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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.
|
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 $U_{q}[sl(2)]$
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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
|
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
|
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.
|
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.
|
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 $A_l, D_l, E_l$.
|
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.
|
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.
|
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.
|
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
|
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.
|
Integrable KdV Systems: Recursion Operators of Degree Four | The recursion operator and bi-Hamiltonian formulation of the Drinfeld-
Sokolov system are given
|
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.
|
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.
|
On the Calogero model with negative harmonic term | The Calogero model with negative harmonic term is shown to be equivalent to
the set of negative harmonic oscillators. Two time-independent canonical
transformations relating both models are constructed: one based on the recent
results concerning quantum Calogero model and one obtained from dynamical
$sL(2,\rr)$ algebra. The two-particle case is discussed in some detail.
|
The symplectic structure of rational Lax pair systems | We consider dynamical systems associated to Lax pairs depending rationnally
on a spectral parameter. We show that we can express the symplectic form in
terms of algebro--geometric data provided that the symplectic structure on L is
of Kirillov type. In particular, in this case the dynamical system is
integrable.
|
Coverings and integrability of the Gauss-Mainardi-Codazzi equations | Using covering theory approach (zero-curvature representations with the gauge
group SL2), we insert the spectral parameter into the Gauss-Mainardi-Codazzi
equations in Tchebycheff and geodesic coordinates. For each choice, four
integrable systems are obtained.
|
A Realization of Discrete Geometry by String Model | A realization of discrete conjugate net is presented by using correlation
functions of strings in a gauge covariant form.
|
The periodic Lax operators $\cL$ of the equations of Benney type II | This text has been withdrawn by the author.
|
Symmetries of Discrete Dynamical Systems Involving Two Species | The Lie point symmetries of a coupled system of two nonlinear
differential-difference equations are investigated. It is shown that in special
cases the symmetry group can be infinite dimensional, in other cases up to 10
dimensional. The equations can describe the interaction of two long molecular
chains, each involving one type of atoms.
|
Spontaneous magnetization of the XXZ Heisenberg spin-1/2 chain | Determinant representations of form factors are used to represent the
spontaneous magnetization of the Heisenberg XXZ chain (Delta >1) on the finite
lattice as the ratio of two determinants. In the thermodynamic limit (the
lattice of infinite length), the Baxter formula is reproduced in the framework
of Algebraic Bethe Ansatz. It is shown that the finite size corrections to the
Baxter formula are exponentially small.
|
Extension of the bilinear formalism to supersymmetric KdV-type equations | Extending the gauge-invariance principle for \tau functions of the standard
bilinear formalism to the supersymmetric case, we define N=1 supersymmetric
Hirota operators. Using them, we bilinearize SUSY KdV-type equations (KdV,
Sawada-Kotera, Hirota-Satsuma). The solutions for multiple collisions of
super-solitons and extension to SUSY sine-Gordon are also discussed.
|
The Cole-Hopf and Miura transformations revisited | An elementary yet remarkable similarity between the Cole-Hopf transformation
relating the Burgers and heat equation and Miura's transformation connecting
the KdV and mKdV equations is studied in detail.
|
The classical Boussinesq hierarchy revisited | We develop a systematic approach to the classical Boussinesq (cBsq) hierarchy
based on an elementary polynomial recursion formalism. Moreover, the gauge
equivalence between the cBsq and AKNS hierarchies is studied in detail and used
to provide an effortless derivation of algebro-geometric solutions and their
theta function representations of the cBsq hierarchy.
|
On some soliton equations in 2+1 dimensions and their 1+1 and/or 2+0
dimensional integrable reductions | Some soliton equation in 2+1 dimensions and their 1+1 and/or dimensional
integrable reductions are considered.
|
Integral equations for the correlation functions of the quantum
one-dimensional Bose gas | The large time and long distance behavior of the temperature correlation
functions of the quantum one-dimensional Bose gas is considered. We obtain
integral equations, which solutions describe the asymptotics. These equations
are closely related to the thermodynamic Bethe Ansatz equations. In the low
temperature limit the solutions of these equations are given in terms of
observables of the model.
|
The relation between the Toda hierarchy and the KdV hierarchy | Under three relations connecting the field variables of Toda flows and that
of KdV flows, we present three new sequences of combination of the equations in
the Toda hierarchy which have the KdV hierarchy as a continuous limit. The
relation between the Poisson structures of the KdV hierarchy and the Toda
hierarchy in continuous limit is also studied.
|
The Lax pairs for the Holt system | By using non-canonical transformation between the Holt system and the
Henon-Heiles system the Lax pairs for all the integrable cases of the Holt
system are constructed from the known Lax representations for the Henon-Heiles
system.
|
The Camassa-Holm Equation: Conserved Quantities and the Initial Value
Problem | Using a Miura-Gardner-Kruskal type construction, we show that the
Camassa-Holm equation has an infinite number of local conserved quantities. We
explore the implications of these conserved quantities for global
well-posedness.
|
Universality of the distribution functions of random matrix theory | This paper first surveys the connection of integrable systems of the Painleve
type to various distribution functions appearing in Wigner-Dyson random matrix
theory. A short discussion is then given of the appearance of these same
distributions in other areas of mathematics.
|
Coupled KdV equations of Hirota-Satsuma type | It is shown that the system of two coupled Korteweg-de Vries equations passes
the Painlev\'e test for integrability in nine distinct cases of its
coefficients. The integrability of eight cases is verified by direct
construction of Lax pairs, whereas for one case it remains unknown.
|
Darboux-type transformations and hyperelliptic curves | We systematically study Darboux-type transformations for the KdV and AKNS
hierarchies and provide a complete account of their effects on hyperelliptic
curves associated with algebro-geometric solutions of these hierarchies.
|
Multi-soliton Solution of the Integrable Coupled Nonlinear Schrodinger
Equation of Manakov Type | The general multi-soliton solution of the integrable coupled nonlinear
Schrodinger equation (NLS) of Manakov type is investigated by using
Zakharov-Shabat (ZS) scheme. We get the bright and dark multi-soliton solution
using inverse scattering method of ZS scheme. Elastic and inelastic collision
of N-solitons solution of the equation are also discussed.
|
A note on the third family of N=2 supersymmetric KdV hierarchies | We propose a hamiltonian formulation of the $N=2$ supersymmetric KP type
hierarchy recently studied by Krivonos and Sorin. We obtain a quadratic
hamiltonian structure which allows for several reductions of the KP type
hierarchy. In particular, the third family of $N=2$ KdV hierarchies is
recovered. We also give an easy construction of Wronskian solutions of the KP
and KdV type equations.
|
On time-dependent symmetries and formal symmetries of evolution
equations | We present the explicit formulae, describing the structure of symmetries and
formal symmetries of any scalar (1+1)-dimensional evolution equation. Using
these results, the formulae for the leading terms of commutators of two
symmetries and two formal symmetries are found. The generalization of these
results to the case of system of evolution equations is also discussed.
|
Modified KP and Discrete KP | The discrete KP, or 1-Toda lattice hierarchy is the same as a properly
defined modified KP hierarchy.
|
Modular Invariants and Generalized Halphen Systems | Generalized Halphen systems are solved in terms of functions that uniformize
genus zero Riemann surfaces, with automorphism groups that are commensurable
with the modular group. Rational maps relating these functions imply subgroup
relations between their automorphism groups and symmetrization relations
between the associated differential systems.
|
Soliton equations in 2+1 dimensions: reductions, bilinearizations and
simplest solutions | Soliton equations in 2+1 and their 1+1 = 2+0 reductions are considered.
|
Baxter's Q-operator for the homogeneous XXX spin chain | Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator
for the homogeneous XXX model as integral operator in standard representation
of SL(2). The connection between Q-operator and local Hamiltonians is
discussed. It is shown that operator of Lipatov's duality symmetry arises
naturally as leading term of the asymptotic expansion of Q-operator for large
values of spectral parameter.
|
Vertex Operator Solutions of 2d Dimensionally Reduced Gravity | We apply algebraic and vertex operator techniques to solve two dimensional
reduced vacuum Einstein's equations. This leads to explicit expressions for the
coefficients of metrics solutions of the vacuum equations as ratios of
determinants. No quadratures are left. These formulas rely on the
identification of dual pairs of vertex operators corresponding to dual metrics
related by the Kramer-Neugebauer symmetry.
|
Functional relations and nested Bethe ansatz for sl(3) chiral Potts
model at q^2=-1 | We obtain the functional relations for the eigenvalues of the transfer matrix
of the sl(3) chiral Potts model for q^2=-1. For the homogeneous model in both
directions a solution of these functional relations can be written in terms of
roots of Bethe ansatz-like equations. In addition, a direct nested Bethe ansatz
has also been developed for this case.
|
Nambu--Poisson reformulation of the finite dimensional dynamical systems | In this paper we introduce a system of nonlinear ordinary differential
equations which in a particular case reduces to Volterra's system. We found in
two simplest cases the complete sets of the integrals of motion using
Nambu--Poisson reformulation of the Hamiltonian dynamics. In these cases we
have solved the systems by quadratures.
|
Integrable mixed vertex models from braid-monoid algebra | We use the braid-monoid algebra to construct integrable mixed vertex models.
The transfer matrix of a mixed SU(N) model is diagonalized by nested Bethe
ansatz approach.
|
Singularity Structure Analysis, Integrability, Solitons and Dromions in
(2+1)-Dimensional Zakharov Equations | The (2+1)-dimensional integrable Zakharov equations and their reductions are
considered
|
Bethe ansatz for the three-layer Zamolodchikov model | This paper is a continuation of our previous work (solv-int/9903001). We
obtain two more functional relations for the eigenvalues of the transfer
matrices for the $sl(3)$ chiral Potts model at $q^2=-1$. This model, up to a
modification of boundary conditions, is equivalent to the three-layer
three-dimensional Zamolodchikov model. From these relations we derive the Bethe
ansatz equations.
|
Multipeakons and a theorem of Stieltjes | A closed form of the multi-peakon solutions of the Camassa-Holm equation is
found using a theorem of Stieltjes on continued fractions. An explicit formula
is obtained for the scattering shifts.
|
A generalization of determinant formulas for the solutions of Painlev\'e
II and XXXIV equations | A generalization of determinant formulas for the classical solutions of
Painlev\'e XXXIV and Painlev\'e II equations are constructed using the
technique of Darboux transformation and Hirota's bilinear formalism. It is
shown that the solutions admit determinant formulas even for the transcendental
case.
|
On the Umemura Polynomials for the Painlev\'e III equation | A determinant expression for the rational solutions of the Painlev\'e III
(P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given.
Degeneration of this determinant expression to that for the rational solutions
of P$_{\rm II}$ is discussed by applying the coalescence procedure.
|
Canonicity of Baecklund transformation: r-matrix approach. I | For the Hamiltonian integrable systems governed by SL(2)-invariant r-matrix
(such as Heisenberg magnet, Toda lattice, nonlinear Schroedinger equation) a
general procedure for constructing Baecklund transformation is proposed. The
corresponding BT is shown to preserve the Poisson bracket. The proof is given
by a direct calculation using the r-matrix expression for the Poisson bracket.
|
The tetrahedral analog of Veneziano amplitude | In solv-int/9812016 it was shown that the Veneziano amplitude in string
theory comes naturally from one of the simplest solutions of the functional
pentagon equation (FPE). More generally, FPE is intimately connected with the
duality condition for scattering processes. Here I find the amplitude that
comes the same way from a solution of the functional tetrahedron equation, with
the duality replaced by the local Yang - Baxter equation.
|
Complex sine-Gordon Equation in Coherent Optical Pulse Propagation | It is shown that the McCall-Hahn theory of self-induced transparency in
coherent optical pulse propagation can be identified with the complex
sine-Gordon theory in the sharp line limit. We reformulate the theory in terms
of the deformed gauged Wess-Zumino-Witten sigma model and address various new
aspects of self-induced transparency.
|
On the M-XX equation | The (2+1)-dimensional integrable M-XX equation is considered.
|
Nonlinear waves, differential resultant, computer algebra and completely
integrable dynamical systems | The hierarchy of integrable equations are considered. The dynamical approach
to the theory of nonlinear waves is proposed. The special solutions(nonlinear
waves) of considered equations are derived. We use powerful methods of computer
algebra such differential resultant and others.
|
Compacton-like Solutions for Modified KdV and other Nonlinear Equations | We present compacton-like solution of the modified KdV equation and compare
its properties with those of the compactons and solitons. We further show that,
the nonlinear Schr{\"o}dinger equation with a source term and other higher
order KdV-like equations also possess compact solutions of the similar form.
|
Integrability Tests for Nonlinear Evolution Equations | Discusses several integrability tests for nonlinear evolution equations.
|
From Agmon-Kannai expansion to Korteweg-de Vries hierarchy | We present a new method for computation of the Korteweg-de Vries hierarchy
via heat invariants of the 1-dimensional Schrodinger operator. As a result new
explicit formulas for the KdV hierarchy are obtained. Our method is based on an
asymptotic expansion of resolvent kernels of elliptic operators due to S.Agmon
and Y.Kannai.
|
Bethe ansatz solution of the closed anisotropic supersymmetric U model
with quantum supersymmetry | The nested algebraic Bethe ansatz is presented for the anisotropic
supersymmetric $U$ model maintaining quantum supersymmetry. The Bethe ansatz
equations of the model are obtained on a one-dimensional closed lattice and an
expression for the energy is given.
|
Solutions of Non-linear Differential and Difference Equations with
Superposition Formulas | Matrix Riccati equations and other nonlinear ordinary differential equations
with superposition formulas are, in the case of constant coefficients, shown to
have the same exact solutions as their group theoretical discretizations.
Explicit solutions of certain classes of scalar and matrix Riccati equations
are presented as an illustration of the general results.
|
New integrable systems of derivative nonlinear Schr\"{o}dinger equations
with multiple components | The Lax pair for a derivative nonlinear Schr\"{o}dinger equation proposed by
Chen-Lee-Liu is generalized into matrix form. This gives new types of
integrable coupled derivative nonlinear Schr\"{o}dinger equations. By virtue of
a gauge transformation, a new multi-component extension of a derivative
nonlinear Schr\"{o}dinger equation proposed by Kaup-Newell is also obtained.
|
The KP Hierarchy in Miwa Coordinates | A systematic reformulation of the KP hierarchy by using continuous Miwa
variables is presented. Basic quantities and relations are defined and
determinantal expressions for Fay's identities are obtained. It is shown that
in terms of these variables the KP hierarchy gives rise to a Darboux system
describing an infinite-dimensional conjugate net.
|
Magnetization waves in Landau-Lifshitz Model | The solutions of the Landau-Lifshitz equation with finite-gap behavior at
infinity are considered. By means of the inverse scattering method the
large-time asymptotics is obtained.
|
$U_q(\hat{sl}_n)$-analog of the XXZ chain with a boundary | We study $U_q(\hat{sl}_n)$ analog of the XXZ spin chain with a boundary
magnetic field h. We construct explicit bosonic formulas of the vacuum vector
and the dual vacuum vector with a boundary magnetic field. We derive integral
formulas of the correlation functions.
|
On the explicit solutions of the elliptic Calogero system | Let $q_1,q_2,...,q_N$ be the coordinates of $N$ particles on the circle,
interacting with the integrable potential $\sum_{j<k}^N\wp(q_j-q_k)$, where
$\wp$ is the Weierstrass elliptic function. We show that every symmetric
elliptic function in $q_1,q_2,...,q_N$ is a meromorphic function in time. We
give explicit formulae for these functions in terms of genus $N-1$ theta
functions.
|
A System with a Recursion Operator but One Higher Local Symmetry of the
Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ | We construct a recursion operator for the system $(u_t,v_t)=(u_4+v^2,1/5
v_4)$, for which only one local symmetry is known and we show that the action
of the recursion operator on $(u_t,v_t)$ is a local function.
|
Soliton Collisions in the Ion Acoustic Plasma Equations | Numerical experiments involving the interaction of two solitary waves of the
ion acoustic plasma equations are described. An exact 2-soliton solution of the
relevant KdV equation was fitted to the initial data, and good agreement was
maintained throughout the entire interaction. The data demonstrates that the
soliton interactions are virtually elastic
|
Paraconformal Structures and Integrable Systems | We consider some natural connections which arise between right-flat (p, q)
paraconformal structures and integrable systems. We find that such systems may
be formulated in Lax form, with a "Lax p-tuple" of linear differential
operators, depending a spectral parameter which lives in (q-1)-dimensional
complex projective space. Generally, the differential operators contain partial
derivatives with respect to the spectral parameter.
|
Reduction of bihamiltonian systems and separation of variables: an
example from the Boussinesq hierarchy | We discuss the Boussinesq system with $t_5$ stationary, within a general
framework for the analysis of stationary flows of n-Gel'fand-Dickey
hierarchies. We show how a careful use of its bihamiltonian structure can be
used to provide a set of separation coordinates for the corresponding
Hamilton--Jacobi equations.
|
p-adic Difference-Difference Lotka-Volterra Equation and Ultra-Discrete
Limit | In this article, we have studied the difference-difference Lotka-Volterra
equations in p-adic number space and its p-adic valuation version. We pointed
out that the structure of the space given by taking the ultra-discrete limit is
the same as that of the $p$-adic valuation space.
|
Integrability of the higher-order nonlinear Schroedinger equation
revisited | Only the known integrable cases of the Kodama-Hasegawa higher-order nonlinear
Schroedinger equation pass the Painleve test. Recent results of Ghosh and Nandy
add no new integrable cases of this equation.
|
On the equivalence of the discrete nonlinear Schr\"odinger equation and
the discrete isotropic Heisenberg magnet | The equivalence of the discrete isotropic Heisenberg magnet (IHM) model and
the discrete nonlinear Schr\"odinger equation (NLSE) given by Ablowitz and
Ladik is shown. This is used to derive the equivalence of their discretization
with the one by Izergin and Korepin. Moreover a doubly discrete IHM is
presented that is equivalent to Ablowitz' and Ladiks doubly discrete NLSE.
|
On A Recently Proposed Relation Between oHS and Ito Systems | The bi-Hamiltonian structure of original Hirota-Satsuma system proposed by
Roy based on a modification of the bi-Hamiltonian structure of Ito system is
incorrect.
|
Integrable supersymmetric correlated electron chain with open boundaries | We construct an extended Hubbard model with open boundaries from a $R$-matrix
based on the $U_q[Osp(2|2)]$ superalgebra. We study the reflection equation and
find two classes of diagonal solutions. The corresponding one-dimensional open
Hamiltonians are diagonalized by means of the Bethe ansatz approach.
|
A construction for R-matrices without difference property in the
spectral parameter | A new construction is given for obtaining R-matrices which solve the
McGuire-Yang-Baxter equation in such a way that the spectral parameters do not
possess the difference property. A discussion of the derivation of the
supersymmetric U model is given in this context such that applied chemical
potential and magnetic field terms can be coupled arbitrarily. As a limiting
case the Bariev model is obtained.
|
Symmetric Linear Backlund Transformation for Discrete BKP and DKP
equation | Proper lattices for the discrete BKP and the discrete DKP equaitons are
determined. Linear B\"acklund transformation equations for the discrete BKP and
the DKP equations are constructed, which possesses the lattice symmetries and
generate auto-B\"acklund transformations
|
Multi-Field Integrable Systems Related to WKI-Type Eigenvalue Problems | Higher flows of the Heisenberg ferromagnet equation and the
Wadati-Konno-Ichikawa equation are generalized into multi-component systems on
the basis of the Lax formulation. It is shown that there is a correspondence
between the multi-component systems through a gauge transformation. An
integrable semi-discretization of the multi-component higher Heisenberg
ferromagnet system is obtained.
|
Miura Map between Lattice KP and its Modification is Canonical | We consider the Miura map between the lattice KP hierarchy and the lattice
modified KP hierarchy and prove that the map is canonical not only between the
first Hamiltonian structures, but also between the second Hamiltonian
structures.
|
Classical Solutions Generating Tree Form-Factors in Yang-Mills,
Sin(h)-Gordon and Gravity | Classical solutions generating tree form-factors are defined and constructed
in various models.
|
Complete integrability of derivative nonlinear Schr\"{o}dinger-type
equations | We study matrix generalizations of derivative nonlinear Schr\"{o}dinger-type
equations, which were shown by Olver and Sokolov to possess a higher symmetry.
We prove that two of them are `C-integrable' and the rest of them are
`S-integrable' in Calogero's terminology.
|
Determinant Formulas for the Toda and Discrete Toda Equations | Determinant formulas for the general solutions of the Toda and discrete Toda
equations are presented. Application to the $\tau$ functions for the Painlev\'e
equations is also discussed.
|
Towards the Lax formulation of SU(2) principal models with nonconstant
metric | The equations that define the Lax pairs for generalized principal chiral
models can be solved for any constant nondegenerate bilinear form on SU(2).
Necessary conditions for the nonconstant metric on SU(2) that define the
integrable models are given.
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Type II vertex operators for the $A_{n-1}^{(1)}$ face model | Presented is a free boson representation of the type II vertex operators for
the $A_{n-1}^{(1)}$ face model. Using the bosonization, we derive some
properties of the type II vertex operators, such as commutation, inversion and
duality relations.
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Discrete Z^a and Painleve equations | A discrete analogue of the holomorphic map z^a is studied. It is given by a
Schramm's circle pattern with the combinatorics of the square grid. It is shown
that the corresponding immersed circle patterns lead to special separatrix
solutions of a discrete Painleve equation. Global properties of these
solutions, as well as of the discrete $z^a$ are established.
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On Construction of Recursion Operators From Lax Representation | In this work we develop a general procedure for constructing the recursion
operators fro non-linear integrable equations admitting Lax representation.
Svereal new examples are given. In particular we find the recursion operators
for some KdV-type of integrable equations.
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On integrable deformations of the spherical top | The motion on the sphere $S^2$ with the potential $V= (x_1x_2x_3)^{-2/3}$ is
considered. The Lax representation and the linearisation procedure for this
two-dimensional integrable system are discussed.
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New integrable string-like fields in 1+1 dimensions | The symmetry classification method is applied to the string-like scalar
fields in two-dimensional space-time. When the configurational space is
three-dimensional and reducible we present the complete list of the systems
admiting higher polynomial symmetries of the 3rd, 4th and 5th-order.
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Lie point symmetries of integrable evolution equations and invariant
solutions | An integrable hierarchies connected with linear stationary Schr\"odinger
equation with energy dependent potentials (in general case) are considered.
Galilei-like and scaling invariance transformations are constructed. A symmetry
method is applied to construct invariant solutions.
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The Complex Bateman Equation | The general solution to the Complex Bateman equation is constructed. It is
given in implicit form in terms of a functional relationship for the unknown
function. The known solution of the usual Bateman equation is recovered as a
special case.
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