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The KP Hierarchy in Miwa Coordinates | 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.
|
Coupled nonlinear Schrodinger equations with cubic-quintic nonlinearity:
Integrability and soliton interaction in non-Kerr media | 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.
|
The continuum limit of sl(N/K) integrable super spin chains | 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.
|
Magnetization waves in Landau-Lifshitz Model | 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.
|
$U_q(\hat{sl}_n)$-analog of the XXZ chain with a boundary | 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.
|
Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili
equations | 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.
|
On the explicit solutions of the elliptic Calogero system | 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.
|
A System with a Recursion Operator but One Higher Local Symmetry of the
Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ | 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.
|
On pulse broadening for optical solitons | 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.
|
Multipeakons and the Classical Moment Problem | 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
|
Soliton Collisions in the Ion Acoustic Plasma Equations | 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
|
Reflection K-Matrices for 19-Vertex Models | 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.
|
On the bilinear equations for Fredholm determinants appearing in random
matrices | 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.
|
Families of quai-bi-Hamiltonian systems and separability | 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.
|
On the discrete and continuous Miura Chain associated with the Sixth
Painlev\'e Equation | 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.
|
On exact solution of a classical 3D integrable model | 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.
|
Paraconformal Structures and Integrable Systems | 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.
|
Reduction of bihamiltonian systems and separation of variables: an
example from the Boussinesq hierarchy | 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.
|
r-matrices for relativistic deformations of integrable systems | 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).
|
p-adic Difference-Difference Lotka-Volterra Equation and Ultra-Discrete
Limit | 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.
|
Integrability of the higher-order nonlinear Schroedinger equation
revisited | 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.
|
Factorization dynamics and Coxeter-Toda lattices | 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.
|
Classical skew orthogonal polynomials and random matrices | 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.
|
On the equivalence of the discrete nonlinear Schr\"odinger equation and
the discrete isotropic Heisenberg magnet | 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.
|
A New Class of Optical Solitons | 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.
|
N=2 local and N=4 nonlocal reductions of supersymmetric KP hierarchy in
N=2 superspace | 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.
|
On A Recently Proposed Relation Between oHS and Ito Systems | 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.
|
Integrable supersymmetric correlated electron chain with open boundaries | 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.
|
New Integrable Coupled Nonlinear Schrodinger Equations | 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.
|
Inter-relationships between orthogonal, unitary and symplectic matrix
ensembles | 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.
|
Integrable Systems in the Infinite Genus Limit | 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.
|
A construction for R-matrices without difference property in the
spectral parameter | 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.
|
Spectral decomposition for the Dirac system associated to the DSII
equation | 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.
|
The symmetric, D-invariant and Egorov reductions of the quadrilateral
lattice | 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.
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Lattice geometry of the Hirota equation | 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.
|
Integrable Discrete Geometry: the Quadrilateral Lattice, its
Transformations and Reductions | 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.
|
Discrete Dubrovin Equations and Separation of Variables for Discrete
Systems | A universal system of difference equations associated with a hyperelliptic
curve is derived constituting the discrete analogue of the Dubrovin equations
arising in the theory of finite-gap integration. The parametrisation of the
solutions in terms of Abelian functions of Kleinian type (i.e. the higher-genus
analogues of the Weierstrass elliptic functions) is discussed as well as the
connections with the method of separation of variables.
|
Symmetric Linear Backlund Transformation for Discrete BKP and DKP
equation | Proper lattices for the discrete BKP and the discrete DKP equaitons are
determined. Linear B\"acklund transformation equations for the discrete BKP and
the DKP equations are constructed, which possesses the lattice symmetries and
generate auto-B\"acklund transformations
|
A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear
Schr\"odinger Equations | A spectral mapping theorem is proved that resolves a key problem in applying
invariant manifold theorems to nonlinear Schr\" odinger type equations. The
theorem is applied to the operator that arises as the linearization of the
equation around a standing wave solution. We cast the problem in the context of
space-dependent nonlinearities that arise in optical waveguide problems. The
result is, however, more generally applicable including to equations in higher
dimensions and even systems. The consequence is that stable, unstable, and
center manifolds exist in the neighborhood of a (stable or unstable) standing
wave, such as a waveguide mode, under simple and commonly verifiable spectral
conditions.
|
Multi-Field Integrable Systems Related to WKI-Type Eigenvalue Problems | Higher flows of the Heisenberg ferromagnet equation and the
Wadati-Konno-Ichikawa equation are generalized into multi-component systems on
the basis of the Lax formulation. It is shown that there is a correspondence
between the multi-component systems through a gauge transformation. An
integrable semi-discretization of the multi-component higher Heisenberg
ferromagnet system is obtained.
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Miura Map between Lattice KP and its Modification is Canonical | We consider the Miura map between the lattice KP hierarchy and the lattice
modified KP hierarchy and prove that the map is canonical not only between the
first Hamiltonian structures, but also between the second Hamiltonian
structures.
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Soliton Cellular Automata Associated With Crystal Bases | We introduce a class of cellular automata associated with crystals of
irreducible finite dimensional representations of quantum affine algebras
U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller
algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for
\hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n
and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering
matrices of two solitons coincide with the combinatorial R matrices of
U'_q(C^{(1)}_{n-1})-crystals.
|
Supersymmetric KP hierarchy in N=1 superspace and its N=2 reductions | A wide class of N=2 reductions of the supersymmetric KP hierarchy in N=1
superspace is described. This class includes a new N=2 supersymmetric
generalization of the Toda chain hierarchy. The Lax pair representations of the
bosonic and fermionic flows, local and nonlocal Hamiltonians, finite and
infinite discrete symmetries, first two Hamiltonian structures and the
recursion operator of this hierarchy are constructed. Its secondary reduction
to new N=2 supersymmetric modified KdV hierarchy is discussed.
|
Non-additive fusion, Hubbard models and non-locality | In the framework of quantum groups and additive R-matrices, the fusion
procedure allows to construct higher-dimensional solutions of the Yang-Baxter
equation. These solutions lead to integrable one-dimensional spin-chain
Hamiltonians. Here fusion is shown to generalize naturally to non-additive
R-matrices, which therefore do not have a quantum group symmetry. This method
is then applied to the generalized Hubbard models. Although the resulting
integrable models are not as simple as the starting ones, the general structure
is that of two spin-(s times s') sl(2) models coupled at the free-fermion
point. An important issue is the probable lack of regular points which give
local Hamiltonians. This problem is related to the existence of second order
zeroes in the unitarity equation, and arises for the XX models of higher spins,
the building blocks of the Hubbard models. A possible connection between some
Lax operators L and R-matrices is noted.
|
Quantum Backlund transformation for the integrable DST model | For the integrable case of the discrete self-trapping (DST) model we
construct a Backlund transformation. The dual Lax matrix and the corresponding
dual Backlund transformation are also found and studied. The quantum analog of
the Backlund transformation (Q-operator) is constructed as the trace of a
monodromy matrix with an infinite-dimensional auxiliary space. We present the
Q-operator as an explicit integral operator as well as describe its action on
the monomial basis. As a result we obtain a family of integral equations for
multivariable polynomial eigenfunctions of the quantum integrable DST model.
These eigenfunctions are special functions of the Heun class which is beyond
the hypergeometric class. The found integral equations are new and they shall
provide a basis for efficient analytical and numerical studies of such
complicated functions.
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Classical Solutions Generating Tree Form-Factors in Yang-Mills,
Sin(h)-Gordon and Gravity | Classical solutions generating tree form-factors are defined and constructed
in various models.
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Integrable ODEs on Associative Algebras | In this paper we give definitions of basic concepts such as symmetries, first
integrals, Hamiltonian and recursion operators suitable for ordinary
differential equations on associative algebras, and in particular for matrix
differential equations. We choose existence of hierarchies of first integrals
and/or symmetries as a criterion for integrability and justify it by examples.
Using our componentless approach we have solved a number of classification
problems for integrable equations on free associative algebras. Also, in the
simplest case, we have listed all possible Hamiltonian operators of low order.
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A Coupled AKNS-Kaup-Newell Soliton Hierarchy | A coupled AKNS-Kaup-Newell hierarchy of systems of soliton equations is
proposed in terms of hereditary symmetry operators resulted from Hamiltonian
pairs. Zero curvature representations and tri-Hamiltonian structures are
established for all coupled AKNS-Kaup-Newell systems in the hierarchy.
Therefore all systems have infinitely many commuting symmetries and
conservation laws. Two reductions of the systems lead to the AKNS hierarchy and
the Kaup-Newell hierarchy, and thus those two soliton hierarchies also possess
tri-Hamiltonian structures.
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Complete integrability of derivative nonlinear Schr\"{o}dinger-type
equations | We study matrix generalizations of derivative nonlinear Schr\"{o}dinger-type
equations, which were shown by Olver and Sokolov to possess a higher symmetry.
We prove that two of them are `C-integrable' and the rest of them are
`S-integrable' in Calogero's terminology.
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Determinant Formulas for the Toda and Discrete Toda Equations | Determinant formulas for the general solutions of the Toda and discrete Toda
equations are presented. Application to the $\tau$ functions for the Painlev\'e
equations is also discussed.
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Towards the Lax formulation of SU(2) principal models with nonconstant
metric | The equations that define the Lax pairs for generalized principal chiral
models can be solved for any constant nondegenerate bilinear form on SU(2).
Necessary conditions for the nonconstant metric on SU(2) that define the
integrable models are given.
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Type II vertex operators for the $A_{n-1}^{(1)}$ face model | Presented is a free boson representation of the type II vertex operators for
the $A_{n-1}^{(1)}$ face model. Using the bosonization, we derive some
properties of the type II vertex operators, such as commutation, inversion and
duality relations.
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Non-symmetry constraints of the AKNS system yielding integrable
Hamiltonian systems | This paper aims to show that there exist non-symmetry constraints which yield
integrable Hamiltonian systems through nonlinearization of spectral problems of
soliton systems, like symmetry constraints. Taking the AKNS spectral problem as
an illustrative example, a class of such non-symmetry constraints is introduced
for the AKNS system, along with two-dimensional integrable Hamiltonian systems
generated from the AKNS spectral problem.
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Discrete Z^a and Painleve equations | A discrete analogue of the holomorphic map z^a is studied. It is given by a
Schramm's circle pattern with the combinatorics of the square grid. It is shown
that the corresponding immersed circle patterns lead to special separatrix
solutions of a discrete Painleve equation. Global properties of these
solutions, as well as of the discrete $z^a$ are established.
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On Construction of Recursion Operators From Lax Representation | In this work we develop a general procedure for constructing the recursion
operators fro non-linear integrable equations admitting Lax representation.
Svereal new examples are given. In particular we find the recursion operators
for some KdV-type of integrable equations.
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On integrable deformations of the spherical top | The motion on the sphere $S^2$ with the potential $V= (x_1x_2x_3)^{-2/3}$ is
considered. The Lax representation and the linearisation procedure for this
two-dimensional integrable system are discussed.
|
New integrable string-like fields in 1+1 dimensions | The symmetry classification method is applied to the string-like scalar
fields in two-dimensional space-time. When the configurational space is
three-dimensional and reducible we present the complete list of the systems
admiting higher polynomial symmetries of the 3rd, 4th and 5th-order.
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Canonical transformations of the extended phase space, Toda lattices and
Stackel family of integrable systems | We consider compositions of the transformations of the time variable and
canonical transformations of the other coordinates, which map completely
integrable system into other completely integrable system. Change of the time
gives rise to transformations of the integrals of motion and the Lax pairs,
transformations of the corresponding spectral curves and R-matrices. As an
example, we consider canonical transformations of the extended phase space for
the Toda lattices and the Stackel systems.
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CPT Symmetries and the Backlund Transformations | We show that the auto-Backlund transformations of the sine-Gordon,
Korteweg-deVries, nonlinear Schrodinger, and Ernst equations are related to
their respective CPT symmetries. This is shown by applying the CPT symmetries
of these equations to the Riccati equations of the corresponding
pseudopotential functions where the fields are allowed to transform into new
solutions while the pseudopotential functions and the Backlund parameter are
held fixed.
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Lie point symmetries of integrable evolution equations and invariant
solutions | An integrable hierarchies connected with linear stationary Schr\"odinger
equation with energy dependent potentials (in general case) are considered.
Galilei-like and scaling invariance transformations are constructed. A symmetry
method is applied to construct invariant solutions.
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The motion of a rigid body in a quadratic potential: an integrable
discretization | The motion of a rigid body in a quadratic potential is an important example
of an integrable Hamiltonian system on a dual to a semidirect product Lie
algebra so(n) x Symm(n). We give a Lagrangian derivation of the corresponding
equations of motion, and introduce a discrete time analog of this system. The
construction is based on the discrete time Lagrangian mechanics on Lie groups,
accompanied with the discrete time Lagrangian reduction. The resulting
multi-valued map (correspondence) on the dual to so(n) x Symm(n) is Poisson
with respect to the Lie-Poisson bracket, and is also completely integrable. We
find a Lax representation based on matrix factorisations, in the spirit of
Veselov-Moser.
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Pfaff tau-functions | Consider the evolution $$ \frac{\pl m_\iy}{\pl t_n}=\Lb^n m_\iy, \frac{\pl
m_\iy}{\pl s_n}=-m_\iy(\Lb^\top)^n, $$ on bi- or semi-infinite matrices
$m_\iy=m_\iy(t,s)$, with skew-symmetric initial data $m_{\iy}(0,0)$. Then,
$m_\iy(t,-t)$ is skew-symmetric, and so the determinants of the successive
"upper-left corners" vanish or are squares of Pfaffians. In this paper, we
investigate the rich nature of these Pfaffians, as functions of t. This problem
is motivated by questions concerning the spectrum of symmetric and symplectic
random matrix ensembles.
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The Complex Bateman Equation | The general solution to the Complex Bateman equation is constructed. It is
given in implicit form in terms of a functional relationship for the unknown
function. The known solution of the usual Bateman equation is recovered as a
special case.
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The General Solution of the Complex Monge-Amp\`ere Equation in two
dimensional space | The general solution to the Complex Monge-Amp\`ere equation in a two
dimensional space is constructed.
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The Complex Bateman Equation in a space of arbitrary dimension | A general solution to the Complex Bateman equation in a space of arbitrary
dimensions is constructed.
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The General Solution of the Complex Monge-Amp\`ere Equation in a space
of arbitrary dimension | A general solution to the Complex Monge-Amp\`ere equation in a space of
arbitrary dimensions is constructed.
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Discrete asymptotic nets and W-congruences in Plucker line geometry | The asymptotic lattices and their transformations are studied within the line
geometry approach. It is shown that the discrete asymptotic nets are
represented by isotropic congruences in the Plucker quadric. On the basis of
the Lelieuvre-type representation of asymptotic lattices and of the discrete
analog of the Moutard transformation, it is constructed the discrete analog of
the W-congruences, which provide the Darboux-Backlund type transformation of
asymptotic lattices.The permutability theorems for the discrete Moutard
transformation and for the corresponding transformation of asymptotic lattices
are established as well. Moreover, it is proven that the discrete W-congruences
are represented by quadrilateral lattices in the quadric of Plucker. These
results generalize to a discrete level the classical line-geometric approach to
asymptotic nets and W-congruences, and incorporate the theory of asymptotic
lattices into more general theory of quadrilateral lattices and their
reductions.
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Darboux Transformation for Supersymmetric KP Hierarchies | We construct Darboux transformations for the super-symmetric KP hierarchies
of Manin--Radul and Jacobian types. We also consider the binary Darboux
transformation for the hierarchies. The iterations of both type of Darboux
transformations are briefly discussed.
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Darboux transformations for a Bogoyavlenskii equation in 2+1 dimensions | We use the singular manifold method to obtain the Lax pair, Darboux
transformations and soliton solutions for a (2+1) dimensional integrable
equation.
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Surfaces of Constant negative Scalar Curvature and the Correpondence
between the Liouvulle and the sine-Gordon Equations | By studying the {\it internal} Riemannian geometry of the surfaces of
constant negative scalar curvature, we obtain a natural map between the
Liouville, and the sine-Gordon equations. First, considering isometric
immersions into the Lobachevskian plane, we obtain an uniform expression for
the general (locally defined) solution of both the equations. Second, we prove
that there is a Lie-B\"acklund transformation interpolating between Liouville
and sine-Gordon. Third, we use isometric immersions into the Lobachevskian
plane to describe sine-Gordon N-solitons explicitly.
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The Structure of the Bazhanov-Baxter Model and a New Solution of the
Tetrahedron Equation | We clarify the structure of the Bazhanov-Baxter model of the 3-dim N-state
integrable model. There are two essential points, i) the cubic symmetries, and
ii) the spherical trigonometry parametrization, to understand the structure of
this model. We propose two approaches to find a candidate as a solution of the
tetrahedron equation, and we find a new solution.
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The complex Toda chains and the simple Lie algebras - solutions and
large time asymptotics -- II | We propose a compact and explicit expression for the solutions of the complex
Toda chains related to the classical series of simple Lie algebras g. The
solutions are parametrized by a minimal set of scattering data for the
corresponding Lax matrix. They are expressed as sums over the weight systems of
the fundamental representations of g and are explicitly covariant under the
corresponding Weyl group action. In deriving these results we start from the
Moser formula for the A_r series and obtain the results for the other classical
series of Lie algebras by imposing appropriate involutions on the scattering
data. Thus we also show how Moser's solution goes into the one of Olshanetsky
and Perelomov. The results for the large-time asymptotics of the A_r -CTC
solutions are extended to the other classical series B_r - D_r. We exhibit also
some `irregular' solutions for the D_{2n+1} algebras whose asymptotic regimes
at t ->\pm\infty are qualitatively different. Interesting examples of bounded
and periodic solutions are presented and the relations between the solutions
for the algebras D_4, B_3 and G_2 $ are analyzed.
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Self-similar solutions of NLS-type dynamical systems | We study self-similar solutions of NLS-type dynamical systems. Lagrangian
approach is used to show that they can be reduced to three canonical forms,
which are related by Miura transformations. The fourth Painleve equation (PIV)
is central in our consideration - it connects Heisenberg model, Volterra model
and Toda model to each other. The connection between the rational solutions of
PIV and Coulomb gas in a parabolic potential is established. We discuss also
the possibility to obtain an exact solution for optical soliton i.e. of the NLS
equation with time-dependent dispersion.
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Self-similarity in Spectral Problems and q-special Functions | Similarity symmetries of the factorization chains for one-dimensional
differential and finite-difference Schr\"odinger equations are discussed.
Properties of the potentials defined by self-similar reductions of these chains
are reviewed. In particular, their algebraic structure, relations to
$q$-special functions, infinite soliton systems, supersymmetry, coherent
states, orthogonal polynomials, one-dimensional Ising chains and random
matrices are outlined.
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Orthogonal and symplectic matrix integrals and coupled KP hierarchy | Orthogonal and symplectic matrix integrals are investigated. It is shown that
the matrix integrals can be considered as a $\tau$-function of the coupled KP
hierarchy, whose solution can be expressed in terms of pfaffians.
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Dressing method and the coupled KP hierarchy | The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by
using the dressing method. It is shown that the coupled KP hierarchy can be
reformulated as a reduced case of the 2-component KP hierarchy.
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Quasi-Lagrangian Systems of Newton Equations | Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$
with an integral of motion quadratic in velocities are studied. These equations
generalize the potential case (when A=I, the identity matrix) and they admit a
curious quasi-Lagrangian formulation which differs from the standard Lagrange
equations by the plus sign between terms. A theory of such quasi-Lagrangian
Newton (qLN) systems having two functionally independent integrals of motion is
developed with focus on two-dimensional systems. Such systems admit a
bi-Hamiltonian formulation and are proved to be completely integrable by
embedding into five-dimensional integrable systems. They are characterized by a
linear, second-order PDE which we call the fundamental equation. Fundamental
equations are classified through linear pencils of matrices associated with qLN
systems. The theory is illustrated by two classes of systems: separable
potential systems and driven systems. New separation variables for driven
systems are found. These variables are based on sets of non-confocal conics. An
effective criterion for existence of a qLN formulation of a given system is
formulated and applied to dynamical systems of the Henon-Heiles type.
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On a Schwarzian PDE associated with the KdV Hierarchy | We present a novel integrable non-autonomous partial differential equation of
the Schwarzian type, i.e. invariant under M\"obius transformations, that is
related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered
as the generating equation for the entire hierarchy of Schwarzian KdV
equations. We present its Lax pair, establish its connection with the SKdV
hierarchy, its Miura relations to similar generating PDEs for the modified and
regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate
that its similarity reductions lead to the {\it full} Painlev\'e VI equation,
i.e. with four arbitary parameters.
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On two aspects of the Painleve analysis | We use the Calogero equation to illustrate the following two aspects of the
Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the
Painleve test for integrability, the singular expansions of its solutions
around characteristic hypersurfaces can be neither single-valued functions of
independent variables nor single-valued functionals of data. Second, if the
truncation of singular expansions of solutions is consistent, the truncation
not necessarily leads to the simplest, or elementary, auto-Backlund
transformation related to the Lax pair.
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Matrix integrals and the geometry of spinors | We obtain the collection of symmetric and symplectic matrix integrals and the
collection of Pfaffian tau-functions, recently described by Peng and Adler and
van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector
of a fermionic Fock space. This fermionic Fock space is the same space as one
constructs to obtain the KP and 1-Toda lattice hierarchy.
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Dispersionless Fermionic KdV | We analyze the dispersionless limits of the Kupershmidt equation, the SUSY
KdV-B equation and the SUSY KdV equation. We present the Lax description for
each of these models and bring out various properties associated with them as
well as discuss open questions that need to be addressed in connection with
these models.
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The KdV equation on a half-line | The initial boundary value problem on a half-line for the KdV equation with
the boundary conditions $u|_{x=0}=a\leq0$, $u_{xx}|_{x=0}=3a^2$ is integrated
by means of the inverse scattering method. In order to find the time evolution
of the scattering matrix it turned out to be sufficient to solve the Riemann
problem on a hyperelliptic curve of genus two, where the conjugation matrices
are effectively defined by initial data.
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Yang-Baxter Algebra for the n-Harmonic Oscillator Realisation of
sp(2n,R) | Using a rational R-matrix associated with the 4 x 4 defining matrix
representation of c_2=sp(4), the Lie algebra of Sp(4), a one-site operator
solution of the associated Yang-Baxter algebra acting in the Fock space of two
harmonic oscillators is derived. This is used to define N-site integrable
systems, which are soluble by a version of the algebraic Bethe ansatz method
without nesting. All essential aspects of the work generalise directly from c_2
to c_n.
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On integrable discretization of the inhomogeneous Ablowitz-Ladik model | An integrable discretization of the inhomogeneous Ablowitz-Ladik model with a
linear force is introduced. Conditions on parameters of the discretization
which are necessary for reproducing Bloch oscillations are obtained. In
particular, it is shown that the step of the discretization must be
comensurable with the period of oscillations imposed by the inhomogeneous
force. By proper choice of the step of the discretization the period of
oscillations of a soliton in the discrete model can be made equal to an integer
number of periods of oscillations in the underline continuous-time lattice.
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Lax pair, Darboux Transformations and solitonic solutions for a (2+1)
dimensional NLSE | In this paper the Singular Manifold Method has allowed us to obtain the Lax
pair, Darboux transformations and tau functions for a non-linear Schr\"odiger
equation in 2+1 dimensions. In this way we can iteratively build different kind
of solutions with solitonic behavior.
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Beyond Nonlinear Schr\"odinger Equation Approximation for an Anharmonic
Chain with Harmonic Long Range Interaction | Multi scales method is used to analyze a nonlinear differential-difference
equation. In order $\epsilon^3$ the NLS equation is found to determine the
space-time evolution of the leading amplitude. In the next order this has to
satisfy a complex mKdV equation (the next in the NLS hierarchy) in order to
eliminate secular terms. The zero dispersion point case is also analyzed and
the relevant equation is a modified NLS equation with a third order derivative
term included
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Singularity confinement and algebraic entropy: the case of the discrete
Painlev\'e equations | We examine the validity of the results obtained with the singularity
confinement integrability criterion in the case of discrete Painlev\'e
equations. The method used is based on the requirement of non-exponential
growth of the homogeneous degree of the iterate of the mapping. We show that
when we start from an integrable autonomous mapping and deautonomise it using
singularity confinement the degrees of growth of the nonautonomous mapping and
of the autonomous one are identical. Thus this low-growth based approach is
compatible with the integrability of the results obtained through singularity
confinement. The origin of the singularity confinement property and its
necessary character for integrability are also analysed.
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Exact Solution of the Quantum Calogero-Gaudin System and of its
q-Deformation | A complete set of commuting observables for the Calogero-Gaudin system is
diagonalized, and the explicit form of the corresponding eigenvalues and
eigenfunctions is derived. We use a purely algebraic procedure exploiting the
co-algebra invariance of the model; with the proper technical modifications
this procedure can be applied to the $q-$deformed version of the model, which
is then also exactly solved.
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N=2 Hamiltonians with sl(2) coalgebra symmetry and their integrable
deformations | Two dimensional classical integrable systems and different integrable
deformations for them are derived from phase space realizations of classical
$sl(2)$ Poisson coalgebras and their $q-$deformed analogues. Generalizations of
Morse, oscillator and centrifugal potentials are obtained. The N=2 Calogero
system is shown to be $sl(2)$ coalgebra invariant and the well-known
Jordan-Schwinger realization can be also derived from a (non-coassociative)
coproduct on $sl(2)$. The Gaudin Hamiltonian associated to such
Jordan-Schwinger construction is presented. Through these examples, it can be
clearly appreciated how the coalgebra symmetry of a hamiltonian system allows a
straightforward construction of different integrable deformations for it.
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Schlesinger transformations for elliptic isomonodromic deformations | Schlesinger transformations are discrete monodromy preserving symmetry
transformations of the classical Schlesinger system. Generalizing well-known
results from the Riemann sphere we construct these transformations for
isomonodromic deformations on genus one Riemann surfaces. Their action on the
system's tau-function is computed and we obtain an explicit expression for the
ratio of the old and the transformed tau-function.
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The averaging of non-local Hamiltonian structures in Whitham's method | We consider the $m$-phase Whitham's averaging method and propose a procedure
of "averaging" of non-local Hamiltonian structures. The procedure is based on
the existence of a sufficient number of local commuting integrals of a system
and gives a Poisson bracket of Ferapontov type for the Whitham's system. The
method can be considered as a generalization of the Dubrovin-Novikov procedure
for the local field-theoretical brackets.
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Form factors of the SU(2) invariant massive Thirring model with boundary
reflection | The SU(2) invariant massive Thirring model with a boundary is considered on
the basis of the vertex operator approach. The bosonic formulae are presented
for the vacuum vector and its dual in the presence of the boundary. The
integral representations are also given for form factors of the present model.
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Canonical transformations of the time for the Toda lattice and the Holt
system | For the Toda lattice and the Holt system we consider properties of canonical
transformations of the extended phase space, which preserve integrability. The
separated variables are invariant under change of the time. On the other hand,
mapping of the time induces transformations of the action-angles variables and
a shift of the generating function of the B\"{a}cklund transformation.
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Introduction to the functions on compact Riemann surfaces and
theta-functions | We collect some classical results related to analysis on the Riemann
surfaces. The notes may serve as an introduction to the field: we suppose that
the reader is familiar only with the basic facts from topology and complex
analysis. the treatment is organized to give a background for further
applications to non-linear differential equations.
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Resonant Bifurcations | We consider dynamical systems depending on one or more real parameters, and
assuming that, for some ``critical'' value of the parameters, the eigenvalues
of the linear part are resonant, we discuss the existence -- under suitable
hypotheses -- of a general class of bifurcating solutions in correspondence to
this resonance. These bifurcating solutions include, as particular cases, the
usual stationary and Hopf bifurcations. The main idea is to transform the given
dynamical system into normal form (in the sense of Poincar\'e-Dulac), and to
impose that the normalizing transformation is convergent, using the convergence
conditions in the form given by A. Bruno. Some specially interesting
situations, including the cases of multiple-periodic solutions, and of
degenerate eigenvalues in the presence of symmetry, are also discussed with
some detail.
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Integrable deformations of oscillator chains from quantum algebras | A family of completely integrable nonlinear deformations of systems of N
harmonic oscillators are constructed from the non-standard quantum deformation
of the sl(2,R) algebra. Explicit expressions for all the associated integrals
of motion are given, and the long-range nature of the interactions introduced
by the deformation is shown to be linked to the underlying coalgebra structure.
Separability and superintegrability properties of such systems are analysed,
and their connection with classical angular momentum chains is used to
construct a non-standard integrable deformation of the XXX hyperbolic Gaudin
system.
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A note on real forms of the complex N=4 supersymmetric Toda chain
hierarchy in real N=2 and N=4 superspaces | Three inequivalent real forms of the complex N=4 supersymmetric Toda chain
hierarchy (Nucl. Phys. B558 (1999) 545, solv-int/9907004) in the real N=2
superspace with one even and two odd real coordinates are presented. It is
demonstrated that the first of them possesses a global N=4 supersymmetry, while
the other two admit a twisted N=4 supersymmetry. A new superfield basis in
which supersymmetry transformations are local is discussed and a manifest N=4
supersymmetric representation of the N=4 Toda chain in terms of a chiral and an
anti-chiral N=4 superfield is constructed. Its relation to the complex N=4
supersymmetric KdV hierarchy is established. Darboux-Backlund symmetries and a
new real form of this last hierarchy possessing a twisted N=4 supersymmetry are
derived.
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Bicomplexes and finite Toda lattices | We associate bicomplexes with the finite Toda lattice and with a finite Toda
field theory in such a way that conserved currents and charges are obtained by
a simple iterative construction.
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Separation of variables for soliton equations via their binary
constrained flows | Binary constrained flows of soliton equations admitting $2\times 2$ Lax
matrices have 2N degrees of freedom, which is twice as many as degrees of
freedom in the case of mono-constrained flows. For their separation of
variables only N pairs of canonical separated variables can be introduced via
their Lax matrices by using the normal method. A new method to introduce the
other N pairs of canonical separated variables and additional separated
equations is proposed. The Jacobi inversion problems for binary constrained
flows are established. Finally, the factorization of soliton equations by two
commuting binary constrained flows and the separability of binary constrained
flows enable us to construct the Jacobi inversion problems for some soliton
hierarchies.
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Multidimensional analogs of geometric s<-->t duality | The usual propetry of s<-->t duality for scattering amplitudes, e.g. for
Veneziano amplitude, is deeply connected with the 2-dimensional geometry. In
particular, a simple geometric construction of such amplitudes was proposed in
a joint work by this author and S.Saito (solv-int/9812016). Here we propose
analogs of one of those amplitudes associated with multidimensional euclidean
spaces, paying most attention to the 3-dimensional case. Our results can be
regarded as a variant of "Regge calculus" intimately connected with ideas of
the theory of integrable models.
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