in the theory of integrable discretizations.


title:
The solution of the Painleve equations as special functions of
  catastrophes, defined by a rejection in these equations of terms with
  derivative

abstract:
  The relation between the Painleve equations and the algebraic equations with
the catastrophe theory point of view are considered. The asymptotic solutions
with respect to the small parameter of the Painleve equations different types
are discussed. The qualitative analysis of the relation between algebraic and
fast oscillating solutions is done for Painleve-2 as an example.


title: Multiscale Analysis of Discrete Nonlinear Evolution Equations

abstract:
  The method of multiscale analysis is constructed for dicrete systems of
evolution equations for which the problem is that of the far behavior of an
input boundary datum. Discrete slow space variables are introduced in a general
setting and the related finite differences are constructed. The method is
applied to a series of representative examples: the Toda lattice, the nonlinear
Klein-Gordon chain, the Takeno system and a discrete version of the
Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a
discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave
resonant interaction system and a discrete modified Volterra model.


title:
On the Hamiltonian and Lagrangian structures of time-dependent
  reductions of evolutionary PDEs

abstract:
  In this paper we study the reductions of evolutionary PDEs on the manifold of
the stationary points of time-dependent symmetries. In particular we describe
how the finite dimensional Hamiltonian structure of the reduced system is
obtained from the Hamiltonian structure of the initial PDE and we construct the
time-dependent Hamiltonian function. We also present a very general Lagrangian
formulation of the procedure of reduction. As an application we consider the
case of the Painleve' equations PI, PII, PIII, PVI and also certain higher
order systems appeared in the theory of Frobenius manifolds.


title:
Asymptotic approach for the rigid condition of appearance of the
  oscillations in the solution of the Painleve-2 equation

abstract:
  The asymptotic solution for the Painleve-2 equation with small parameter is
considered. The solution has algebraic behavior before point $t_*$ and fast
oscillating behavior after the point $t_*$. In the transition layer the
behavior of the asymptotic solution is more complicated. The leading term of
the asymptotics satisfies the Painleve-1 equation and some elliptic equation
with constant coefficients, where the solution of the Painleve-1 equation has
poles. The uniform smooth asymptotics are constructed in the interval,
containing the critical point $t_*$.


title: Modified KP and Discrete KP

abstract:
  The discrete KP, or 1-Toda lattice hierarchy is the same as a properly
defined modified KP hierarchy.


title: A critical Ising model on the Labyrinth

abstract:
  A zero-field Ising model with ferromagnetic coupling constants on the
so-called Labyrinth tiling is investigated. Alternatively, this can be regarded
as an Ising model on a square lattice with a quasi-periodic distribution of up
to eight different coupling constants. The duality transformation on this
tiling is considered and the self-dual couplings are determined. Furthermore,
we analyze the subclass of exactly solvable models in detail parametrizing the
coupling constants in terms of four rapidity parameters. For those, the
self-dual couplings correspond to the critical points which, as expected,
belong to the Onsager universality class.


title:
Lax Pair Formulation and Multi-soliton Solution of the Integrable Vector
  Nonlinear Schrodinger Equation

abstract:
  The integrable vector nonlinear Schrodinger (vector NLS) equation is
investigated by using Zakharov-Shabat (ZS) scheme. We get a Lax pair
formulation of the vector NLS model. Multi-soliton solution of the equation is
also derived by using inverse scattering method of ZS scheme. We also find that
there is an elastic and inelastic collision of the bright and dark
multi-solitons of the system.

categories: solv-int adap-org hep-th math-ph math.DS math.MP nlin.AO nlin.PS nlin.SI patt-sol


title:
Orthonormal Polynomials on the Unit Circle and Spatially Discrete
  Painlev\'e II Equation

abstract:
  We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+
>...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z))
dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient
$\kappa_n$ is found to satisfy a difference-differential (spatially discrete)
equation which is further proved to approach a third order differential
equation by double scaling. The third order differential equation is equivalent
to the Painlev\'e II equation. The leading coefficient and second leading
coefficient of $\phi_n(z)$ can be expressed asymptotically in terms of the
Painlev\'e II function.


title: Modular Invariants and Generalized Halphen Systems

abstract:
  Generalized Halphen systems are solved in terms of functions that uniformize
genus zero Riemann surfaces, with automorphism groups that are commensurable
with the modular group. Rational maps relating these functions imply subgroup
relations between their automorphism groups and symmetrization relations
between the associated differential systems.

categories: solv-int hep-th math-ph math.DS math.GR math.MP math.NT nlin.SI


title: Picard-Fuchs Equations, Hauptmoduls and Integrable Systems

abstract:
  The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing
functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs
equations for families of elliptic curves and associated surfaces. The
inhomogeneous Picard-Fuchs equations associated to elliptic integrals with
varying endpoints are derived and used to determine solutions of equations that
are algebraically related to a class of Painlev\'e VI equations.

categories: solv-int hep-th math-ph math.AG math.GR math.MP math.NT nlin.SI


title:
Soliton equations in 2+1 dimensions: reductions, bilinearizations and
  simplest solutions

abstract: Soliton equations in 2+1 and their 1+1 = 2+0 reductions are considered.


title: Baxter's Q-operator for the homogeneous XXX spin chain

abstract:
  Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator
for the homogeneous XXX model as integral operator in standard representation
of SL(2). The connection between Q-operator and local Hamiltonians is
discussed. It is shown that operator of Lipatov's duality symmetry arises
naturally as leading term of the asymptotic expansion of Q-operator for large
values of spectral parameter.


title: Discrete analogues of the Liouville equation

abstract:
  The notion of Laplace invariants is transferred to the lattices and discrete
equations which are difference analogs of hyperbolic PDE's with two independent
variables. The sequence of Laplace invariants satisfy the discrete analog of
twodimensional Toda lattice. The terminating of this sequence by zeroes is
proved to be the necessary condition for existence of the integrals of the
equation under consideration. The formulae are presented for the higher
symmetries of the equations possessing integrals. The general theory is
illustrated by examples of difference analogs of Liouville equation.


title: Vertex Operator Solutions of 2d Dimensionally Reduced Gravity

abstract:
  We apply algebraic and vertex operator techniques to solve two dimensional
reduced vacuum Einstein's equations. This leads to explicit expressions for the
coefficients of metrics solutions of the vacuum equations as ratios of
determinants. No quadratures are left. These formulas rely on the
identification of dual pairs of vertex operators corresponding to dual metrics
related by the Kramer-Neugebauer symmetry.


title:
Functional relations and nested Bethe ansatz for sl(3) chiral Potts
  model at q^2=-1

abstract:
  We obtain the functional relations for the eigenvalues of the transfer matrix
of the sl(3) chiral Potts model for q^2=-1. For the homogeneous model in both
directions a solution of these functional relations can be written in terms of
roots of Bethe ansatz-like equations. In addition, a direct nested Bethe ansatz
has also been developed for this case.


title: Nambu--Poisson reformulation of the finite dimensional dynamical systems

abstract:
  In this paper we introduce a system of nonlinear ordinary differential
equations which in a particular case reduces to Volterra's system. We found in
two simplest cases the complete sets of the integrals of motion using
Nambu--Poisson reformulation of the Hamiltonian dynamics. In these cases we
have solved the systems by quadratures.


title: New Integrable Models from Fusion

abstract:
  Integrable multistate or multiflavor/color models were recently introduced.
They are generalizations of models corresponding to the defining
representations of the U_q(sl(m)) quantum algebras. Here I show that a similar
generalization is possible for all higher dimensional representations. The
R-matrices and the Hamiltonians of these models are constructed by fusion. The
sl(2) case is treated in some detail and the spin-0 and spin-1 matrices are
obtained in explicit forms. This provides in particular a generalization of the
Fateev-Zamolodchikov Hamiltonian.

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


title: Finite genus solutions for the Ablowitz-Ladik hierarchy

abstract:
  The question of constructing the finite genus quasiperiodic solutions for the
Ablowitz-Ladik hierarchy (ALH) is considered by establishing relations between
the ALH and the Fay's identity for the theta-functions. It is shown that using
a limiting procedure one can derive from the latter an infinite number of
differential identities which can be arranged as an infinite set of
differential-difference equations coinciding with the equations of the ALH, and
that the original Fay's identity can be rewritten in a form similar to the
functional equations representing the ALH which have been derived in the
previous works of the author. This provides an algorithm for obtaining some
class of quasiperiodic solutions for the ALH, which can be viewed as an
alternative to the inverse scattering transform or the algebro-geometrical
approach.


title: The Pfaff lattice and skew-orthogonal polynomials

abstract:
  Consider a semi-infinite skew-symmetric moment matrix, $m_{\iy}$ evolving
according to the vector fields $\pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} ,$ where
$\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\iy}:= Q^{-1}
J Q^{\top -1} $ leads to the so-called Pfaff Lattice, which is integrable, by
virtue of the AKS theorem, for a splitting involving the affine symplectic
algebra. The tau-functions for the system are shown to be pfaffians and the
wave vectors skew-orthogonal polynomials; we give their explicit form in terms
of moments. This system plays an important role in symmetric and symplectic
matrix models and in the theory of random matrices (beta=1 or 4).


title: Integrable mixed vertex models from braid-monoid algebra

abstract:
  We use the braid-monoid algebra to construct integrable mixed vertex models.
The transfer matrix of a mixed SU(N) model is diagonalized by nested Bethe
ansatz approach.


title: Integrable vs Nonintegrable Geodesic Soliton Behavior

abstract:
  We study confined solutions of certain evolutionary partial differential
equations (pde) in 1+1 space-time. The pde we study are Lie-Poisson Hamiltonian
systems for quadratic Hamiltonians defined on the dual of the Lie algebra of
vector fields on the real line. These systems are also Euler-Poincare equations
for geodesic motion on the diffeomorphism group in the sense of the Arnold
program for ideal fluids, but where the kinetic energy metric is different from
the L2 norm of the velocity. These pde possess a finite-dimensional invariant
manifold of particle-like (measure-valued) solutions we call ``pulsons.'' We
solve the particle dynamics of the two-pulson interaction analytically as a
canonical Hamiltonian system for geodesic motion with two degrees of freedom
and a conserved momentum. The result of this two-pulson interaction for
rear-end collisions is elastic scattering with a phase shift, as occurs with
solitons. In contrast, head-on antisymmetric collisons of pulsons tend to form
singularities.


title:
Singularity Structure Analysis, Integrability, Solitons and Dromions in
  (2+1)-Dimensional Zakharov Equations

abstract:
  The (2+1)-dimensional integrable Zakharov equations and their reductions are
considered


title: Symmetric random matrices and the Pfaff lattice

abstract:
  Consider a symmetric (finite) matrix ensemble, with a certain probability
distribution. What is the probability that the spectrum belongs to a certain
interval or union of intervals on the real line? In this paper, we show that,
upon introducing an appropriate time parameter, this probability is intimately
related to Pfaffians, which as a vector satisfy the so-called Pfaff lattice.
The latter is a particular reduction of the 2d-Toda lattice. In particular,
they satisfy a KP-like equation, but with a right hand side, depending on
nearest neighbors. They also satisfy Virasoro constraints, which combined with
the KP-like equation lead to inductive equations for the probabilities.


title: Bethe ansatz for the three-layer Zamolodchikov model

abstract:
  This paper is a continuation of our previous work (solv-int/9903001). We
obtain two more functional relations for the eigenvalues of the transfer
matrices for the $sl(3)$ chiral Potts model at $q^2=-1$. This model, up to a
modification of boundary conditions, is equivalent to the three-layer
three-dimensional Zamolodchikov model. From these relations we derive the Bethe
ansatz equations.


title: Multipeakons and a theorem of Stieltjes

abstract:
  A closed form of the multi-peakon solutions of the Camassa-Holm equation is
found using a theorem of Stieltjes on continued fractions. An explicit formula
is obtained for the scattering shifts.


title: On integrability test for ultradiscrete equations

abstract:
  We consider an integrability test for ultradiscrete equations based on the
singularity confinement analysis for discrete equations. We show how
singularity pattern of the test is transformed into that of ultradiscrete
equation. The ultradiscrete solution pattern can be interpreted as a perturbed
solution. We can also check an integrability of a given equation by a
perturbation growth of a solution, namely Lyapunov exponent. Therefore,
singularity confinement test and Lyapunov exponent are related each other in
ultradiscrete equations and we propose an integrability test from this point of
view.


title:
Integrable semi-discretization of the coupled nonlinear Schr\"{o}dinger
  equations

abstract:
  A system of semi-discrete coupled nonlinear Schr\"{o}dinger equations is
studied. To show the complete integrability of the model with multiple
components, we extend the discrete version of the inverse scattering method for
the single-component discrete nonlinear Schr\"{o}dinger equation proposed by
Ablowitz and Ladik. By means of the extension, the initial-value problem of the
model is solved. Further, the integrals of motion and the soliton solutions are
constructed within the framework of the extension of the inverse scattering
method.


title:
A generalization of determinant formulas for the solutions of Painlev\'e
  II and XXXIV equations

abstract:
  A generalization of determinant formulas for the classical solutions of
Painlev\'e XXXIV and Painlev\'e II equations are constructed using the
technique of Darboux transformation and Hirota's bilinear formalism. It is
shown that the solutions admit determinant formulas even for the transcendental
case.


title: On the Umemura Polynomials for the Painlev\'e III equation

abstract:
  A determinant expression for the rational solutions of the Painlev\'e III
(P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given.
Degeneration of this determinant expression to that for the rational solutions
of P$_{\rm II}$ is discussed by applying the coalescence procedure.


title: Canonicity of Baecklund transformation: r-matrix approach. I

abstract:
  For the Hamiltonian integrable systems governed by SL(2)-invariant r-matrix
(such as Heisenberg magnet, Toda lattice, nonlinear Schroedinger equation) a
general procedure for constructing Baecklund transformation is proposed. The
corresponding BT is shown to preserve the Poisson bracket. The proof is given
by a direct calculation using the r-matrix expression for the Poisson bracket.


title: Canonicity of Baecklund transformation: r-matrix approach. II

abstract:
  This is the second part of the paper devoted to the general proof of
canonicity of Baecklund transformation (BT) for a Hamiltonian integrable system
governed by SL(2)-invariant r-matrix. Introducing an extended phase space from
which the original one is obtained by imposing a 1st kind constraint, we are
able to prove the canonicity of BT in a new way. The new proof allows to
explain naturally the fact why the gauge transformation matrix M associated to
the BT has the same structure as the Lax operator L. The technique is
illustrated on the example of the DST chain.


title: Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

abstract:
  The real, nonsingular elliptic solutions of the Korteweg-deVries equation are
studied through the time dynamics of their poles in the complex plane. The
dynamics of these poles is governed by a dynamical system with a constraint.
This constraint is shown to be solvable for any finite number of poles located
in the fundamental domain of the elliptic function, often in many different
ways. Special consideration is given to those elliptic solutions that have a
real nonsingular soliton limit.


title: Discrete equations and the singular manifold method

abstract:
  The Painleve expansion for the second Painleve equation (PII) and fourth
Painleve equation (PIV) have two branches. The singular manifold method
therefore requires two singular manifolds. The double singular manifold method
is used to derive Miura transformations from PII and PIV to modified Painleve
type equations for which auto-Backlund transformations are obtained. These
auto-Backlund transformations can be used to obtain discrete equations.


title: Backlund transformations for many-body systems related to KdV

abstract:
  We present Backlund transformations (BTs) with parameter for certain
classical integrable n-body systems, namely the many-body generalised
Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the
fact that all these systems may be obtained as particular reductions
(stationary or restricted flows) of the KdV hierarchy; alternatively they may
be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide
exact time-discretizations of the original (continuous) systems, preserving the
Lax matrix and hence all integrals of motion, and satisfy the spectrality
property with respect to the Backlund parameter.


title:
Recurrent procedure for the determination of the Free Energy
  $\epsilon^{2}$-expansion in the Topological String Theory

abstract:
  We present here the iteration procedure for the determination of free energy
$\epsilon^{2}$-expansion using the theory of KdV - type equations. In our
approach we use the conservation laws for KdV - type equations depending
explicitly on times $t_{1}, t_{2}, ...$ to find the $\epsilon^{2}$-expansion of
$u(x,t_{1},t_{2},...)$ after the infinite number of shifts of $u(x,0,0,...)
\equiv x$ along $t_{1}, t_{2}, ...$ in recurrent form. The formulas for the
free energy expansion are just obtained then as a result of quite simple
integration procedure applied to $u_{n}(x)$.


title: The tetrahedral analog of Veneziano amplitude

abstract:
  In solv-int/9812016 it was shown that the Veneziano amplitude in string
theory comes naturally from one of the simplest solutions of the functional
pentagon equation (FPE). More generally, FPE is intimately connected with the
duality condition for scattering processes. Here I find the amplitude that
comes the same way from a solution of the functional tetrahedron equation, with
the duality replaced by the local Yang - Baxter equation.


title:
Suppression and Enhancement of Soliton Switching During Interaction in
  Periodically Twisted Birefringent Fiber

abstract:
  Soliton interaction in periodically twisted birefringent optical fibers has
been analysed analytically with refernce to soliton switching. For this purpose
we construct the exact general two-soliton solution of the associated coupled
system and investigate its asymptotic behaviour. Using the results of our
analytical approach we point out that the interaction can be used as a switch
to suppress or to enhance soliton switching dynamics, if one injects
multi-soliton as an input pulse in the periodically twisted birefringent fiber.


title: Complex sine-Gordon Equation in Coherent Optical Pulse Propagation

abstract:
  It is shown that the McCall-Hahn theory of self-induced transparency in
coherent optical pulse propagation can be identified with the complex
sine-Gordon theory in the sharp line limit. We reformulate the theory in terms
of the deformed gauged Wess-Zumino-Witten sigma model and address various new
aspects of self-induced transparency.


title: Conservation Laws in Higher-Order Nonlinear Optical Effects

abstract:
  Conservation laws of the nonlinear Schr\"{o}dinger equation are studied in
the presence of higher-order nonlinear optical effects including the
third-order dispersion and the self-steepening. In a context of group theory,
we derive a general expression for infinitely many conserved currents and
charges of the coupled higher-order nonlinear Schr\"{o}dinger equation. The
first few currents and charges are also presented explicitly. Due to the
higher-order effects, conservation laws of the nonlinear Schr\"{o}dinger
equation are violated in general. The differences between the types of the
conserved currents for the Hirota and the Sasa-Satsuma equations imply that the
higher-order terms determine the inherent types of conserved quantities for
each integrable cases of the higher-order nonlinear Schr\"{o}dinger equation.


title:
Painlev\'{e} analysis of the coupled nonlinear Schr\"{o}dinger equation
  for polarized optical waves in an isotropic medium

abstract:
  Using the Painlev\'{e} analysis, we investigate the integrability properties
of a system of two coupled nonlinear Schr\"{o}dinger equations that describe
the propagation of orthogonally polarized optical waves in an isotropic medium.
Besides the well-known integrable vector nonlinear Schr\"{o}dinger equation, we
show that there exist a new set of equations passing the Painlev\'{e} test
where the self and cross phase modulational terms are of different magnitude.
We introduce the Hirota bilinearization and the B\"{a}cklund transformation to
obtain soliton solutions and prove integrability by making a change of
variables. The conditions on the third-order susceptibility tensor $\chi^{(3)}
$ imposed by these new integrable equations are explained.


title:
Kinetic and Transport Equations for Localized Excitations in Sine-Gordon
  Model

abstract:
  We analyze the kinetic behavior of localized excitations - solitons,
breathers and phonons - in Sine-Gordon model. Collision integrals for all type
of localized excitation collision processes are constructed, and the kinetic
equations are derived. We analyze the kinetic behavior of localized excitations
- solitons, breathers and phonons - in Sine-Gordon model. Collision integrals
for all type of localized excitation collision processes are constructed, and
the kinetic equations are derived. We prove that the entropy production in the
system of localized excitations takes place only in the case of inhomogeneous
distribution of these excitations in real and phase spaces. We derive transport
equations for soliton and breather densities, temperatures and mean velocities
i.e. show that collisions of localized excitations lead to creation of
diffusion, thermoconductivity and intrinsic friction processes. The diffusion
coefficients for solitons and breathers, describing the diffusion processes in
real and phase spaces, are calculated. It is shown that diffusion processes in
real space are much faster than the diffusion processes in phase space.


title: On the M-XX equation

abstract: The (2+1)-dimensional integrable M-XX equation is considered.


title:
Differential equations and duality in massless integrable field theories
  at zero temperature

abstract:
  Functional relations play a key role in the study of integrable models. We
argue in this paper that for massless field theories at zero temperature, these
relations can in fact be interpreted as monodromy relations. Combined with a
recently discovered duality, this gives a way to bypass the Bethe ansatz, and
compute directly physical quantities as solutions of a linear differential
equation, or as integrals over a hyperelliptic curve. We illustrate these ideas
in details in the case of the $c=1$ theory, and the associated boundary
sine-Gordon model.


title: Solitons in a 3d integrable model

abstract:
  Equations of motion for a classical 3d discrete model, whose auxialiary
system is a linear system, are investigated. The Lagrangian form of the
equations of motion is derived. The Lagrangian variables are a triplet of "tau
functions". The equations of motion for the Triplet of Tau functions are Three
Trilinear equations. Simple solitons for the trilinear equations are given.
Both the dispersion relation and the phase shift reflect the triplet structure
of equations.


title:
Invariant Modules and the Reduction of Nonlinear Partial Differential
  Equations to Dynamical Systems

abstract:
  We completely characterize all nonlinear partial differential equations
leaving a given finite-dimensional vector space of analytic functions
invariant. Existence of an invariant subspace leads to a re duction of the
associated dynamical partial differential equations to a system of ordinary
differential equations, and provide a nonlinear counterpart to quasi-exactly
solvable quantum Hamiltonians. These results rely on a useful extension of the
classical Wronskian determinant condition for linear independence of functions.
In addition, new approaches to the characterization o f the annihilating
differential operators for spaces of analytic functions are presented.


title:
Nonlinear waves, differential resultant, computer algebra and completely
  integrable dynamical systems

abstract:
  The hierarchy of integrable equations are considered. The dynamical approach
to the theory of nonlinear waves is proposed. The special solutions(nonlinear
waves) of considered equations are derived. We use powerful methods of computer
algebra such differential resultant and others.


title:
Korteweg-de Vries hierarchy and related completely integrable systems:
  I. Algebro-geometrical approach

abstract:
  We consider complementary dynamical systems related to stationary Korteweg-de
Vries hierarchy of equations. A general approach for finding elliptic solutions
is given. The solutions are expressed in terms of Novikov polynomials in
general quais-periodic case. For periodic case these polynomials coincide with
Hermite and Lam\'e polynomials. As byproduct we derive $2\times 2$ matrix Lax
representation for Rosochatius-Wojciechiwski, Rosochatius, second flow of
stationary nonlinear vectro Schr\"{o}dinger equations and complex Neumann
system.


title:
Quasi-Periodic and Periodic Solutions for Systems of Coupled Nonlinear
  SCHR\"Odinger Equations

abstract:
  We consider travelling periodic and quasiperiodic wave solutions of a set of
coupled nonlinear Schr\"odimger equations. In fibre optics these equations can
be used to model single mode fibers with strong birefringence and two-mode
optical fibres. Recently these equations appear as modes, which describe
pulse-pulse interaction in wavelength-division-multiplexed channels of optical
fiber transmission systems. Two phase quasi-periodic solutions for integrable
Manakov system are given in tems of two-dimensional Kleinian functions. The
reduction of quasi-periodic solutions to elliptic functions is dicussed. New
solutions in terms of generalized Hermite polynomilas, which are associated
with two-gap Treibich-Verdier potentials are found.


title:
Soliton solutions, Liouville integrability and gauge equivalence of Sasa
  Satsuma equation

abstract:
  Exact integrability of the Sasa Satsuma eqation (SSE) in the Liouville sense
is established by showing the existence of an infinite set of conservation
laws. The explicit form of the conserved quantities in term of the fields are
obtained by solving the Riccati equation for the associated 3x3 Lax operator.
The soliton solutions in particular, one and two soliton solutions, are
constructed by the Hirota's bilinear method. The one soliton solutions is also
compared with that found through the inverse scattering method. The gauge
equivalence of the SSE with a generalized Landau Lifshitz equation is
established with the explicit construction o


title: Optical solitons in higher order nonlinear Schrodinger equation

abstract:
  We show the complete integrability and the existence of optical solitons of
higher order nonlinear Schrodinger equation by inverse scattering method for a
wide range of values of coefficients. This is achieved first by invoking a
novel connection between the integrability of a nonlinear evolution equation
and the dimensions of a family of matrix Lax pairs. It is shown that Lax pairs
of different dimensions lead to the same evolution equation only with the
coefficients of the terms in different integer ratios. Optical solitons, thus
obtained by inverse scattering method, have been found by solving an n
dimensional eigenvalue problem.


title: Compacton-like Solutions for Modified KdV and other Nonlinear Equations

abstract:
  We present compacton-like solution of the modified KdV equation and compare
its properties with those of the compactons and solitons. We further show that,
the nonlinear Schr{\"o}dinger equation with a source term and other higher
order KdV-like equations also possess compact solutions of the similar form.


title:
Inverse scattering method and vector higher order nonlinear Schrodinger
  equation

abstract:
  A generalised inverse scattering method has been developed for arbitrary n
dimensional Lax equations. Subsequently, the method has been used to obtain N
soliton solutions of a vector higher order nonlinear Schrodinger equation,
proposed by us. It has been shown that under suitable reduction, vector higher
order nonlinear Schrodinger equation reduces to higher order nonlinear
Schrodinger equation. The infinite number of conserved quantities have been
obtained by solving a set of coupled Riccati equation. A gauge equivalence is
shown between the vector higher order nonlinear Schrodinger equation and the
generalized Landau Lifshitz equation and the Lax pair for the latter equation
has also been constructed in terms of the spin field, establishing direct
integrability of the spin system.


title: Integrability Tests for Nonlinear Evolution Equations

abstract: Discusses several integrability tests for nonlinear evolution equations.


title:
Mappings preserving locations of movable poles: a new extension of the
  truncation method to ordinary differential equations

abstract:
  The truncation method is a collective name for techniques that arise from
truncating a Laurent series expansion (with leading term) of generic solutions
of nonlinear partial differential equations (PDEs). Despite its utility in
finding Backlund transformations and other remarkable properties of integrable
PDEs, it has not been generally extended to ordinary differential equations
(ODEs). Here we give a new general method that provides such an extension and
show how to apply it to the classical nonlinear ODEs called the Painleve
equations. Our main new idea is to consider mappings that preserve the
locations of a natural subset of the movable poles admitted by the equation. In
this way we are able to recover all known fundamental Backlund transformations
for the equations considered. We are also able to derive Backlund
transformations onto other ODEs in the Painleve classification.


title:
Multi-Component Matrix KP Hierarchies as Symmetry-Enhanced Scalar KP
  Hierarchies and Their Darboux-B"acklund Solutions

abstract:
  We show that any multi-component matrix KP hierarchy is equivalent to the
standard one-component (scalar) KP hierarchy endowed with a special infinite
set of abelian additional symmetries, generated by squared eigenfunction
potentials. This allows to employ a special version of the familiar
Darboux-B"acklund transformation techniques within the ordinary scalar KP
hierarchy in the Sato formulation for a systematic derivation of explicit
multiple-Wronskian tau-function solutions of all multi-component matrix KP
hierarchies.


title: From Agmon-Kannai expansion to Korteweg-de Vries hierarchy

abstract:
  We present a new method for computation of the Korteweg-de Vries hierarchy
via heat invariants of the 1-dimensional Schrodinger operator. As a result new
explicit formulas for the KdV hierarchy are obtained. Our method is based on an
asymptotic expansion of resolvent kernels of elliptic operators due to S.Agmon
and Y.Kannai.


title:
Bethe ansatz solution of the closed anisotropic supersymmetric U model
  with quantum supersymmetry

abstract:
  The nested algebraic Bethe ansatz is presented for the anisotropic
supersymmetric $U$ model maintaining quantum supersymmetry. The Bethe ansatz
equations of the model are obtained on a one-dimensional closed lattice and an
expression for the energy is given.


title:
Solutions of Non-linear Differential and Difference Equations with
  Superposition Formulas

abstract:
  Matrix Riccati equations and other nonlinear ordinary differential equations
with superposition formulas are, in the case of constant coefficients, shown to
have the same exact solutions as their group theoretical discretizations.
Explicit solutions of certain classes of scalar and matrix Riccati equations
are presented as an illustration of the general results.


title:
New integrable systems of derivative nonlinear Schr\"{o}dinger equations
  with multiple components

abstract:
  The Lax pair for a derivative nonlinear Schr\"{o}dinger equation proposed by
Chen-Lee-Liu is generalized into matrix form. This gives new types of
integrable coupled derivative nonlinear Schr\"{o}dinger equations. By virtue of
a gauge transformation, a new multi-component extension of a derivative
nonlinear Schr\"{o}dinger equation proposed by Kaup-Newell is also obtained.


title: The KP Hierarchy in Miwa Coordinates

abstract:
  A systematic reformulation of the KP hierarchy by using continuous Miwa
variables is presented. Basic quantities and relations are defined and
determinantal expressions for Fay's identities are obtained. It is shown that
in terms of these variables the KP hierarchy gives rise to a Darboux system
describing an infinite-dimensional conjugate net.


title:
Coupled nonlinear Schrodinger equations with cubic-quintic nonlinearity:
  Integrability and soliton interaction in non-Kerr media

abstract:
  We propose an integrable system of coupled nonlinear Schrodinger equations
with cubic-quintic terms describing the effects of quintic nonlinearity on the
ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair,
conserved quantities and exact soliton solutions for the proposed integrable
model are given. Explicit form of two-solitons are used to study soliton
interaction showing many intriguing features including inelastic (shape
changing) scattering. Another novel system of coupled equations with
fifth-degree nonlinearity is derived, which represents vector generalization of
the known chiral-soliton bearing system.


title: The continuum limit of sl(N/K) integrable super spin chains

abstract:
  I discuss in this paper the continuum limit of integrable spin chains based
on the superalgebras sl(N/K). The general conclusion is that, with the full
``supersymmetry'', none of these models is relativistic. When the supersymmetry
is broken by the generator of the sub u(1), Gross Neveu models of various types
are obtained. For instance, in the case of sl(N/K) with a typical fermionic
representation on every site, the continuum limit is the GN model with N colors
and K flavors. In the case of sl(N/1) and atypical representations of spin j, a
close cousin of the GN model with N colors, j flavors and flavor anisotropy is
obtained. The Dynkin parameter associated with the fermionic root, while
providing solutions to the Yang Baxter equation with a continuous parameter,
thus does not give rise to any new physics in the field theory limit.
  These features are generalized to the case where an impurity is embedded in
the system.


title: Magnetization waves in Landau-Lifshitz Model

abstract:
  The solutions of the Landau-Lifshitz equation with finite-gap behavior at
infinity are considered. By means of the inverse scattering method the
large-time asymptotics is obtained.


title: $U_q(\hat{sl}_n)$-analog of the XXZ chain with a boundary

abstract:
  We study $U_q(\hat{sl}_n)$ analog of the XXZ spin chain with a boundary
magnetic field h. We construct explicit bosonic formulas of the vacuum vector
and the dual vacuum vector with a boundary magnetic field. We derive integral
formulas of the correlation functions.


title:
Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili
  equations

abstract:
  We study characteristic Cauchy problems for the Korteweg-deVries (KdV)
equation $u_t=uu_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation
$u_{yy}=\bigl(u_{xxx}+uu_x+u_t\bigr)_x$ with holomorphic initial data
possessing nonnegative Taylor coefficients around the origin. For the KdV
equation with initial value $u(0,x)=u_0(x)$, we show that there is no solution
holomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\mathbb C}^2$ unless
$u_0(x)=a_0+a_1x$. This also furnishes a nonexistence result for a class of
$y$-independent solutions of the KP equation. We extend this to $y$-dependent
cases by considering initial values given at $y=0$, $u(t,x,0)=u_0(x,t)$,
$u_y(t,x,0)=u_1(x,t)$, where the Taylor coefficients of $u_0$ and $u_1$ around
$t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic
solution around the origin in ${\mathbb C}^3$ unless $u_0$ and $u_1$ are
polynomials of degree 2 or lower.


title: On the explicit solutions of the elliptic Calogero system

abstract:
  Let $q_1,q_2,...,q_N$ be the coordinates of $N$ particles on the circle,
interacting with the integrable potential $\sum_{j<k}^N\wp(q_j-q_k)$, where
$\wp$ is the Weierstrass elliptic function. We show that every symmetric
elliptic function in $q_1,q_2,...,q_N$ is a meromorphic function in time. We
give explicit formulae for these functions in terms of genus $N-1$ theta
functions.


title:
A System with a Recursion Operator but One Higher Local Symmetry of the
  Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$

abstract:
  We construct a recursion operator for the system $(u_t,v_t)=(u_4+v^2,1/5
v_4)$, for which only one local symmetry is known and we show that the action
of the recursion operator on $(u_t,v_t)$ is a local function.


title: On pulse broadening for optical solitons

abstract:
  Pulse broadening for optical solitons due to birefringence is investigated.
We present an analytical solution which describes the propagation of solitons
in birefringent optical fibers. The special solutions consist of a combination
of purely solitonic terms propagating along the principal birefringence axes
and soliton-soliton interaction terms. The solitonic part of the solutions
indicates that the decay of initially localized pulses could be due to
different propagation velocities along the birefringence axes. We show that the
disintegration of solitonic pulses in birefringent optical fibers can be caused
by two effects. The first effect is similar as in linear birefringence and is
related to the unequal propagation velocities of the modes along the
birefringence axes. The second effect is related to the nonlinear
soliton-soliton interaction between the modes, which makes the solitonic
pulse-shape blurred.


title: Multipeakons and the Classical Moment Problem

abstract:
  Classical results of Stieltjes are used to obtain explicit formulas for the
peakon-antipeakon solutions of the Camassa-Holm equation. The closed form
solution is expressed in terms of the orthogonal polynomials of the related
classical moment problem. It is shown that collisions occur only in
peakon-antipeakon pairs, and the details of the collisions are analyzed using
results {}from the moment problem. A sharp result on the steepening of the
slope at the time of collision is given. Asymptotic formulas are given, and the
scattering shifts are calculated explicitly


title: Soliton Collisions in the Ion Acoustic Plasma Equations

abstract:
  Numerical experiments involving the interaction of two solitary waves of the
ion acoustic plasma equations are described. An exact 2-soliton solution of the
relevant KdV equation was fitted to the initial data, and good agreement was
maintained throughout the entire interaction. The data demonstrates that the
soliton interactions are virtually elastic


title: Reflection K-Matrices for 19-Vertex Models

abstract:
  We derive and classify all regular solutions of the boundary Yang-Baxter
equation for 19-vertex models known as Zamolodchikov-Fateev or $A_{1}^{(1)}$
model, Izergin-Korepin or $A_{2}^{(2)}$ model, sl(2|1) model and osp(2|1)
model. We find that there is a general solution for $A_{1}^{(1)}$ and sl(2|1)
models. In both models it is a complete K-matrix with three free parameters.
For the $A_{2}^{(2)}$ and osp(2|1) models we find three general solutions,
being two complete reflection K-matrices solutions and one incomplete
reflection K-matrix solution with some null entries. In both models these
solutions have two free parameters. Integrable spin-1 Hamiltonians with general
boundary interactions are also presented. Several reduced solutions from these
general solutions are presented in the appendices.

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


title:
On the bilinear equations for Fredholm determinants appearing in random
  matrices

abstract:
  It is shown how the bilinear differential equations satisfied by Fredholm
determinants of integral operators appearing as spectral distribution functions
for random matrices may be deduced from the associated systems of nonautonomous
Hamiltonian equations satisfied by auxiliary canonical phase space variables
introduced by Tracy and Widom. The essential step is to recast the latter as
isomonodromic deformation equations for families of rational covariant
derivative operators on the Riemann sphere and interpret the Fredholm
determinants as isomonodromic $\tau$-functions.


title: Families of quai-bi-Hamiltonian systems and separability

abstract:
  It is shown how to construct an infinite number of families of
quasi-bi-Hamiltonian (QBH) systems by means of the constrained flows of soliton
equations. Three explicit QBH structures are presented for the first three
families of the constrained flows. The Nijenhuis coordinates defined by the
Nijenhuis tensor for the corresponding families of QBH systems are proved to be
exactly the same as the separated variables introduced by means of the Lax
matrices for the constrained flows.


title:
On the discrete and continuous Miura Chain associated with the Sixth
  Painlev\'e Equation

abstract:
  A Miura chain is a (closed) sequence of differential (or difference)
equations that are related by Miura or B\"acklund transformations. We describe
such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from
\pvi itself, a Schwarzian version as well as a second-order second-degree
ordinary differential equation (ODE). As a byproduct we derive an
auto-B\"acklund transformation, relating two copies of \pvi with different
parameters. We also establish the analogous ordinary difference equations in
the discrete counterpart of the chain. Such difference equations govern
iterations of solutions of \pvi under B\"acklund transformations. Both discrete
and continuous equations constitute a larger system which include partial
difference equations, differential-difference equations and partial
differential equations, all associated with the lattice Korteweg-de Vries
equation subject to similarity constraints.


title: On exact solution of a classical 3D integrable model

abstract:
  We investigate some classical evolution model in the discrete 2+1 space-time.
A map, giving an one-step time evolution, may be derived as the compatibility
condition for some systems of linear equations for a set of auxiliary linear
variables. Dynamical variables for the evolution model are the coefficients of
these systems of linear equations. Determinant of any system of linear
equations is a polynomial of two numerical quasimomenta of the auxiliary linear
variables. For one, this determinant is the generating functions of all
integrals of motion for the evolution, and on the other hand it defines a high
genus algebraic curve. The dependence of the dynamical variables on the
space-time point (exact solution) may be expressed in terms of theta functions
on the jacobian of this curve. This is the main result of our paper.


title: Paraconformal Structures and Integrable Systems

abstract:
  We consider some natural connections which arise between right-flat (p, q)
paraconformal structures and integrable systems. We find that such systems may
be formulated in Lax form, with a "Lax p-tuple" of linear differential
operators, depending a spectral parameter which lives in (q-1)-dimensional
complex projective space. Generally, the differential operators contain partial
derivatives with respect to the spectral parameter.


title:
Reduction of bihamiltonian systems and separation of variables: an
  example from the Boussinesq hierarchy

abstract:
  We discuss the Boussinesq system with $t_5$ stationary, within a general
framework for the analysis of stationary flows of n-Gel'fand-Dickey
hierarchies. We show how a careful use of its bihamiltonian structure can be
used to provide a set of separation coordinates for the corresponding
Hamilton--Jacobi equations.


title: r-matrices for relativistic deformations of integrable systems

abstract:
  We include the relativistic lattice KP hierarchy, introduced by Gibbons and
Kupershmidt, into the $r$-matrix framework. An $r$-matrix account of the
nonrelativistic lattice KP hierarchy is also provided for the reader's
convenience. All relativistic constructions are regular one-parameter
perturbations of the nonrelativistic ones. We derive in a simple way the linear
Hamiltonian structure of the relativistic lattice KP, and find for the first
time its quadratic Hamiltonian structure. Amasingly, the latter turns out to
coincide with its nonrelativistic counterpart (a phenomenon, known previously
only for the simplest case of the relativistic Toda lattice).


title:
p-adic Difference-Difference Lotka-Volterra Equation and Ultra-Discrete
  Limit

abstract:
  In this article, we have studied the difference-difference Lotka-Volterra
equations in p-adic number space and its p-adic valuation version. We pointed
out that the structure of the space given by taking the ultra-discrete limit is
the same as that of the $p$-adic valuation space.


title:
Integrability of the higher-order nonlinear Schroedinger equation
  revisited

abstract:
  Only the known integrable cases of the Kodama-Hasegawa higher-order nonlinear
Schroedinger equation pass the Painleve test. Recent results of Ghosh and Nandy
add no new integrable cases of this equation.

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


title: Factorization dynamics and Coxeter-Toda lattices

abstract:
  It is shown that the factorization relation on simple Lie groups with
standard Poisson Lie structure restricted to Coxeter symplectic leaves gives an
integrable dynamical system. This system can be regarded as a discretization of
the Toda flow. In case of $SL_n$ the integrals of the factorization dynamics
are integrals of the relativistic Toda system. A substantial part of the paper
is devoted to the study of symplectic leaves in simple complex Lie groups, its
Borel subgroups and their doubles.


title: Classical skew orthogonal polynomials and random matrices

abstract:
  Skew orthogonal polynomials arise in the calculation of the $n$-point
distribution function for the eigenvalues of ensembles of random matrices with
orthogonal or symplectic symmetry. In particular, the distribution functions
are completely determined by a certain sum involving the skew orthogonal
polynomials. In the cases that the eigenvalue probability density function
involves a classical weight function, explicit formulas for the skew orthogonal
polynomials are given in terms of related orthogonal polynomials, and the
structure is used to give a closed form expression for the sum. This theory
treates all classical cases on an equal footing, giving formulas applicable at
once to the Hermite, Laguerre and Jacobi cases.


title:
On the equivalence of the discrete nonlinear Schr\"odinger equation and
  the discrete isotropic Heisenberg magnet

abstract:
  The equivalence of the discrete isotropic Heisenberg magnet (IHM) model and
the discrete nonlinear Schr\"odinger equation (NLSE) given by Ablowitz and
Ladik is shown. This is used to derive the equivalence of their discretization
with the one by Izergin and Korepin. Moreover a doubly discrete IHM is
presented that is equivalent to Ablowitz' and Ladiks doubly discrete NLSE.


title: A New Class of Optical Solitons

abstract:
  Existence of a new class of soliton solutions is shown for higher order
nonlinear Schrodinger equation, describing thrid order dispersion, Kerr effect
and stimulated Raman scattering. These new solutions have been obtaiened by
invoking a group of nonlinear transformations acting on localised stable
solutions. Stability of these solutions has been studied for different values
of the arbitrary coefficients, involved in the recursion relation and
consequently, different values of coefficient lead to different transmission
rates for almost same input power. Another series solution containing even
powers of localised stable solution is shown to exist for higher order
nonlinear Schrodinger equation.


title:
N=2 local and N=4 nonlocal reductions of supersymmetric KP hierarchy in
  N=2 superspace

abstract:
  A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its
reductions which are characterized by a finite number of fields are described.
This class includes the one-dimensional reduction of the two-dimensional
N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry
-- the N=4 Toda chain hierarchy -- which may be relevant in the construction of
supersymmetric matrix models. The Lax pair representations of the bosonic and
fermionic flows, corresponding local and nonlocal Hamiltonians, finite and
infinite discrete symmetries, the first two Hamiltonian structures and the
recursion operator connecting all evolution equations and the Hamiltonian
structures of the N=4 Toda chain hierarchy are constructed in explicit form.
Its secondary reduction to the N=2 supersymmetric alpha=-2 KdV hierarchy is
discussed.


title: On A Recently Proposed Relation Between oHS and Ito Systems

abstract:
  The bi-Hamiltonian structure of original Hirota-Satsuma system proposed by
Roy based on a modification of the bi-Hamiltonian structure of Ito system is
incorrect.


title: Integrable supersymmetric correlated electron chain with open boundaries

abstract:
  We construct an extended Hubbard model with open boundaries from a $R$-matrix
based on the $U_q[Osp(2|2)]$ superalgebra. We study the reflection equation and
find two classes of diagonal solutions. The corresponding one-dimensional open
Hamiltonians are diagonalized by means of the Bethe ansatz approach.


title: New Integrable Coupled Nonlinear Schrodinger Equations

abstract:
  Two types of integrable coupled nonlinear Schrodinger (NLS) equations are
derived by using Zakharov-Shabat (ZS) dressing method.The Lax pairs for the
coupled NLS equations are also investigated using the ZS dressing method. These
give new types of the integrable coupled NLS equations with certain additional
terms. Then, the exact solutions of the new types are obtained. We find that
the solution of these new types do not always produce a soliton solution even
they are the kind of the integrable NLS equations.


title:
Inter-relationships between orthogonal, unitary and symplectic matrix
  ensembles

abstract:
  We consider the following problem: When do alternate eigenvalues taken from a
matrix ensemble themselves form a matrix ensemble? More precisely, we classify
all weight functions for which alternate eigenvalues from the corresponding
orthogonal ensemble form a symplectic ensemble, and similarly classify those
weights for which alternate eigenvalues from a union of two orthogonal
ensembles forms a unitary ensemble. Also considered are the $k$-point
distributions for the decimated orthogonal ensembles.


title: Integrable Systems in the Infinite Genus Limit

abstract:
  We provide an elementary approach to integrable systems associated with
hyperelliptic curves of infinite genus. In particular, we explore the extent to
which the classical Burchnall-Chaundy theory generalizes in the infinite genus
limit, and systematically study the effect of Darboux transformations for the
KdV hierarchy on such infinite genus curves. Our approach applies to
complex-valued periodic solutions of the KdV hierarchy and naturally identifies
the Riemann surface familiar from standard Floquet theoretic considerations
with a limit of Burchnall-Chaundy curves.


title:
A construction for R-matrices without difference property in the
  spectral parameter

abstract:
  A new construction is given for obtaining R-matrices which solve the
McGuire-Yang-Baxter equation in such a way that the spectral parameters do not
possess the difference property. A discussion of the derivation of the
supersymmetric U model is given in this context such that applied chemical
potential and magnetic field terms can be coupled arbitrarily. As a limiting
case the Bariev model is obtained.


title:
Spectral decomposition for the Dirac system associated to the DSII
  equation

abstract:
  A new (scalar) spectral decomposition is found for the Dirac system in two
dimensions associated to the focusing Davey--Stewartson II (DSII) equation.
Discrete spectrum in the spectral problem corresponds to eigenvalues embedded
into a two-dimensional essential spectrum. We show that these embedded
eigenvalues are structurally unstable under small variations of the initial
data. This instability leads to the decay of localized initial data into
continuous wave packets prescribed by the nonlinear dynamics of the DSII
equation.


title:
The symmetric, D-invariant and Egorov reductions of the quadrilateral
  lattice

abstract:
  We present a detailed study of the geometric and algebraic properties of the
multidimensional quadrilateral lattice (a lattice whose elementary
quadrilaterals are planar; the discrete analogue of a conjugate net) and of its
basic reductions. To make this study, we introduce the notions of forward and
backward data, which allow us to give a geometric meaning to the tau-function
of the lattice, defined as the potential connecting these data. Together with
the known circular lattice (a lattice whose elementary quadrilaterals can be
inscribed in circles; the discrete analogue of an orthogonal conjugate net) we
introduce and study two other basic reductions of the quadrilateral lattice:
the symmetric lattice, for which the forward and backward data coincide, and
the D-invariant lattice, characterized by the invariance of a certain natural
frame along the main diagonal. We finally discuss the Egorov lattice, which is,
at the same time, symmetric, circular and D-invariant. The integrability
properties of all these lattices are established using geometric, algebraic and
analytic means; in particular we present a D-bar formalism to construct large
classes of such lattices. We also discuss quadrilateral hyperplane lattices and
the interplay between quadrilateral point and hyperplane lattices in all the
above reductions.


title: Lattice geometry of the Hirota equation

abstract:
  Geometric interpretation of the Hirota equation is presented as equation
describing the Laplace sequence of two-dimensional quadrilateral lattices.
Different forms of the equation are given together with their geometric
interpretation: (i) the discrete coupled Volterra system for the coefficients
of the Laplace equation, (ii) the gauge invariant form of the Hirota equation
for projective invariants of the Laplace sequence, (iii) the discrete Toda
system for the rotation coefficients, (iv) the original form of the Hirota
equation for the tau-function of the quadrilateral lattice.


title:
Integrable Discrete Geometry: the Quadrilateral Lattice, its
  Transformations and Reductions

abstract:
  We review recent results on Integrable Discrete Geometry. It turns out that
most of the known (continuous and/or discrete) integrable systems are
particular symmetries of the quadrilateral lattice, a multidimensional lattice
characterized by the planarity of its elementary quadrilaterals. Therefore the
linear property of planarity seems to be a basic geometric property underlying
integrability. We present the geometric meaning of its tau-function, as the
potential connecting its forward and backward data. We present the theory of
transformations of the quadrilateral lattice, which is based on the discrete
analogue of the theory of rectilinear congruences. In particular, we discuss
the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental
transformations and their interrelations. We also show how the sequence of
Laplace transformations of a quadrilateral surface is described by the discrete
Toda system. We finally show that these classical transformations are strictly
related to the basic operators associated with the quantum field theoretical
formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review
the properties of quadrilateral hyperplane lattices, which play an interesting
role in the reduction theory, when the introduction of additional geometric
structures allows to establish a connection between point and hyperplane
lattices. We present and fully characterize some geometrically distinguished
reductions of the quadrilateral lattice, like the symmetric, circular and
Egorov lattices; we review also basic geometric results of the theory of
quadrilateral lattices in quadrics, and the corresponding analogue of the
Ribaucour reduction of the fundamental transformation.