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On the point transformations for the equation $y''= P + 3Qy' + 3R{y'}^2
+ S{y'}^3$ | For the equations $y''=P(x,y) + 3Q(x,y)y' + 3R(x,y){y'}^2 + S(x,y){y'}^3$ the
problem of equivalence is considered. Some classical results are resumed in
order to prepare the background for the study of special subclass of such
equations, which arises in the theory of dynamical systems admitting the normal
shift.
|
An analytic description of the vector constrained KP hierarchy | In this paper we give a geometric description in terms of the Grassmann
manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the
vector $k$-constrained KP hierarchy. We also show in a geometric way that these
hierarchies are equivalent to Krichever's general rational reductions of the KP
hierarchy.
|
The geometry of spinors and the multicomponent BKP and DKP hierarchies | We develop a formalism of multicomponent BKP hierarchies using elementary
geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence
all their reductions like KdV, NLS, AKNS or DS) are reductions of the
multicomponent BKP.
|
On the Liouville transformation and exactly-solvable Schrodinger
equations | The present article discusses the connection between exactly-solvable
Schrodinger equations and the Liouville transformation. This transformation
yields a large class of exactly-solvable potentials, including the
exactly-solvable potentials introduced by Natanzon. As well, this class is
shown to contain two new families of exactly solvable potentials.
|
An Extension of the KdV Hierarchy Arising from a Representation of a
Toroidal Lie Algebra | In this article we show how to construct hierarchies of partial differential
equations from the vertex operator representations of toroidal Lie algebras. In
the smallest example - rank 2 toroidal cover of $sl_2$ - we obtain an extension
of the KdV hierarchy. We use the action of the corresponding
infinite-dimensional group to construct solutions for these non-linear PDEs.
|
Matrix Integrals, Toda symmetries, Virasoro constraints, and orthogonal
polynomials | The relationship is made between matrix integrals, Toda master-symmetries,
Virasoro constraints and orthogonal polynomials.
|
The Distribution of the Largest Eigenvalue in the Gaussian Ensembles | The focus of this survey paper is on the distribution function for the
largest eigenvalue in the finite N Gaussian ensembles (GOE,GUE,GSE) in the edge
scaling limit of N->infinity. These limiting distribution functions are
expressible in terms of a particular Painleve II function. Comparisons are made
with finite N simulations as well as a discussion of the universality of these
distribution functions.
|
Trace Formula for a System of Particles with Elliptic Potential | We consider classical particles on the line with the Weierstrass $\wp$
function as potential. This system parameterizes special solutions of the KP
equation. We derive the trace formula which relates the Hamiltonian of the
particle system to the residues of some Abelian differential (meromorphic
one-form) on the spectral curve. Such formula is important for the construction
action-angle variables and study invariant Gibbs' states.
|
Nonlinear Quasiclassics and the Painlev\'e Equations | Problem of asymptotic description for global solutions to the six Painleve
equations was investigated. Elliptic anzatzes and appropriate modulation
equations were written out.
|
The Constrained MKP Hierarchy and the Generalized Kupershmidt-Wilson
Theorem | The constrained Modified KP hierarchy is considered from the viewpoint of
modification. It is shown that its second Poisson bracket, which has a rather
complicated form, is associated to a vastly simpler bracket via Miura-type map.
The similar results are established for a natural reduction of MKP.
|
The constrained modified KP hierarchy and the generalized Miura
transformations | In this letter, we consider the second Hamiltonian structure of the
constrained modified KP hierarchy. After mapping the Lax operator to a pure
differential operator the second structure becomes the sum of the second and
the third Gelfand-Dickey brackets defined by this differential operator. We
simplify this Hamiltonian structure by factorizing the Lax operator into linear
terms.
|
B\"acklund transformation for the Krichever-Novikov equation | The B\"acklund transformation and its nonlinear superposition principle are
presented for the Krichever-Novikov equation $u_t= u_{xxx} - {3/(2u_x)}
(u^2_{xx} - r(u)) + cu_x, r^{(5)}=0$.
|
Two-dimensional soliton cellular automaton of deautonomized Toda-type | A deautonomized version of the two-dimensional Toda lattice equation is
presented. Its ultra-discrete analogue and soliton solutions are also
discussed.
|
Generating function of correlators in the sl_2 Gaudin model | For the sl_2 Gaudin model (degenerated quantum integrable XXX spin chain) an
exponential generating function of correlators is calculated explicitely. The
calculation relies on the Gauss decomposition for the SL_2 loop group. From the
generating function a new explicit expression for the correlators is derived
from which the determinant formulas for the norms of Bethe eigenfunctions due
to Richardson and Gaudin are obtained.
|
d=2, N=2 Superconformally Covariant Operators and Super W-Algebras | We construct and classify superconformally covariant differential operators
defined on N=2 super Riemann surfaces. By contrast to the N=1 theory, these
operators give rise to partial rather than ordinary differential equations
which leads to novel features for their matrix representation. The latter is
applied to the derivation of N=2 super W-algebras in terms of N=2 superfields.
|
Discrete Levy Transformations and Casorati Determinant Solutions of
Quadrilateral Lattices | Sequences of discrete Levy and adjoint Levy transformations for the
multidimensional quadrilateral lattices are studied. After a suitable number of
iterations we show how all the relevant geometrical features of the transformed
quadrilateral lattice can be expressed in terms of multi-Casorati determinants.
As an example we dress the Cartesian lattice.
|
The Camassa-Holm Equation: A Loop Group Approach | A map is presented that associates with each element of a loop group a
solution of an equation related by a simple change of coordinates to the
Camassa-Holm (CH) Equation. Certain simple automorphisms of the loop group give
rise to Backlund transformations of the equation. These are used to find
2-soliton solutions of the CH equation, as well as some novel singular
solutions.
|
Determinant Structure of the Rational Solutions for the Painlev\'e IV
Equation | Rational solutions for the Painlev\'e IV equation are investigated by Hirota
bilinear formalism. It is shown that the solutions in one hierarchy are
expressed by 3-reduced Schur functions, and those in another two hierarchies by
Casorati determinant of the Hermite polynomials, or by special case of the
Schur polynomials.
|
Correlation functions for a strongly correlated boson system | The correlation functions for a strongly correlated exactly solvable
one-dimensional boson system on a finite chain as well as in the thermodynamic
limit are calculated explicitly. This system which we call the phase model is
the strong coupling limit of the integrable q-boson hopping model. The results
are presented as determinants.
|
Shock waves in one-dimensional Heisenberg ferromagnets | We use SU(2) coherent state path integral formulation with the stationary
phase approximation to investigate, both analytically and numerically, the
existence of shock waves in the one- dimensional Heisenberg ferromagnets with
anisotropic exchange interaction. As a result we show the existence of shock
waves of two types,"bright" and "dark", which can be interpreted as moving
magnetic domains.
|
Two-body Elliptic Model in proper variables: Lie-algebraic forms and
their discretizations | Two Lie algebraic forms of the 2-body Elliptic Calogero model are presented.
Translation-invariant and dilatation-invariant discretizations of the model are
obtained.
|
Equations of the reaction-diffusion type with a loop algebra structure | A system of equations of the reaction-diffusion type is studied in the
framework of both the direct and the inverse prolongation structure. We find
that this system allows an incomplete prolongation Lie algebra, which is used
to find the spectral problem and a whole class of nonlinear field equations
containing the original ones as a special case.
|
On classical string configurations | Equations which define classical configurations of strings in $R^3$ are
presented in a simple form. General properties as well as particular classes of
solutions of these equations are considered.
|
A new explicit expression for the Korteweg-De Vries hierarchy | We derive an improved fully explicit expression for the right-hand sides of
the matrix KdV hierarchy using the relation to the heat kernel of the
one-dimensional Schr\"odinger operator. Our method of "matrix elements"
produces, moreover, an explicit expression for the powers of a
Schr\"odinger-like differential operator of any order.
|
A Conjectured R-Matrix | A new spectral parameter independent R-matrix (that depends on all of the
dynamical variables) is proposed for the elliptic Calogero-Moser models.
Necessary and sufficient conditions for this R-matrix to exist reduce to an
equality between determinants of matrices involving elliptic functions. The
needed identity appears new and is still unproven in full generality: we
present it as a conjecture.
|
D-modules and Darboux transformations | A method of G. Wilson for generating commutative algebras of ordinary
differential operators is extended to higher dimensions. Our construction,
based on the theory of D-modules, leads to a new class of examples of
commutative rings of partial differential operators with rational spectral
varieties. As an application, we briefly discuss their link to the bispectral
problem and to the theory of lacunas.
|
Equal-time temperature correlators of the one-dimensional Heisenberg XY
chain | Representations as determinants of $M\times M$ dimensional matrices are
obtained for equal-time temperature correlators of the anisotropic Heisenberg
XY chain. These representations are simple deformations of the answers for the
isotropic XX0 chain. In the thermodynamic limit, the correlators are expressed
in terms of the Fredholm determinants of linear integral operators.
|
Darboux Transformations for SUSY Integrable Systems | Several types of Darboux transformations for supersymmetric integrable
systems such as the Manin-Radul KdV, Mathieu KdV and SUSY sine-Gordon equations
are considered. We also present solutions such as supersolitons and superkinks.
|
Additional symmetries of the Zakharov-Shabat hierarchy, String equation
and Isomonodromy | Isomonodromic deformations are nothing but symmetries of the Zakharov-Shabat
(isospectral) hierarchy, both the basic ones (belonging to the hierarchy) and
additional, restricted to the submanifold of solutions to the string equation.
|
On the Integrability of the One-Dimensional Open XYZ Spin Chain | The Lax pair for the one-dimensional open XYZ spin chain is constructed, this
shows that the system is completely integrable .
|
Comment on ``Equal-time temperature correlators of the one-dimensional
Heisenberg XY chain'', preprint solv-int/9710028 | In the comment we give references to our papers where the problem was solved
for more general case of time-dependent finite temperature correlators.
|
A new class of completely integrable quantum spin chains | A large (infinitely-dimensional) class of completely integrable (possibly
non-autonomous) spin chains is discovered associated to an infinite-dimensional
Lie Algebra of infinite rank. The complete set of integrals of motion is
constructed explicitly, as well as their eigenstates and spectra. As an example
we outline kicked Ising model: Ising chain periodically kicked with transversal
magnetic field.
|
The nondynamical r-matrix structure of the elliptic Calogero-Moser model | In this paper, we construct a new Lax operator for the elliptic
Calogero-Moser model with N=2. The nondynamical r-matrix structure of this Lax
operator is also studied .
The relation between our Lax operator and the Lax operator given by Krichever
is also obtained.
|
Integrable Coupled KdV Systems | We give the conditions for a system of N- coupled Korteweg de Vries(KdV) type
of equations to be integrable. Recursion operators of each subclasses are also
given. All examples for N=2 are explicitly given.
|
On the geometry of point-expansions for certain class of differential
equations of the second order | Second order ordinary differential equations of the form $y'' = P(x,y)
+ 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered
and their point-expansions are constructed. Geometrical structures connected
with these expansions are described.
|
Extended N=2 supersymmetric matrix (1,s)-KdV hierarchies | We propose the Lax operators for N=2 supersymmetric matrix generalization of
the bosonic (1,s)-KdV hierarchies. The simplest examples - the N=2
supersymmetric a=4 KdV and a=5/2 Boussinesq hierarchies - are discussed in
detail.
|
The New Identity for the Scattering Matrx of Exactly Solvable Models | We discovered a simple quadratic equation, which relates scattering phases of
particles on Fermi surface. We consider one dimensional Bose gas and XXZ
Heisenberg spin chain.
|
Integrability in 3+1 Dimensions: Relaxing a Tetrahedron Relation | I propose a scheme of constructing classical integrable models in 3+1
discrete dimensions, based on a relaxed version of the problem of factorizing a
matrix into the product of four matrices of a special form.
|
The XXC Models | A class of recently introduced multi-states XX models is generalized to
include a deformation parameter. This corresponds to an additional
nearest-neighbor CC interaction in the defining quadratic hamiltonian. Complete
integrability of the one-dimensional models is shown in the context of the
quantum inverse scattering method. The new R-matrix is derived. The
diagonalization of the XXC models is carried out using the algebraic Bethe
Ansatz.
|
Canonical gauge equivalences of the sAKNS and sTB hierarchies | We study the gauge transformations between the supersymmetric AKNS (sAKNS)
and supersymmetric two-boson (sTB) hierarchies. The Hamiltonian nature of these
gauge transformations is investigated, which turns out to be canonical. We also
obtain the Darboux-Backlund transformations for the sAKNS hierarchy from these
gauge transformations.
|
The averaging of Hamiltonian structures in discrete variant of Whitham
method | Paper is devoted to the construction of averaging procedure of Hamiltonian
structures in discrete Whitham method. The procedure is analogous to
Dubrovin-Novikov procedure of averaging of local field-theoretical Poisson
brackets and gives the Poisson bracket of Hydrodynamic Type starting from
Poisson bracket for a discrete chain.
|
Boundary K-matrices and the Lax pair for 1D open XYZ spin-chain | We analysis the symmetries of the reflection equation for open $XYZ$ model
and find their solutions $K^{\pm}$ case by case. In the general open boundary
conditions, the Lax pair for open one-dimensional $XYZ$ spin-chain is given.
|
Toda-Darboux maps and vertex operators | The purpose of this paper is to study Toda-Darboux transforms, i.e., Darboux
transforms for operators L(t) flowing according to the Toda lattice. Each
element of the null-space $L(t)-z$ specifies a factorization for all t and thus
a Toda-Darboux transform on $L(t)$. The Toda-Darboux map induces a
transformation on the tau-vectors, given by a certain vertex operator, and on
eigenfunctions, given by a Wronskian. .
|
Sigma Models and Minimal Surfaces | The correspondance is established between the sigma models, the minimal
surfaces and the Monge-Ampere equation. The Lax -Pairs of the minimality
condition of the minimal surfaces and the Monge-Ampere equations are given.
Existance of infinitely many nonlocal conservation laws is shown and some
Backlund transformations are also given.
|
Quadratically integrable geodesic flows on the torus and on the Klein
bottle | In the present paper we prove, that if the geodesic flow of a metric G on the
torus T is quadratically integrable, then the torus T isometrically covers a
torus with a Liouville metric on it, and describe the set of quadratically
integrable geodesic flows on the Klein bottle.
|
Polarization scattering by soliton-soliton collisions | Collision of two solitons of the Manakov system is analytically studied.
Existence of a complete polarization mode switching regime is proved and the
parameters of solitons prepared for polarization switching are found.
|
A Symmetric Generalization of Linear B\"acklund Transformation
associated with the Hirota Bilinear Difference Equation | The Hirota bilinear difference equation is generalized to discrete space of
arbitrary dimension. Solutions to the nonlinear difference equations can be
obtained via B\"acklund transformation of the corresponding linear problems.
|
Temperature correlators in the two-component one-dimensional gas | The quantum nonrelativistic two-component Bose and Fermi gases with the
infinitely strong point-like coupling between particles in one space dimension
are considered. Time and temperature dependent correlation functions are
represented in the thermodynamic limit as Fredholm determinants of integrable
linear integral operators.
|
Asymptotics of a class of Fredholm determinants | In this expository article we describe the asymptotics of certain Fredholm
determinants which provide solutions to the cylindrical Toda equations, and we
explain how these asymptotics are derived. The connection with Fredholm
determinants arising in the theory of random matrices, and their asymptotics,
are also discussed.
|
Dynamical boundary conditions for integrable lattices | Some special solutions to the reflection equation are considered. These
boundary matrices are defined on the common quantum space with the other
operators in the chain. The relations with the Drinfeld twist are discussed.
|
Asymptotics of perturbed soliton for Davey--Stewartson II equation | It is shown that, under a small perturbation of lump (soliton) for
Davey--Stewartson (DS-II) equation, the scattering data gain the nonsoliton
structure. As a result, the solution has the form of Fourier type integral.
Asymptotic analysis shows that, in spite of dispertion, the principal term of
the asymptotic expansion for the solution has the solitary wave form up to
large time.
|
Functional Tetrahedron Equation | We describe a scheme of constructing classical integrable models in
2+1-dimensional discrete space-time, based on the functional tetrahedron
equation - equation that makes manifest the symmetries of a model in local
form. We construct a very general "block-matrix model" together with its
algebro-geometric solutions, study its various particular cases, and also
present a remarkably simple scheme of quantization for one of those cases.
|
Perturbation theory for the modified nonlinear Schr{\"o}dinger solitons | The perturbation theory based on the Riemann-Hilbert problem is developed for
the modified nonlinear Schr{\"o}dinger equation which describes the propagation
of femtosecond optical pulses in nonlinear single-mode optical fibers. A
detailed analysis of the adiabatic approximation to perturbation-induced
evolution of the soliton parameters is given. The linear perturbation and the
Raman gain are considered as examples.
|
Universal formats for nonlinear dynamical systems | It is demonstrated that very general nonlinear dynamical systems covering all
cases arising in practice can be brought down to rate equations of chemical
kinetics
|
Computation of conservation laws for nonlinear lattices | An algorithm to compute polynomial conserved densities of polynomial
nonlinear lattices is presented. The algorithm is implemented in Mathematica
and can be used as an automated integrability test. With the code diffdens.m,
conserved densities are obtained for several well-known lattice equations. For
systems with parameters, the code allows one to determine the conditions on
these parameters so that a sequence of conservation laws exist.
|
On the Lakshmanan and gauge equivalent counterpart of the
Myrzakulov-VIII equation | The Lakshmanan equivalent counterparts of the some Myrzakulov equations are
found.
|
Chiral Solitons in Generalized Korteweg-de Vries Equations | Generalizations of the Korteweg-de Vries equation are considered, and some
explicit solutions are presented. There are situations where solutions engender
the interesting property of being chiral, that is, of having velocity
determined in terms of the parameters that define the generalized equation,
with a definite sign.
|
Lax pairs for N=2,3 Supersymmetric KdV Equations and their Extensions | We present the Lax operator for the N=3 KdV hierarchy and consider its
extensions. We also construct a new infinite family of N=2 supersymmetric
hierarchies by exhibiting the corresponding super Lax operators. The new
realization of N=4 supersymmetry on the two general N=2 superfields, bosonic
spin 1 and fermionic spin 1/2, is discussed.
|
Surfaces, curves and the Lakshmanan equivalent counterparts of the some
Myrzakulov equations | The Lakshmanan equivalent counterparts of the some Myrzakulov equations are
found.
|
Supersymmetric Drinfeld-Sokolov reduction | The Drinfeld-Sokolov construction of integrable hierarchies, as well as its
generalizations, may be extended to the case of loop superalgebras. A
sufficient condition on the algebraic data for the resulting hierarchy to be
invariant under supersymmetry transformation is given. The method used is a
construction of the hierarchies in superspace, where supersymmetry is manifest.
Several examples are discussed.
|
3D symplectic map | Quantum 3D R-matrix in the classical (i.e. functional) limit gives a
symplectic map of dynamical variables. The corresponding 3D evolution model is
considered. An auxiliary problem for it is a system of linear equations playing
the role of the monodromy matrix in 2D models. A generating function for the
integrals of motion is constructed as a determinant of the auxiliary system.
|
The nondynamical r-matrix structure of the elliptic
Ruijsenaars-Schneider model with N=2 | We demonstrate that in a certain gauge the elliptic Ruijsenaars-Shneider
model with N=2 admits a nondynamical r-matrix structure and the corresponding
classical r-matrix is the same as that of its non-relativistic counterpart
(Calogero-Moser model) in the same gauge.The relation between our
(classical)Lax operator and the Lax operator given by Ruijsenaars is also
obtained.
|
Solitons, Surfaces, Curves, and the Spin Description of Nonlinear
Evolution Equations | The briefly review on the common spin description of the nonlinear evolution
equations.
|
Motion of Curves on Two Dimensional Surfaces and Soliton Equations | A connection is established between the soliton equations and curves moving
in a three dimensional space $V_{3}$. The sign of the self-interacting terms of
the soliton equations are related to the signature of $V_{3}$. It is shown that
there corresponds a moving curve to each soliton equations.
|
Extension of Hereditary Symmetry Operators | Two models of candidates for hereditary symmetry operators are proposed and
thus many nonlinear systems of evolution equations possessing infinitely many
commutative symmetries may be generated. Some concrete structures of hereditary
symmetry operators are carefully analyzed on the base of the resulting general
conditions and several corresponding nonlinear systems are explicitly given out
as illustrative examples.
|
Finsler-Geometrical Approach to the Studying of Nonlinear Dynamical
Systems | A two dimensional Finsler space associated with the differential equation
$y''=Y_3 y'^3+Y_2 y'^2+Y_1 y'+Y_0$ is characterized by a tensor equation and
called the Douglas space. An application to the Lorenz nonlinear dynamical
equation is discussed from the standpoint of Finsler geometry.
|
The Painlev\'e Integrability Test | The Painlev\'e test is a widely applied and quite successful technique to
investigate the integrability of nonlinear ODEs and PDEs by analyzing the
singularity structure of the solutions. The test is named after the French
mathematician Paul Painlev\'e ....
|
To the Gel'fand-Tsetlin realization of irreducible representations of
classical semisimple algebras | It is shown that the Gel'fand-Tsetlin realization of irreducible
representations of the $A_n$ algebra is directly connected with a linear
exactly integrable system in the n-dimensional space. General solution for this
system is explicitly given.
|
A Class of Coupled KdV systems and Their Bi-Hamiltonian Formulations | A Hamiltonian pair with arbitrary constants is proposed and thus a sort of
hereditary operators is resulted. All the corresponding systems of evolution
equations possess local bi-Hamiltonian formulation and a special choice of the
systems leads to the KdV hierarchy. Illustrative examples are given.
|
The solution of the N=(0|2) superconformal f-Toda lattice | The general solution of the two-dimensional integrable generalization of the
f-Toda chain with fixed ends is explicitly presented in terms of matrix
elements of various fundamental representations of the SL(n|n-1) supergroup.
The dominant role of the representation theory of graded Lie algebras in the
problem of constructing integrable mappings and lattices is demonstrated.
|
Towards second order Lax pairs to discrete Painlev\'e equations of first
degree | We investigate the question of finding discrete Lax pairs for the six
discrete Painlev\'e equations (Pn). The choice we make is to discretize the
pairs of Garnier, once converted to matricial form.
|
Rules of discretization for Painlev\'e equations | The discrete Painlev\'e property is precisely defined, and basic
discretization rules to preserve it are stated. The discrete Painlev\'e test is
enriched with a new method which perturbs the continuum limit and generates
infinitely many no-log conditions. A general, direct method is provided to
search for discrete Lax pairs.
|
All generalized SU(2) chiral models have spectral dependent Lax
formulation | The equations that define the Lax pairs for generalized principal chiral
models can be solved for any nondegenerate bilinear form on $su(2)$. The
solution is dependent on one free variable that can serve as the spectral
parameter.
|
On lump instability of Davey--Stewartson II equation | We show that lumps (solitons) of the Davey--Stewartson II equation fail under
small perturbations of initial data.
|
The system of three vortexes of two dimensional ideal hydrodinamics as a
new example of the (integrable) Nambu- Poisson mechanics | A Nambu-Poisson formulation of the system of three ordinary differential
equations describing dynamics of three vortexes of the ideal two-dimensional
hydrodynamics is given. The system is integrated by quadratures.
|
On the exact solutions of the Bianchi IX cosmological model in the
proper time | It has recently been argued that there might exist a four-parameter analytic
solution to the Bianchi IX cosmological model, which would extend the
three-parameter solution of Belinskii et al. to one more arbitrary constant. We
perform the perturbative Painlev\'e test in the proper time variable, and
confirm the possible existence of such an extension.
|
Determinant formula for the six-vertex model with reflecting end | Using the Quantum Inverse Scattering Method for the XXZ model with open
boundary conditions, we obtained the determinant formula for the six vertex
model with reflecting end.
|
The Gambier Mapping, Revisited | We examine critically the Gambier equation and show that it is the generic
linearisable equation containing, as reductions, all the second-order equations
which are integrable through linearisation. We then introduce the general
discrete form of this equation, the Gambier mapping, and present conditions for
its integrability. Finally, we obtain the reductions of the Gambier mapping,
identify their integrable forms and compute their continuous limits.
|
Again, Linearizable Mappings | We examine a family of 3-point mappings that include mappings solvable
through linearization. The different origins of mappings of this type are
examined: projective equations and Gambier systems. The integrable cases are
obtained through the application of the singularity confinement criterion and
are explicitly integrated.
|
The Gel'fand-Tsetlin Selection Rules and Representations of Quantum
Algebras | The problem of construction of irreducible representations of quantum $A^q_n$
algebras is solved at the level of explicit integration of the linear
(inhomogeneous) system in finite differences in the n-dimensional space. The
general solution of this system is given explicitly and particular ones, which
correspond to the irreducible representations are selected.
|
Knizhnik-Zamolodchikov-Bernard equations connected with the eight-vertex
model | Using quasiclassical limit of Baxter's 8 - vertex R - matrix, an elliptic
generalization of the Knizhnik-Zamolodchikov equation is constructed. Via
Off-Shell Bethe ansatz an integrable representation for this equation is
obtained. It is shown that there exists a gauge transformation connecting this
equation with Knizhnik-Zamolodchikov-Bernard equation for SU(2)-WZNW model on
torus.
|
A nonlinear indentity for the scattering phase of integrable models | A nonlinear identity for the scattering phase of quantum integrable models is
proved.
|
On the Miura and Backlund transformations associated with the
supersymmetric Gelfand-Dickey bracket | The supersymmetric version of the Miura and B\"acklund transformations
associated with the supersymmetric Gelfand-Dickey bracket are investigated from
the point of view of the Kupershmidt-Wilson theorem.
|
Hidden Algebra of Three-Body Integrable Systems | It is shown that all 3-body quantal integrable systems that emerge in the
Hamiltonian reduction method possess the same hidden algebraic structure. All
of them are given by a second degree polynomial in generators of an
infinite-dimensional Lie algebra of differential operators. It leads to new
families of the orthogonal polynomials in two variables.
|
Reduced Vectorial Ribaucour Transformation for the Darboux-Egoroff
Equations | The vectorial fundamental transformation for the Darboux equations is reduced
to the symmetric case. This is combined with the orthogonal reduction of Lame
type to obtain those vectorial Ribaucour transformations which preserve the
Egoroff reduction. We also show that a permutability property holds for all
these transformations. Finally, as an example, we apply these transformations
to the Cartesian background.
|
On Grassmannian Description of the Constrained KP Hierarchy | This note develops an explicit construction of the constrained KP hierarchy
within the Sato Grassmannian framework. Useful relations are established
between the kernel elements of the underlying ordinary differential operator
and the eigenfunctions of the associated KP hierarchy as well as between the
related bilinear concomitant and the squared eigenfunction potential.
|
From Ramond Fermions to Lame Equations for Orthogonal Curvilinear
Coordinates | We show how Ramond free neutral Fermi fields lead to a $\tau$-function theory
of BKP type which describes iso-orthogonal deformations of systems of ortogonal
curvilinear coordinates. We also provide a vertex operator representation for
the classical Ribaucour transformation.
|
Connection formulae for degenerated asymptotic solutions of the fourth
Painleve equation | All possible 1-parametric classical and transcendent degenerated solutions of
the fourth Painleve equation with the corresponding connection formulae of the
asymptotic parameters are described.
|
On integrability of a (2+1)-dimensional perturbed Kdv equation | A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma
and B. Fuchssteiner, is proven to pass the Painlev\'e test for integrability
well, and its 4$\times $4 Lax pair with two spectral parameters is found. The
results show that the Painlev\'e classification of coupled KdV equations by A.
Karasu should be revised.
|
Pfaffian form of the Grammian determinant solutions of the BKP hierarchy | The Grammian determinant type solutions of the KP hierarchy, obtained through
the vectorial binary Darboux transformation, are reduced, imposing suitable
differential constraint on the transformation data, to Pfaffian solutions of
the BKP hierarchy.
|
Pfaffian Solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY
sine-Gordon Equations | We reduce the vectorial binary Darboux transformation for the Manin-Radul
supersymmetric KdV system in such a way that it preserves the
Manin-Radul-Mathieu supersymmetric KdV equation reduction. Expressions in terms
of bosonic Pfaffians are provided for transformed solutions and wave functions.
We also consider the implications of these results for the supersymmetric
sine-Gordon equation.
|
Initial boundary value problem on a half-line for the MKdV equation | Initial boundary value problem on a half-line for the Modified KdV equation
is considered with the boundary conditions equal to zero at the origin and
initial condition chosen arbitrary decreasing rapidly enough and this problem
is plunged into the scheme of the inverse scattering method. Here the inverse
scattering problem is reduced to the Riemann problem on a system of rays on the
complex plane.
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Separation of Variables in the Elliptic Gaudin Model | For the elliptic Gaudin model (a degenerate case of XYZ integrable spin
chain) a separation of variables is constructed in the classical case. The
corresponding separated coordinates are obtained as the poles of a suitably
normalized Baker-Akhiezer function. The classical results are generalized to
the quantum case where the kernel of separating integral operator is
constructed. The simplest one-degree-of-freedom case is studied in detail.
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Non-classical symmetries and the singular manifold method: A further two
examples | This paper discusses two equations with the conditional Painleve property.
The usefulness of the singular manifold method as a tool for determining the
non-classical symmetries that reduce the equations to ordinary differential
equations with the Painleve property is confirmed once more
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An Approach to Master Symmetries of Lattice Equations | An approach to master symmetries of lattice equations is proposed by the use
of discrete zero curvature equation. Its key is to generate non-isospectral
flows from the discrete spectral problem associated with a given lattice
equation. A Volterra-type lattice hierarchy and the Toda lattice hierarchy are
analyzed as two illustrative examples.
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Polynomial rings of the chiral $SU(N)_{2}$ models | Via explicit diagonalization of the chiral $SU(N)_{2}$ fusion matrices, we
discuss the possibility of representing the fusion ring of the chiral SU(N)
models, at level K=2, by a polynomial ring in a single variable when $N$ is odd
and by a polynomial ring in two variables when $N$ is even.
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From One-Component KP Hierarchy to Two-Component KP Hierarchy and Back | We show that the system of the standard one-component KP hierarchy endowed
with a special infinite set of abelian additional symmetries, generated by
squared eigenfunction potentials, is equivalent to the two-component KP
hierarchy.
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Finite gap integration of a discrete Euler top | In [1] new discretizations of the Euler top have been found. They can be
discribed with a Lax pair with a spectral parameter on an elliptic curve. This
is used in this paper to perform a finite gap integration.
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On The KMS Condition for the critical Ising model | Using the KMS condition and exchange algebras we discuss the monodromy and
modular properties of two-point KMS states of the critical Ising model.
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On fusion algebra of chiral $SU(N)_{k}$ models | We discuss some algebraic setting of chiral $SU(N)_{k}$ models in terms of
the statistical dimensions of their fields. In particular, the conformal
dimensions and the central charge of the chiral $SU(N)_{k}$ models are
calculated from their braid matrices. Futhermore, at level K=2, we present the
characteristic polynomials of their fusion matrices in a factored form.
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