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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.
|
Integrable Extensions of N=2 Supersymmetric KdV Hierarchy Associated
with the Nonuniqueness of the Roots of the Lax operator | We preesent a new supersymmetric integrable extensions of the a=4,N=2 KdV
hierarchy. The root of the supersymmetric Lax operator of the KdV equation is
generalized, by including additional fields. This generalized root generate new
hierarchy of integrable equations, for which we investigate the hamiltonian
structure. In special case our system describes the interaction of the KdV
equation with the two MKdV equations.
|
Exact Kink Solitons in the Presence of Diffusion, Dispersion, and
Polynomial Nonlinearity | We describe exact kink soliton solutions to nonlinear partial differential
equations in the generic form u_{t} + P(u) u_{x} + \nu u_{xx} + \delta u_{xxx}
= A(u), with polynomial functions P(u) and A(u) of u=u(x,t), whose generality
allows the identification with a number of relevant equations in physics. We
emphasize the study of chirality of the solutions, and its relation with
diffusion, dispersion, and nonlinear effects, as well as its dependence on the
parity of the polynomials $P(u)$ and $A(u)$ with respect to the discrete
symmetry $u\to-u$. We analyze two types of kink soliton solutions, which are
also solutions to 1+1 dimensional phi^{4} and phi^{6} field theories.
|
Multiplicity A_m Models | Models generalizing the su(2) XX spin-chain were recently introduced. These
XXC models also have an underlying su(2) structure. Their construction method
is shown to generalize to the chains based on the fundamental representations
of the A_m Lie algebras. Integrability of the new models is shown in the
context of the quantum inverse scattering method. Their R-matrix is found and
shown to yield a representation of the Hecke algebra. The diagonalization of
the transfer matrices is carried out using the algebraic Bethe Ansatz. I
comment on eventual generalizations and possible links to reaction-diffusion
processes.
|
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.
|
Quantization of cohomology in semi-simple Lie algebras | The space of realizations of a finite-dimensional Lie algebra by first order
differential operators is naturally isomorphic to H^1 with coefficients in the
module of functions. The condition that a realization admits a
finite-dimensional invariant subspace of functions seems to act as a kind of
quantization condition on this H^1. It was known that this quantization of
cohomology holds for all realizations on 2-dimensional homogeneous spaces, but
the extent to which quantization of cohomology is true in general was an open
question. The present article presents the first known counter-examples to
quantization of cohomology; it is shown that quantization can fail even if the
Lie algebra is semi-simple, and even if the homogeneous space in question is
compact. A explanation for the quantization phenomenon is given in the case of
semi-simple Lie algebras. It is shown that the set of classes in H^1 that admit
finite-dimensional invariant subspaces is a semigroup that lies inside a
finitely-generated abelian group. In order for this abelian group be a discrete
subset of H^1, i.e. in order for quantization to take place, some extra
conditions on the isotropy subalgebra are required. Two different instances of
such necessary conditions are presented.
|
Singular sector of the KP hierarchy, $\bar{\partial}$-operators of
non-zero index and associated integrable systems | Integrable hierarchies associated with the singular sector of the KP
hierarchy, or equivalently, with $\dbar$-operators of non-zero index are
studied. They arise as the restriction of the standard KP hierarchy to
submanifols of finite codimension in the space of independent variables. For
higher $\dbar$-index these hierarchies represent themselves families of
multidimensional equations with multidimensional constraints. The
$\dbar$-dressing method is used to construct these hierarchies. Hidden KdV,
Boussinesq and hidden Gelfand-Dikii hierarchies are considered too.
|
Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP
Hierarchy | We use ideas of the geometry of bihamiltonian manifolds, developed by
Gel'fand and Zakharevich, to study the KP equations. In this approach they have
the form of local conservation laws, and can be traded for a system of ordinary
differential equations of Riccati type, which we call the Central System. We
show that the latter can be linearized by means of a Darboux covering, and we
use this procedure as an alternative technique to construct rational solutions
of the KP equations.
|
Imprimitively generated Lie-algebraic Hamiltonians and separation of
variables | Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose
domain is a subset of the Euclidean plane admits a separation of variables. A
proof of this conjecture is given in those cases where the generating
Lie-algebra acts imprimitively. The general form of the conjecture is false. A
counter-example is given based on the trigonometric Olshanetsky-Perelomov
potential corresponding to the A_2 root system.
|
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.
|
Phonon Scattering by Breathers in the Discrete Nonlinear Schroedinger
Equation | Linear theory for phonon scattering by discrete breathers in the discrete
nonlinear Schroedinger equation using the transfer matrix approach is
presented. Transmission and reflection coefficients are obtained as a function
of the wave vector of the input phonon. The occurrence of a nonzero
transmission, which in fact becomes perfect for a symmetric breather, is shown
to be connected with localized eigenmodes thresholds. In the weak-coupling
limit, perfect reflection are shown to exist, which requires two scattering
channels. A necessary condition for a system to have a perfect reflection is
also considered in a general context.
|
M\"obius invariant integrable lattice equations associated with KP and
2DTL hierarchies | Integrable lattice equations arising in the context of singular manifold
equations for scalar, multicomponent KP hierarchies and 2D Toda lattice
hierarchy are considered. These equation generate the corresponding continuous
hierarchy of singular manifold equations, its B\"acklund transformations and
different forms of superposition principles. They possess rather special form
of compatibility representation. The distinctive feature of these equations is
invariance under the action of M\"obius transformation. Geometric
interpretation of these discrete equations is given.
|
Geometric B\"acklund--Darboux transformations for the KP hierarchy | We shown that, if you have two planes in the Segal-Wilson Grassmannian that
have an intersection of finite codimension, then the corresponding solutions of
the KP hierarchy are linked by B\"acklund-Darboux transformations (BDT). The
pseudodifferential operator that performs this transformation is shown to be
built up in a geometric way from elementary BDT's and is given here in a closed
form. The geometric description of elementary DBT's requires that one has a
geometric interpretation of the dual wavefunctions involved. This is done here
with the help of a suitable algebraic characterization of the wavefunction. The
BDT's also induce transformations of the tau-function associated to a plane in
the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric
characterization of the BDT'ss that preserves these subsystems of the KP
hierarchy. This generalizes the classical Darboux-transformations. we also
determine an explicit expression for the squared eigenfunction potentials. Next
a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy.
It is shown that infinite flags in the Grassmannian yield solutions of the
latter hierarchy. these flags can be constructed by means of BDT's, starting
from some plane. Other applications of these BDT's are a geometric way to
characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy
and the construction of a vast collection of orthogonal polynomials, playing a
role in matrix models.
|
DNA Transcription Mechanism with a Moving Enzyme | Previous numerical investigations of an one-dimensional DNA model with an
extended modified coupling constant by transcripting enzyme are integrated to
longer time and demonstrated explicitly the trapping of breathers by DNA chains
with realistic parameters obtained from experiments. Furthermore, collective
coordinate method is used to explain a previously observed numerical evidence
that breathers placed far from defects are difficult to trap, and the motional
effect of RNA-polymerase is investigated.
|
Extending Hamiltonian Operators to Get Bi-Hamiltonian Coupled KdV
Systems | An analysis of extension of Hamiltonian operators from lower order to higher
order of matrix paves a way for constructing Hamiltonian pairs which may result
in hereditary operators. Based on a specific choice of Hamiltonian operators of
lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a
consequence of bi-Hamiltonian structure, they all possess infinitely many
symmetries and infinitely many conserved densities.
|
Schlesinger Transformations for Linearisable Equations | We introduce the Schlesinger transformations of the Gambier equation. The
latter can be written, in both the continuous and discrete cases, as a system
of two coupled Riccati equations in cascade involving an integer parameter n.
In the continuous case the parameter appears explicitly in the equation while
in the discrete case it corresponds to the number of steps for singularity
confinement. Two Schlesinger transformations are obtained relating the
solutions for some value $n$ to that corresponding to either n+1 or n+2.
|
Degenerate Frobenius manifolds and the bi-Hamiltonian structure of
rational Lax equations | The bi-Hamiltonian structure of certain multi-component integrable systems,
generalizations of the dispersionless Toda hierarchy, is studies for systems
derived from a rational Lax function. One consequence of having a rational
rather than a polynomial Lax function is that the corresponding bi-Hamiltonian
structures are degenerate, i.e. the metric which defines the Hamiltonian
structure has vanishing determinant. Frobenius manifolds provide a natural
setting in which to study the bi-Hamiltonian structure of certain classes of
hydrodynamic systems. Some ideas on how this structure may be extanded to
include degenerate bi-Hamiltonian structures, such as those given in the first
part of the paper, are given.
|
'Universality' of the Ablowitz-Ladik hierarchy | The aim of this paper is to summarize some recently obtained relations
between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It
has been shown that solutions of finite subsystems of the ALH can be used to
derive a wide range of solutions for, e.g., the 2D Toda lattice, nonlinear
Schr\"odinger, Davey-Stewartson, Kadomtsev-Petviashvili (KP) and some other
equations. Similar approach has been used to construct new integrable models:
O(3,1) and multi-field sigma models. Such 'universality' of the ALH becomes
more transparent in the framework of the Hirota's bilinear method. The ALH,
which is usually considered as an infinite set of differential-difference
equations, has been presented as a finite system of functional-difference
equations, which can be viewed as a generalization of the famous bilinear
identities for the KP tau-functions.
|
Dubrovin equations and integrable systems on hyperelliptic curves | We introduce the most general version of Dubrovin-type equations for divisors
on a hyperelliptic curve of arbitrary genus, and provide a new argument for
linearizing the corresponding completely integrable flows. Detailed
applications to completely integrable systems, including the KdV, AKNS, Toda,
and the combined sine-Gordon and mKdV hierarchies, are made. These
investigations uncover a new principle for 1+1-dimensional integrable soliton
equations in the sense that the Dubrovin equations, combined with appropriate
trace formulas, encode all hierarchies of soliton equations associated with
hyperelliptic curves. In other words, completely integable hierarchies of
soliton equations determine Dubrovin equations and associated trace formulas
and, vice versa, Dubrovin-type equations combined with trace formulas permit
the construction of hierarchies of soliton equations.
|
On the integrability of nonlinear partial differential equations | We investigate the integrability of Nonlinear Partial Differential Equations
(NPDEs). The concepts are developed by firstly discussing the integrability of
the KdV equation. We proceed by generalizing the ideas introduced for the KdV
equation to other NPDEs. The method is based upon a linearization principle
which can be applied on nonlinearities which have a polynomial form. We
illustrate the potential of the method by finding solutions of the (coupled)
nonlinear Schr\"{o}dinger equation and the Manakov equation which play an
important role in optical fiber communication. Finally, it is shown that the
method can also be generalized to higher-dimensions.
|
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.
|
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
|
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.
|
$A_n^{(1)}$ Toda Solitons: a Relation between Dressing transformations
and Vertex Operators | Affine Toda equations based on simple Lie algebras arise by imposing zero
curvature condition on a Lax connection which belongs to the corresponding loop
Lie algebra in the principal gradation. In the particular case of $A_n^{(1)}$
Toda models, we exploit the symmetry of the underlying linear problem to
calculate the dressing group element which generates arbitrary $N$-soliton
solution from the vacuum. Starting from this result we recover the vertex
operator representation of the soliton tau functions.
|
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.
|
Berezinian Construction of Super-Solitons in Supersymmetric Constrained
KP Hierarchies | We consider a broad class of consistently reduced Manin-Radul supersymmetric
KP hierarchies (MR-SKP) which are supersymmetric analogs of the ordinary
bosonic constrained KP models. Compatibility of these reductions with the MR
fermionic isospectral flows is achieved via appropriate modification of the
latter preserving their (anti-)commutation algebra. Unlike the general
unconstrained MR-SKP case, Darboux-Backlund transformations do preserve the
fermionic isospectral flows of the reduced MR-SKP hierarchies. This allows for
a systematic derivation of explicit Berezinian solutions for the
super-tau-functions (super-solitons) for these models.
|
Relationship Between the Energy Eigenstates of Calogero-Sutherland
Models With Oscillator and Coulomb-like Potentials | We establish a simple algebraic relationship between the energy eigenstates
of the rational Calogero-Sutherland model with harmonic oscillator and
Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in
both of these models which plays a crucial role in such an identification.
Further, we show that our analysis is in fact valid for any many-particle
system in arbitrary dimensions whose potential term (apart from the oscillator
or the Coulomb-like potential) is a homogeneous function of coordinates of
degree -2. The explicit coordinate transformation which maps the Coulomb-like
problem to the oscillator one has also been determined in some specific cases.
|
Darboux transformations for twisted so(p,q) system and local isometric
immersion of space forms | For the n-dimensional integrable system with a twisted so(p,q) reduction,
Darboux transformations given by Darboux matrices of degree 2 are constructed
explicitly. These Darboux transformations are applied to the local isometric
immersion of space forms with flat normal bundle and linearly independent
curvature normals to give the explicit expression of the position vector. Some
examples are given from the trivial solutions and standard imbedding T^n\to
R^{2n}.
|
Equations of Geodesic Deviation and the Inverse Scattering Transform | Solutions of equations of geodesic deviation in three- and four- dimensional
spaces obtained by the inverse scattering transform are considered. It is shown
that in the case of three-dimensional space solutions of geodesic deviation
equations are reduced to solutions of the well-known Zakharov-Shabat problem.
In four- dimensional space system of geodesic deviation equations is associated
with $3\times 3$ matrix Schr\"{o}dinger equation, and dependence on parameters
defined by the nonlinear equations of three-wave interaction.
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The construction of Frobenius manifolds from KP tau-functions | Frobenius manifolds (solutions of WDVV equations) in canonical coordinates
are determined by the system of Darboux-Egoroff equations. This system of
partial differential equations appears as a specific subset of the
$n$-component KP hierarchy. KP representation theory and the related Sato
infinite Grassmannian are used to construct solutions of this Darboux-Egoroff
system and the related Frobenius manifolds. Finally we show that for these
solutions Dubrovin's isomonodromy tau-function can be expressed in the KP
tau-function.
|
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.
|
Binary Nonlinearization of AKNS Spectral Problem under Higher-Order
Symmetry Constraints | Binary nonlinearization of AKNS spectral problem is extended to the cases of
higher-order symmetry constraints. The Hamiltonian structures, Lax
representations, $r$-matrices and integrals of motion in involution are
explicitly proposed for the resulting constrained systems in the cases of the
first four orders. The obtained integrals of motion are proved to be
functionally independent and thus the constrained systems are completely
integrable in the Liouville sense.
|
Localized solitons of hyperbolic su(N) AKNS system | Using the nonlinear constraint and Darboux transformation methods, the
(m_1,...,m_N) localized solitons of the hyperbolic su(N) AKNS system are
constructed. Here "hyperbolic su(N)" means that the first part of the Lax pair
is F_y=JF_x+U(x,y,t)F where J is constant real diagonal and U^*=-U. When
different solitons move in different velocities, each component U_{ij} of the
solution U has at most m_i m_j peaks as t tends to infinity. This corresponds
to the (M,N) solitons for the DSI equation. When all the solitons move in the
same velocity, U_{ij} still has at most m_i m_j peaks if the phase differences
are large enough.
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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.
|
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.
|
Inversion of the linearized Korteweg-deVries equation at the
multi-soliton solutions | Uniform estimates for the decay structure of the $n$-soliton solution of the
Korteweg-deVries equation are obtained. The KdV equation, linearized at the
$n$-soliton solution is investigated in a class $\WW$ consisting of sums of
travelling waves plus an exponentially decaying residual term. An analog of the
kernel of the time-independent equation is proposed, leading to solvability
conditions on the inhomogeneous term. Estimates on the inversion of the
linearized KdV equation at the $n$-soliton are obtained.
|
On the Solution of a Painlev\'e III Equation | In a 1977 paper of McCoy, Tracy and Wu there appeared for the first time the
solution of a Painlev\'e equation in terms of Fredholm determinants of integral
operators. This equation is $\psi''(t)+t^{-1}\psi'(t)=(1/2) \sinh 2\psi+2\alpha
t^{-1} \sinh\psi$, a special case of the Painlev\'e III equation. The proof in
the cited paper is complicated, and the purpose of this note is to give a more
straightforward one. First we give an equivalent formulation of the solution in
terms of the kernel ${e^{-t (x+x^{-1})/2}\over x+y}\Big|{x-1\over
x+1}\Big|^{2\alpha}$. There are already in the literature relatively simple
proofs of the fact that when $\alpha=0$ Fredholm determinants of this kernel
give solutions to the equation. We extend this result here to general $\alpha$.
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Legendre transformations on the triangular lattice | The main purpose of the paper is to demonstrate that condition of invariance
with respect to the Legendre transformations allows effectively isolate the
class of integrable difference equations on the triangular lattice, which can
be considered as discrete analogues of relativistic Toda type lattices. Some of
obtained equations are new, up to the author knowledge. As an example, one of
them is studied in more details, in particular, its higher continuous
symmetries and zero curvature representation are found.
|
On Some One-Parameter Families of Three-Body Problems in One Dimension:
Exchange Operator Formalism in Polar Coordinates and Scattering Properties | We apply the exchange operator formalism in polar coordinates to a
one-parameter family of three-body problems in one dimension and prove the
integrability of the model both with and without the oscillator potential. We
also present exact scattering solution of a new family of three-body problems
in one dimension.
|
On Explicit Parametrisation of Spectral Curves for Moser-Calogero
Particles and its Applications | The system of $N$ classical particles on the line with the Weierstrass $\wp$
function as potential is known to be completely integrable. Recently D'Hoker
and Phong found a beautiful parameterization by the polynomial of degree $N$ of
the space of Riemann surfaces associated with this system. In the trigonometric
limit of the elliptic potential these Riemann surfaces degenerate into rational
curves. The D'Hoker-Phong polynomial in the limit describes the intersection
points of the rational curves. We found an explicit determinant representation
of the polynomial in the trigonometric case. We consider applications of this
result to the theory of Toeplitz determinants and to geometry of the spectral
curves. We also prove our earlier conjecture on the asymptotic behavior of the
ratio of two symplectic volumes when the number of particles tends to infinity.
|
Hypercomplex Integrable Systems | In this paper we study hypercomplex manifolds in four dimensions. Rather than
using an approach based on differential forms, we develop a dual approach using
vector fields. The condition on these vector fields may then be interpreted as
Lax equations, exhibiting the integrability properties of such manifolds. A
number of different field equations for such hypercomplex manifolds are
derived, one of which is in Cauchy-Kovaleskaya form which enables a formal
general solution to be given. Various other properties of the field equations
and their solutions are studied, such as their symmetry properties and the
associated hierarchy of conservation laws.
|
Bethe ansatz solution of a closed spin 1 XXZ Heisenberg chain with
quantum algebra symmetry | A quantum algebra invariant integrable closed spin 1 chain is introduced and
analysed in detail. The Bethe ansatz equations as well as the energy
eigenvalues of the model are obtained. The highest weight property of the Bethe
vectors with respect to U_q(sl(2)) is proved.
|
A Bethe ansatz solution for the closed $U_{q}[sl(2)]$ Temperley-Lieb
quantum spin chains | We solve the spectrum pf the closed Temperley-Lieb quantum spin chains using
the coordinate Bethe ansatz. These Hamiltonians are invariante under the
quantum group $U_{q}[sl(2)]$
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Osp(1|2) Off-shell Bethe Ansatz Equations | The semiclassical limit of the algebraic quantum inverse scattering method is
used to solve the theory of the Gaudin model. Via Off-shell Bethe ansatz
equations of an integrable representation of the graded osp(1|2) vertex model
we find the spectrum of N-1 independent Hamiltonians of Gaudin. Integral
representations of the N-point correlators are presented as solutions of the
Knizhnik-Zamolodchikov equation. These results are extended for highest
representations of the osp(1|2) Gaudin algebra.
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Algebro-Geometric Solutions of the Boussinesq Hierarchy | We continue a recently developed systematic approach to the Bousinesq (Bsq)
hierarchy and its algebro-geometric solutions. Our formalism includes a
recursive construction of Lax pairs and establishes associated
Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations
for analogs of Dirichlet and Neumann divisors. The principal aim of this paper
is a detailed theta function representation of all algebro-geometric
quasi-periodic solutions and related quantities of the Bsq hierarchy.
|
Elliptic Algebro-Geometric Solutions of the KdV and AKNS Hierarchies -
An Analytic Approach | We provide an overview of elliptic algebro-geometric solutions of the KdV and
AKNS hierarchies, with special emphasis on Floquet theoretic and spectral
theoretic methods. Our treatment includes an effective characterization of all
stationary elliptic KdV and AKNS solutions based on a theory developed by
Hermite and Picard.
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From 2D Integrable Systems to Self-Dual Gravity | We explain how to construct solutions to the self-dual Einstein vacuum
equations from solutions of various two-dimensional integrable systems by
exploiting the fact that the Lax formulations of both systems can be embedded
in that of the self-dual Yang--Mills equations. We illustrate this by
constructing explicit self-dual vacuum metrics on $\R^2\times \Sigma$, where
$\Sigma$ is a homogeneous space for a real subgroup of $SL(2, \C)$ associated
with the two-dimensional system.
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A comparison of two discrete mKdV equations | We consider here two discrete versions of the modified KdV equation. In one
case, some solitary wave solutions, B\"acklund transformations and integrals of
motion are known. In the other one, only solitary wave solutions were given,
and we supply the corresponding results for this equation. We also derive the
integrability of the second equation and give a transformation which links the
two models.
|
Linear r-Matrix Algebra for a Hierarchy of One-Dimensional Particle
Systems Separable in Parabolic Coordinates | We consider a hierarchy of many-particle systems on the line with polynomial
potentials separable in parabolic coordinates. The first non-trivial member of
this hierarchy is a generalization of an integrable case of the H\'enon-Heiles
system. We give a Lax representation in terms of $2\times 2$ matrices for the
whole hierarchy and construct the associated linear r-matrix algebra with the
r-matrix dependent on the dynamical variables. A Yang-Baxter equation of
dynamical type is proposed. Classical integration in a particular case is
carried out and quantization of the system is discussed with the help of
separation variables. This paper was published in the rary issues: Sfb 288
Preprint No. 110, Berlin and Nonlinear Mathematical Physics, {\bf 1(3)},
275-294 (1994)
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Algebraic Structure of Discrete Zero Curvature Equations and Master
Symmetries of Discrete Evolution Equations | An algebraic structure related to discrete zero curvature equations is
established. It is used to give an approach for generating master symmetries of
first degree for systems of discrete evolution equations and an answer to why
there exist such master symmetries. The key of the theory is to generate
nonisospectral flows $(\lambda_t=\lambda ^l, l\ge0)$ from the discrete spectral
problem associated with a given system of discrete evolution equations. Three
examples are given.
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Symmetries and exact solutions of some integrable Haldane-Shastry like
spin chains | By using a class of `anyon like' representations of permutation algebra,
which pick up nontrivial phase factors while interchanging the spins of two
lattice sites, we construct some integrable variants of $SU(M)$ Haldane-Shastry
(HS) spin chain. Lax pairs and conserved quantities for these spin chains are
also found and it is established that these models exhibit multi-parameter
deformed or nonstandard variants of $Y(gl_M)$ Yangian symmetry. Moreover, by
projecting the eigenstates of Dunkl operators in a suitable way, we derive a
class of exact eigenfunctions for such HS like spin chain and subsequently
conjecture that these exact eigenfunctions would lead to the highest weight
states associated with a multi-parameter deformed or nonstandard variant of
$Y(gl_M)$ Yangian algebra. By using this conjecture, and acting descendent
operator on the highest weight states associated with a nonstandard $Y(gl_2)$
Yangian algebra, we are able to find out the complete set of eigenvalues and
eigenfunctions for the related HS like spin-${1\over 2}$ chain. It turns out
that some additional energy levels, which are forbidden due to a selection rule
in the case of SU(2) HS model, interestingly appear in the spectrum of above
mentioned HS like spin chain having nonstandard $Y(gl_2)$ Yangian symmetry.
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The Generalised mKdV Equations for Level -3 of $\hat{sl}_2$ | A certain generalisation of the hierarchy of mKdV equations (modified KdV),
which forms an integrable system, is studied here. This generalisation is based
on a Lax operator associated to the equations, with principal components of
degrees between -3 and 0. The results are the following ones: 1) an isomorphism
between the space of jets of the system and a quotient of ${Sl}_2({\CC}((t)))$;
2) the fact that the monodromy matrixes of the Lax operators have, morover,
Poisson brackets given by the trigonometric r-matrix; 3) a definition of the
action of screening operators on the densities; 4) an identification of the
intersection of the kernel with the integrals of motion.
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The complex geometry of Lagrange top | We prove that the heavy symmetric top (Lagrange, 1788) linearizes on a
two-dimensional non-compact algebraic group -- the generalized Jacobian of an
elliptic curve with two points identified. This leads to a transparent
description of its complex and real invariant level sets. We also deduce, by
making use of a Baker-Akhiezer function, simple explicit formulae for the
general solution of Lagrange top.
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Lax Pairs for Integrable Lattice Systems | This paper studies the structure of Lax pairs associated with integrable
lattice systems (where space is a one-dimensional lattice, and time is
continuous). It describes a procedure for generating examples of such systems,
and emphasizes the features that are needed to obtain equations which are local
on the spatial lattice.
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A Bilinear Approach to Discrete Miura Transformations | We present a systematic approach to the construction of Miura transformations
for discrete Painlev\'e equations. Our method is based on the bilinear
formalism and we start with the expression of the nonlinear discrete equation
in terms of $\tau$-functions. Elimination of $\tau$-functions from the
resulting system leads to another nonlinear equation, which is a ``modified''
version of the original equation. The procedure therefore yields Miura
transformations. In this letter, we illustrate this approach by reproducing
previously known Miura transformations and constructing new ones.
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Generalized Jacobians of spectral curves and completely integrable
systems | Consider an ordinary differential equation which has a Lax pair
representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a
fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such
an equation can be considered as a completely integrable complex Hamiltonian
system. We show that the generic complex invariant manifold {A(x): det(A(x)-y
I)= P(x,y)} of this Lax pair is an affine part of a non-compact commutative
algebraic group---the generalized Jacobian of the spectral curve {(x,y):
P(x,y)=0} with its points at "infinity" identified. Moreover, for suitable
B(x), the Hamiltonian vector field defined by the Lax pairon the generalized
Jacobian is translation--invariant. We provide two examples in which the above
result applies.
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Integrable open boundary conditions for XXC models | The XXC models are multistate generalizations of the well known spin 1/2 XXZ
model. These integrable models share a common underlying su(2) structure. We
derive integrable open boundary conditions for the hierarchy of conserved
quantities of the XXC models . Due to lack of crossing unitarity of the
R-matrix, we develop specific methods to prove integrability. The symmetry of
the spectrum is determined.
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Tau-function for discrete sine-Gordon equation and quantum R-matrix | We prove that the tau-function of the integrable discrete sine-Gordon model
apart from the "standard" bilinar identities obeys a number of "non-standard"
ones. They can be combined into a bivector 3-dimensional difference equation
which is shown to contain Hirota's difference analogue of the sine-Gordon
equation and both auxiliary linear problems for it. We observe that this
equation is most naturally written in terms of the quantum R-matrix for the XXZ
spin chain and looks then like a relation of the "vertex-face correspondence"
type.
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Zeros of the Jimbo, Miwa, Ueno tau function | We introduce a family of local deformations for meromorphic connections on
the Riemann sphere in the neighborhood of a higher rank (simple) singularity.
Following a scheme introduced by Malgrange we use these local models to prove
that the zeros of the tau function introduced by Jimbo, Miwa and Ueno occur
precisely at those points in the deformation space at which a certain
Birkhoff-Riemann- Hilbert problem fails to have a solution.
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Long range interaction corrections on the quantum vibronic soliton | Self-localized modes in a quantum vibronic system, with long range
interaction of Kac-Baker type and interacting nonlinearly with an acoustical
phonon bath, is studied. One works in the coherent state approximation.
Following a procedure of Sarker and Krumhansl, the problem is reduced to a
nearest neighbours one. In the continuum limit the localized state satisfy a
mKdV equation. An approximate expression for its frequency is found.
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Classical Integrable 2-dim Models Inspired by SUSY Quantum Mechanics | A class of integrable 2-dim classical systems with integrals of motion of
fourth order in momenta is obtained from the quantum analogues with the help of
deformed SUSY algebra. With similar technique a new class of potentials
connected with Lax method is found which provides the integrability of
corresponding 2-dim hamiltonian systems. In addition, some integrable 2-dim
systems with potentials expressed in elliptic functions are explored.
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Quelques applications de l'Ansatz de Bethe (Some applications of the
Bethe Ansatz) | The Bethe Ansatz is a method that is used in quantum integrable models in
order to solve them explicitly. This method is explained here in a general
framework, which applies to 1D quantum spin chains, 2D statistical lattice
models (vertex models) and relativistic field theories with 1 space dimension
and 1 time dimension. The connection with quantum groups is expounded. Several
applications are then presented. Finite size corrections are calculated via two
methods: The Non-Linear Integral Equations, which are applied to the study of
the states of the affine Toda model with imaginary coupling, and their
interpolation between the high energy (ultra-violet) and low energy (infra-red)
regions; and the Thermodynamic Bethe Ansatz Equations, along with the
associated Fusion Equations, which are used to determine the thermodynamic
properties of the generalized multi-channel Kondo model. The latter is then
studied in more detail, still using the Bethe Ansatz and quantum groups, so as
to characterize the spectrum of the low energy excitations.
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Miura Transformations for Integrable Evolution Equations of the Form
$u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ | The paper is withdrawn due to an error in Section 3.2. The remaining of the
results are included in the preprint solv-int/9605004.
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Fermionic flows and tau function of the N=(1|1) superconformal Toda
lattice hierarchy | An infinite class of fermionic flows of the N=(1|1) superconformal Toda
lattice hierarchy is constructed and their algebraic structure is studied. We
completely solve the semi-infinite N=(1|1) Toda lattice and chain hierarchies
and derive their tau functions, which may be relevant for building
supersymmetric matrix models. Their bosonic limit is also discussed.
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Pseudo-orthogonal groups and integrable dynamical systems in two
dimensions | Integrable systems in low dimensions, constructed through the symmetry
reduction method, are studied using phase portrait and variable separation
techniques. In particular, invariant quantities and explicit periodic solutions
are determined. Widely applied models in Physics are shown to appear as
particular cases of the method.
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Lax pair tensors in arbitrary dimensions | A recipe is presented for obtaining Lax tensors for any n-dimensional
Hamiltonian system admitting a Lax representation of dimension n. Our approach
is to use the Jacobi geometry and coupling-constant metamorphosis to obtain a
geometric Lax formulation. We also exploit the results to construct integrable
spacetimes, satisfying the weak energy condition.
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Generating Quadrilateral and Circular Lattices in KP Theory | The bilinear equations of the $N$-component KP and BKP hierarchies and a
corresponding extended Miwa transformation allow us to generate quadrilateral
and circular lattices from conjugate and orthogonal nets, respectively. The
main geometrical objects are expressed in terms of Baker functions.
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Stochastic Soliton Lattices | We introduce a new concept, Stochastic Soliton Lattice, as a random process
generated by a finite-gap potential of the Shroedinger operator. We study the
basic properties of this stochastic process and consider its KdV evolution
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On Discretizations of the Vector Nonlinear Schrodinger Equation | Two discretizations of the vector nonlinear Schrodinger (NLS) equation are
studied. One of these discretizations, referred to as the symmetric system, is
a natural vector extension of the scalar integrable discrete NLS equation. The
other discretization, referred to as the asymmetric system, has an associated
linear scattering pair. General formulae for soliton solutions of the
asymmetric system are presented. Formulae for a constrained class of solutions
of the symmetric system may be obtained. Numerical studies support the
hypothesis that the symmetric system has general soliton solutions.
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The Lax operators $\cal L$ of the Benney type equations bound with the
circle | The Lax operators of the Benney type equations are studied on the circle. The
vectors fields of the Lax operators are showed to commute with each other
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Asymptotics of the Fredholm determinant associated with the correlation
functions of the quantum Nonlinear Schrodinger equation | The correlation functions of the quantum nonlinear Schrodinger equation can
be presented in terms of a Fredholm determinant. The explicit expression for
this determinant is found for the large time and long distance.
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Algebraic Exact Solvability of trigonometric-type Hamiltonians
associated to root systems | In this article, we study and settle several structural questions concerning
the exact solvability of the Olshanetsky-Perelomov quantum Hamiltonians
corresponding to an arbitrary root system. We show that these operators can be
written as linear combinations of certain basic operators admitting infinite
flags of invariant subspaces, namely the Laplacian and the logarithmic gradient
of invariant factors of the Weyl denominator. The coefficients of the
constituent linear combination become the coupling constants of the final
model. We also demonstr ate the $L^2$ completeness of the eigenfunctions
obtained by this procedure, and describe a straight-forward recursive procedure
based on the Freudenthal multiplicity formula for constructing the
eigenfunctions explicitly.
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Discrete time Lagrangian mechanics on Lie groups, with an application to
the Lagrange top | We develop the theory of discrete time Lagrangian mechanics on Lie groups,
originated in the work of Veselov and Moser, and the theory of Lagrangian
reduction in the discrete time setting. The results thus obtained are applied
to the investigation of an integrable time discretization of a famous
integrable system of classical mechanics, -- the Lagrange top. We recall the
derivation of the Euler--Poinsot equations of motion both in the frame moving
with the body and in the rest frame (the latter ones being less widely known).
We find a discrete time Lagrange function turning into the known continuous
time Lagrangian in the continuous limit, and elaborate both descriptions of the
resulting discrete time system, namely in the body frame and in the rest frame.
This system naturally inherits Poisson properties of the continuous time
system, the integrals of motion being deformed. The discrete time Lax
representations are also found. Kirchhoff's kinetic analogy between elastic
curves and motions of the Lagrange top is also generalised to the discrete
context.
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Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation | The purpose of this paper is to present a class of particular solutions of a
C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry
reduction. Using the subgroups of similitude group reduced ordinary
differential equations of second order and their solutions by a singularity
analysis are classified. In particular, it has been shown that whenever they
have the Painlev\'e property, they can be transformed to standard forms by
Moebius transformations of dependent variable and arbitrary smooth
transformations of independent variable whose solutions, depending on the
values of parameters, are expressible in terms of either elementary functions
or Jacobi elliptic functions.
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On Integrability and Chaos in Discrete Systems | The scalar nonlinear Schrodinger (NLS) equation and a suitable discretization
are well known integrable systems which exhibit the phenomena of ``effective''
chaos. Vector generalizations of both the continuous and discrete system are
discussed. Some attention is directed upon the issue of the integrability of a
discrete version of the vector NLS equation.
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Hirota bilinear forms with 2-toroidal symmetry | In this note, we compute Hirota bilinear forms arising from both homogeneous
and principal realization of vertex representations of 2-toroidal Lie algebras
of type $A_l, D_l, E_l$.
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Bethe ansatz solution of the anisotropic correlated electron model
associated with the Temperley-Lieb algebra | A recently proposed strongly correlated electron system associated with the
Temperley-Lieb algebra is solved by means of the coordinate Bethe ansatz for
periodic and closed boundary conditions.
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Quantum 2+1 evolution model | A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.
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On Darboux-B\"acklund Transformations for the Q-Deformed Korteweg-de
Vries Hierarchy | We study Darboux-B\"acklund transformations (DBTs) for the $q$-deformed
Korteweg-de Vries hierarchy by using the $q$-deformed pseudodifferential
operators. We identify the elementary DBTs which are triggered by the gauge
operators constructed from the (adjoint) wave functions of the hierarchy.
Iterating these elementary DBTs we obtain not only $q$-deformed Wronskian-type
but also binary-type representations of the tau-function to the hierarchy.
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A unified treatment of cubic invariants at fixed and arbitrary energy | Cubic invariants for two-dimensional Hamiltonian systems are investigated
using the Jacobi geometrization procedure. This approach allows for a unified
treatment of invariants at both fixed and arbitrary energy. In the geometric
picture the invariant generally corresponds to a third rank Killing tensor,
whose existence at a fixed energy value forces the metric to satisfy a
nonlinear integrability condition expressed in terms of a Kahler potential.
Further conditions, leading to a system of equations which is overdetermined
except for singular cases, are added when the energy is arbitrary. As solutions
to these equations we obtain several new superintegrable cases in addition to
the previously known cases. We also discover a superintegrable case where the
cubic invariant is of a new type which can be represented by an energy
dependent linear invariant. A complete list of all known systems which admit a
cubic invariant at arbitrary energy is given.
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Canonical variables for multiphase solutions of the KP equation | The KP equation has a large family of quasiperiodic multiphase solutions.
These solutions can be expressed in terms of Riemann-theta functions. In this
paper, a finite-dimensional canonical Hamiltonian system depending on a finite
number of parameters is given for the description of each such solution. The
Hamiltonian systems are completely integrable in the sense of Liouville. In
effect, this provides a solution of the initial-value problem for the
theta-function solutions. Some consequences of this approach are discussed.
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On a class of dynamical systems both quasi-bi-Hamiltonian and
bi-Hamiltonian | It is shown that a class of dynamical systems (encompassing the one recently
considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both
quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the
separability of these systems; the second one is obtained trough a non
canonical map whose form is directly suggested by the associated Nijenhuis
tensor.
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Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation
variables | We discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi
separability of a few dynamical systems. They are shown to admit, in their
natural phase space, a quasi-bi-Hamiltonian formulation of Pfaffian type. This
property allows us to straightforwardly recover a set of separation variables
for the corresponding Hamilton-Jacobi equation.
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Spectral Difference Equations Satisfied by KP Soliton Wavefunctions | The Baker-Akhiezer (wave) functions corresponding to soliton solutions of the
KP hierarchy are shown to satisfy eigenvalue equations for a commutative ring
of translational operators in the spectral parameter. In the rational limit,
these translational operators converge to the differential operators in the
spectral parameter previously discussed as part of the theory of
"bispectrality". Consequently, these translational operators can be seen as
demonstrating a form of bispectrality for the non-rational solitons as well.
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Q-deformed KP hierarchy: Its additional symmetries and infinitesimal
B\"acklund transformations | We study the additional symmetries associated with the $q$-deformed
Kadomtsev-Petviashvili ($q$-KP) hierarchy. After identifying the resolvent
operator as the generator of the additional symmetries, the $q$-KP hierarchy
can be consistently reduced to the so-called $q$-deformed constrained KP
($q$-cKP) hierarchy. We then show that the additional symmetries acting on the
wave function can be viewed as infinitesimal B\"acklund transformations by
acting the vertex operator on the tau-function of the $q$-KP hierarchy. This
establishes the Adler-Shiota-van Moerbeke formula for the $q$-KP hierarchy.
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Darboux Transformations and solutions for an equation in 2+1 dimensions | Painleve analysis and the singular manifold method are the tools used in this
paper to perform a complete study of an equation in 2+1 dimensions. This
procedure has allowed us to obtain the Lax pair, Darboux transformation and tau
functions in such a way that a plethora of different solutions with solitonic
behavior can be constructed iteratively
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Two Integrable Systems Related to Hyperbolic Monopoles | Monopoles on hyperbolic 3-space were introduced by Atiyah in 1984. This
article describes two integrable systems which are closely related to
hyperbolic monopoles: a one-dimensional lattice equation (the Braam-Austin or
discrete Nahm equation), and a soliton system in (2+1)-dimensional
anti-deSitter space-time.
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Integrable KdV Systems: Recursion Operators of Degree Four | The recursion operator and bi-Hamiltonian formulation of the Drinfeld-
Sokolov system are given
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B\"acklund transformations for the second Painlev\'e hierarchy: a
modified truncation approach | The second Painlev\'e hierarchy is defined as the hierarchy of ordinary
differential equations obtained by similarity reduction from the modified
Korteweg-de Vries hierarchy. Its first member is the well-known second
Painlev\'e equation, P2.
In this paper we use this hierarchy in order to illustrate our application of
the truncation procedure in Painlev\'e analysis to ordinary differential
equations. We extend these techniques in order to derive auto-B\"acklund
transformations for the second Painlev\'e hierarchy. We also derive a number of
other B\"acklund transformations, including a B\"acklund transformation onto a
hierarchy of P34 equations, and a little known B\"acklund transformation for P2
itself.
We then use our results on B\"acklund transformations to obtain, for each
member of the P2 hierarchy, a sequence of special integrals.
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Nonlinear Evolution Equations Invariant Under Schroedinger Group in
three-dimensional Space-time | A classification of all possible realizations of the Galilei,
Galilei-similitude and Schroedinger Lie algebras in three-dimensional
space-time in terms of vector fields under the action of the group of local
diffeomorphisms of the space $\R^3\times\C$ is presented. Using this result a
variety of general second order evolution equations invariant under the
corresponding groups are constructed and their physical significance are
discussed.
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The supersymmetric Camassa-Holm equation and geodesic flow on the
superconformal group | We study a family of fermionic extensions of the Camassa-Holm equation.
Within this family we identify three interesting classes: (a) equations, which
are inherently hamiltonian, describing geodesic flow with respect to an H^1
metric on the group of superconformal transformations in two dimensions, (b)
equations which are hamiltonian with respect to a different hamiltonian
structure and (c) supersymmetric flow equations. Classes (a) and (b) have no
intersection, but the intersection of classes (a) and (c) gives a candidate for
a new supersymmetric integrable system. We demonstrate the Painlev\'e property
for some simple but nontrivial reductions of this system.
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Duality between integrable Stackel systems | For the Stackel family of the integrable systems a non-canonical
transformation of the time variable is considered. This transformation may be
associated to the ambiguity of the Abel map on the corresponding hyperelliptic
curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax
representations and the dynamical r-matrix algebras are constructed. As an
examples the Henon-Heiles systems, integrable Holt potentials and the
integrable deformations of the Kepler problem are discussed in detail.
|
Integrable boundary conditions for nonlinear lattices | Integrable boundary conditions in 1+1 and 2+1 dimensions are discussed from
the higher symmetries point of view. Boundary conditions consistent with the
discrete Landau-Lifshitz model and infinite 2D Toda lattice are represented.
|
The Coupled Modified Korteweg-de Vries Equations | Generalization of the modified KdV equation to a multi-component system, that
is expressed by $(\partial u_i)/(\partial t) + 6 (\sum_{j,k=0}^{M-1} C_{jk} u_j
u_k) (\partial u_i)/(\partial x) + (\partial^3 u_{i})/(\partial x^3) = 0, i=0,
1, ..., M-1 $, is studied. We apply a new extended version of the inverse
scattering method to this system. It is shown that this system has an infinite
number of conservation laws and multi-soliton solutions. Further, the initial
value problem of the model is solved.
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On Calogero wave functions | Two properties of Calogero wave functions for rational Calogero models are
studied: (i) the representation of the wave functions in terms of the
exponential of Lassalle operators, (ii) the $sL(2,\rr)$ structure of the
Calogero--Moser wave functions.
|