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solv-int/9808015

Harold Widom

Harold Widom (University of California, Santa Cruz)

On the Solution of a Painlev\'e III Equation

LaTeX file. 9 pages

solv-int math.FA nlin.SI

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 $\alpha$.

solv-int/9808016

Vsevolod Adler

V.E. Adler (Ufa Institute of Mathematics, Ufa, Russia)

Legendre transformations on the triangular lattice

13 pages, latex

Functional Analysis and Its Applications, 2000, Volume 34, Issue 1, pp 1-9

10.1007/BF02467062

solv-int nlin.SI

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.

solv-int/9808017

Christiane Quesne

Avinash Khare, C. Quesne

On Some One-Parameter Families of Three-Body Problems in One Dimension: Exchange Operator Formalism in Polar Coordinates and Scattering Properties

10 pages, LaTeX, no figure

Phys. Lett. A 250 (1998) 33-38

10.1016/S0375-9601(98)00752-X

ULB/229/CQ/98/7, IOP-BBSR/98-26

solv-int hep-th nlin.SI quant-ph

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.

solv-int/9808018

Kirill Vaninsky

K. L. Vaninsky

On Explicit Parametrisation of Spectral Curves for Moser-Calogero Particles and its Applications

20 pages, 6 pictures

Int.Math.Res.Not. 10 (1999) 509-529

solv-int hep-ph hep-th nlin.SI

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.

solv-int/9808019

James D. E. Grant

J.D.E. Grant and I.A.B. Strachan

Hypercomplex Integrable Systems

Latex file, 19 pages

Nonlinearity 12 (1999) 1247

10.1088/0951-7715/12/5/302

solv-int hep-th nlin.SI

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.

solv-int/9809001

Angela Foerster

Jon Links, Angela Foerster and Michael Karowski

Bethe ansatz solution of a closed spin 1 XXZ Heisenberg chain with quantum algebra symmetry

13 pages, LaTeX, to appear in J. Math. Phys

10.1063/1.532701

solv-int nlin.SI

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.

solv-int/9809002

Antonio Lima Santos

A. Lima-Santos and R.C.T. Ghiotto

A Bethe ansatz solution for the closed $U_{q}[sl(2)]$ Temperley-Lieb quantum spin chains

12 pages, tcilatex

J.Phys.A:Math.Gen.31 (1998)505-512

10.1088/0305-4470/31/2/011

solv-int cond-mat.str-el hep-th nlin.SI

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 $U_{q}[sl(2)]$

solv-int/9809003

Antonio Lima Santos

A. Lima-Santos

Osp(1|2) Off-shell Bethe Ansatz Equations

21 pages, LaTex, no significant changes

Nucl. Phys. B543 (1999)499-517

10.1016/S0550-3213(98)00861-X

solv-int hep-th nlin.SI

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.

solv-int/9809004

Fritz Gesztesy

Ronnie Dickson, Fritz Gesztesy, Karl Unterkofler

Algebro-Geometric Solutions of the Boussinesq Hierarchy

LaTeX, 48 pages

10.1142/S0129055X9900026X

solv-int nlin.SI

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.

solv-int/9809005

Fritz Gesztesy

Fritz Gesztesy and Rudi Weikard

Elliptic Algebro-Geometric Solutions of the KdV and AKNS Hierarchies - An Analytic Approach

LaTeX, 46 pages, to appear in Bull. A.M.S

solv-int nlin.SI

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.

solv-int/9809006

Maciej Dunajski

M. Dunajski, L.J. Mason and N.M.J. Woodhouse

From 2D Integrable Systems to Self-Dual Gravity

9 pages, LaTex, no figures

J.Phys.A: Math.Gen 31 (1998) 6019-6028

10.1088/0305-4470/31/28/015

solv-int nlin.SI

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.

solv-int/9809007

C. Chandre

C. Chandre

A comparison of two discrete mKdV equations

2 pages, REVTeX

Physica Scripta 55, 129 (1997)

10.1088/0031-8949/55/2/001

solv-int nlin.SI

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.

solv-int/9809008

Victor Enolskii

J C Eilbeck, V Z Enol'skii, V B Kuznetsov, D V Leykin

Linear r-Matrix Algebra for a Hierarchy of One-Dimensional Particle Systems Separable in Parabolic Coordinates

plain LaTeX, 28 pages

solv-int nlin.SI

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)

solv-int/9809009

Wen-Xiu Ma

Wen-Xiu Ma and Benno Fuchssteiner

Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations

24 pages, LaTex, revised

10.1063/1.532872

solv-int nlin.SI

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.

solv-int/9809010

Bireswar Basu-Mallick

B. Basu-Mallick

Symmetries and exact solutions of some integrable Haldane-Shastry like spin chains

35 pages, latex, no figures, minor type errors are corrected, version to appear in Nucl. Phys. B

10.1016/S0550-3213(98)00784-6

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

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.

solv-int/9809011

Antoine Balan

A.Balan (Ecole Polytechnique)

The Generalised mKdV Equations for Level -3 of $\hat{sl}_2$

solv-int nlin.SI

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.

solv-int/9809012

Gavrilov

Lubomir Gavrilov and Angel Zhivkov

The complex geometry of Lagrange top

LaTex, 26 pages

L'Enseignement Mathematique, tome 44 (1998) p.133-170

solv-int nlin.SI

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.

solv-int/9809013

R. S. Ward

R. S. Ward

Lax Pairs for Integrable Lattice Systems

15 pages, plainTeX, to appear in J Math Phys

10.1063/1.532772

DTP 98/59

solv-int nlin.SI

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.

solv-int/9809014

Nalini Joshi

N. Joshi, A. Ramani, B. Grammaticos

A Bilinear Approach to Discrete Miura Transformations

7 pages in TeX, to appear in Phys. Letts. A

10.1016/S0375-9601(98)00624-0

solv-int nlin.SI

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.

solv-int/9810001

Gavrilov

Lubomir Gavrilov

Generalized Jacobians of spectral curves and completely integrable systems

20 pages, LaTex2e, to appear in Math. Zeitschrift

Math. Zeitschrift, 230, 487-508 (1999)

solv-int nlin.SI

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.

solv-int/9810002

Daniel Arnaudon

D. Arnaudon and Z. Maassarani

Integrable open boundary conditions for XXC models

Latex2e, 10 pages

JHEP 10 (1998) 024

10.1088/1126-6708/1998/10/024

LAPTH-695/98, LAVAL-PHY-22/98

solv-int cond-mat.str-el math-ph math.MP nlin.SI

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.

solv-int/9810003

Anton Zabrodin

A.Zabrodin

Tau-function for discrete sine-Gordon equation and quantum R-matrix

14 pages, latex

ITEP-TH-55/98

solv-int hep-th nlin.SI

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.

solv-int/9810004

John Palmer

John Palmer

Zeros of the Jimbo, Miwa, Ueno tau function

59 pages

10.1063/1.533112

solv-int nlin.SI

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.

solv-int/9810005

Anca Visinescu

D. Grecu, A.S. C\^arstea, Anca Visinescu

Long range interaction corrections on the quantum vibronic soliton

Proceedings of the Conference "Path-Integral from pev to TeV. 50 years from Feynman's paper", Florence, August 1998, 4 pages, latex, sprocl.sty

IFA-FT-437(1998)

solv-int cond-mat nlin.SI

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.

solv-int/9810006

Alexandr Andrianov

A.A.Andrianov, M.V.Ioffe, D.N.Nishnianidze (St.Petersburg Univ.)

Classical Integrable 2-dim Models Inspired by SUSY Quantum Mechanics

19 pages, LaTeX, final version to be published in J.Phys.A

J.Phys.A32:4641,1999

10.1088/0305-4470/32/25/307

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

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.

solv-int/9810007

Paul Zinn-Justin

P. Zinn-Justin

Quelques applications de l'Ansatz de Bethe (Some applications of the Bethe Ansatz)

PhD dissertation. In French. the articles are not included (they're already on the archive)

solv-int nlin.SI

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.

solv-int/9810008

Doc. Dr. Ayse Humeyra Bilge

Ayse Humeyra Bilge and Fatma Ozdemir

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

solv-int nlin.SI

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.

solv-int/9810009

Olaf Lechtenfeld

Olaf Lechtenfeld and Alexander Sorin

Fermionic flows and tau function of the N=(1|1) superconformal Toda lattice hierarchy

11 pages, no figures, revised version published in Nucl. Phys. B

Nucl.Phys. B557 (1999) 535-547

10.1016/S0550-3213(99)00063-2

ITP-UH-23/98, JINR E2-98-285

solv-int hep-th math-ph math.MP nlin.SI

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.

solv-int/9810010

Juan A. Calzada

J.A. Calzada, M.A. del Olmo, M.A. Rodriguez

Pseudo-orthogonal groups and integrable dynamical systems in two dimensions

32 pages,revtex

10.1063/1.532768

solv-int nlin.SI

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.

solv-int/9810011

Martin Goliath

Martin Goliath, Max Karlovini, and Kjell Rosquist

Lax pair tensors in arbitrary dimensions

8 pages, uses IOP style files. Minor correction. Submitted to J. Phys A

10.1088/0305-4470/32/18/311

solv-int gr-qc nlin.SI

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.

solv-int/9810012

Manuel Manas

Adam Doliwa, Manuel Manas and Luis Martinez Alonso

Generating Quadrilateral and Circular Lattices in KP Theory

20 pages, 1 figure, LaTeX2e with AMSLaTeX, Babel, graphicx and psfrag packages

10.1016/S0375-9601(99)00579-4

solv-int nlin.SI

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.

solv-int/9810013

Gennady El

Gennady A. El, Alexander L.Krylov

Stochastic Soliton Lattices

11 pages. To be published in Proceedings of the International Conference `Solitons, Geometry and Topology: on the Crossroads', Moscow, 1998

solv-int nlin.SI

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

solv-int/9810014

A. David Trubatch

M. J. Ablowitz, Y. Ohta, A. D. Trubatch

On Discretizations of the Vector Nonlinear Schrodinger Equation

16 pages, 1 figure, 1 table

10.1016/S0375-9601(99)00048-1

APPM 349

solv-int nlin.SI

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.

solv-int/9810015

Antoine Balan

Antoine Balan (Ecole Polytechnique)

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

5 pages, LaTex, no figure

solv-int nlin.SI

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

solv-int/9810016

Nikita A. Slavnov

N. A. Slavnov (Steklov Mathematical Institute, Moscow, Russia)

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

14 pages, Latex, no figures

MI-98-06

solv-int nlin.SI

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.

solv-int/9810017

Robert Milson

N. Kamran and R. Milson

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

14 pages

10.1063/1.533012

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

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.

solv-int/9810018

Yuri B. Suris

A.I.Bobenko, Yu.B.Suris (Technische Universitaet Berlin)

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

LaTeX 2e, 44 pages, 1 figure

Commun. Math. Phys., 1999, V. 204, p. 147-188

10.1007/s002200050642

SFB288 preprint No. 345

solv-int nlin.SI

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.

solv-int/9810019

Faruk Gungor

F. Gungor (Istanbul Technical University)

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

16 pages, no figures, revised version

Physica Scripta, Vol. 61, 385-390, 2000

10.1238/Physica.Regular.061a00385

solv-int nlin.SI

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.

solv-int/9810020

A. David Trubatch

M. J. Ablowitz, Y. Ohta, A. D. Trubatch

On Integrability and Chaos in Discrete Systems

14 pages, 1 figure, 2 tables

APPM 350

solv-int nlin.SI

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.

solv-int/9811001

Minoru Wakimoto

K. Iohara (Kyoto Univ.), Y. Saito (Hiroshima Univ.), M. Wakimoto (Kyushu Univ.)

Hirota bilinear forms with 2-toroidal symmetry

11 pages, AMS-latex file

10.1016/S0375-9601(99)00093-6

solv-int nlin.SI

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 $A_l, D_l, E_l$.

solv-int/9811002

Angela Foerster

A. Lima-Santos, I. Roditi and A. Foerster

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

21 pages

Int. Jour. of Mod. Phys. A, Vol. 13, No. 25 (1998) 4309-4324

10.1142/S0217751X98002080

solv-int nlin.SI

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.

solv-int/9811003

Sergei M. Sergeev

S. M. Sergeev

Quantum 2+1 evolution model

LaTeX, 37pages

10.1088/0305-4470/32/30/313

Branch of INP preprint No. 98-02

solv-int nlin.SI

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.

solv-int/9811004

Ming-Hsien Tu

Ming-Hsien Tu, Jiin-Chang Shaw and Chin-Rong Lee

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

12 pages, Revtex, no figures

Lett. Math. Phys., 49(1): 33-45, July 1999

solv-int nlin.SI

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.

solv-int/9811005

Max Karlovini

Max Karlovini and Kjell Rosquist

A unified treatment of cubic invariants at fixed and arbitrary energy

16 pages, LaTeX2e, slightly revised version. To appear in J. Math. Phys. vol 41, pp 370-384 (2000)

10.1063/1.533137

SUITP 98-21

solv-int nlin.SI

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.

solv-int/9811006

Bernard Deconinck

Bernard Deconinck

Canonical variables for multiphase solutions of the KP equation

52 papes, 3 figures, uses psfig, latexsym

solv-int nlin.SI

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.

solv-int/9811007

G. Tondo

C. Morosi, G. Tondo

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

11 pages, AMS-LaTex 1.1

Phys. Lett. A 247 (1998), 59-64

10.1016/S0375-9601(98)00543-X

solv-int nlin.SI

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.

solv-int/9811008

G. Tondo

G. Tondo, C. Morosi

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

12 pages, no figures, LaTeX 2.09, to be published in Report on Mathematical Physics

10.1016/S0034-4877(99)80167-0

solv-int nlin.SI

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.

solv-int/9811009

Alex Kasman

Alex Kasman

Spectral Difference Equations Satisfied by KP Soliton Wavefunctions

to appear in "Inverse Problems"

10.1088/0266-5611/14/6/008

solv-int nlin.SI

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.

solv-int/9811010

Ming-Hsien Tu

Ming-Hsien Tu

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

9 pages, Revtex, no figures

Lett. Math. Phys., 49(2): 95-103, July 1999

solv-int nlin.SI

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.

solv-int/9811011

Pilar G. Estevez

Pilar Garcia Estevez

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

LaTeX 16 pages with 6 figures. Journal of Mathematical Physics (to appear)

University of Salamanca (SPAIN) preprint NLPG 25/98

solv-int nlin.SI

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

solv-int/9811012

R. S. Ward

R S Ward

Two Integrable Systems Related to Hyperbolic Monopoles

12 pages, plainTeX

DTP-98/77

solv-int nlin.SI

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.

solv-int/9811013

Metin Gurses

Metin Gurses and Atalay Karasu

Integrable KdV Systems: Recursion Operators of Degree Four

Latex File, to be published in Physucs Letters A

10.1016/S0375-9601(98)00910-4

solv-int nlin.SI

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

solv-int/9811014

Nalini Joshi

Peter A. Clarkson, Nalini Joshi and Andrew Pickering

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

12 pages in LaTeX 2.09 (uses ioplppt.sty), to appear in Inverse Problems

10.1088/0266-5611/15/1/019

solv-int nlin.SI

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.

solv-int/9811015

Faruk Gungor

Faruk Gungor

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

J. Phys. A: Math. and Gen. 32 (1999) 977-988

10.1088/0305-4470/32/6/010

solv-int nlin.SI

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.

solv-int/9811016

Chand Devchand

Chandrashekar Devchand and Jeremy Schiff

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

14 pages, latex file

J.Math.Phys. 42 (2001) 260-273

10.1063/1.1330196

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

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.

solv-int/9812001

Red Hat Linux User

Andrey Tsiganov

Duality between integrable Stackel systems

LaTeX2e, 18 pages

J. Phys. A, Math. Gen. 32, No.45, 7965-7982, (1999)

10.1088/0305-4470/32/45/311

solv-int nlin.SI

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.

solv-int/9812002

Ihabib

I.T.Habibullin and A.N.Vil'danov

Integrable boundary conditions for nonlinear lattices

11 pages

solv-int nlin.SI

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.

solv-int/9812003

Takayuki Tsuchida

T. Tsuchida, M. Wadati

The Coupled Modified Korteweg-de Vries Equations

26 pages, LaTex209 file, uses jpsj.sty

J. Phys. Soc. Jpn. 67 (1998) 1175-1187

10.1143/JPSJ.67.1175

solv-int nlin.SI

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.

solv-int/9812004

Pawel Maslanka

C.Gonera, P.Kosi\'nski, M.Majewski, P.Ma\'slanka

On Calogero wave functions

10 pages LaTeX2e file

solv-int nlin.SI

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.

solv-int/9812005

Pawel Maslanka

C.Gonera, M.Majewski, P.Ma\'slanka

On the Calogero model with negative harmonic term

6 pages LaTeX2e file

solv-int nlin.SI

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.

solv-int/9812006

Pierre van Moerbeke

M. Adler, T. Shiota, and P. van Moerbeke

Random matrices, Virasoro algebras, and noncommutative KP

56 pages

Duke Math Journal, 94, pp. 379-431, 1998

solv-int nlin.SI

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

solv-int/9812007

Robert Conte

R. Conte (CEA Saclay)

Perturbative methods for the Painlev\'e test

11 pages, no figure, standard Latex, to appear in the proceedings of ``Nonlinear dynamics: integrability and chaos'', Tiruchirapalli, 12--16 Feb 1998, ed. S. Daniel

S98/048

solv-int nlin.SI

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

solv-int/9812008

Robert Conte

R. Conte (CEA Saclay)

Various truncations in Painlev\'e analysis of PDEs

16 pages, no figure, standard Latex, to appear in the proceedings of ``Nonlinear dynamics: integrability and chaos'', Tiruchirapalli, 12--16 Feb 1998, ed. S. Daniel

S98/047

solv-int nlin.SI

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

solv-int/9812009

Michel Talon

O. Babelon, M. Talon

The symplectic structure of rational Lax pair systems

8 pages, no figure, Latex

10.1016/S0375-9601(99)00298-4

PAR LPTHE 98-52

solv-int nlin.SI

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.

solv-int/9812010

Michal Marvan

Joseph Krasil'shchik and Michal Marvan

Coverings and integrability of the Gauss-Mainardi-Codazzi equations

15 pages, LaTeX 2e

DIPS-8/98, ESI 639 (1998), GA 10/1998

solv-int nlin.SI

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.

solv-int/9812011

F. Nijhoff

F. W. Nijhoff, A. Ramani, B. Grammaticos and Y. Ohta

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

60+2 pages, Latex2.09, uses equations.sty

solv-int nlin.SI

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

solv-int/9812012

Zoran Rajilic

S. Lekic, S. Galamic, Z. Rajilic

Optical Fiber Communications:Group of the Nonlinear Transformations

LaTex2e, eps figure, Presented at the conference "Physics-21",St. Petersburg 1998

solv-int nlin.SI

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

solv-int/9812013

Lafortune

S.Lafortune, P.Winternitz and C.R.Menyuk

Solutions to the Optical Cascading Equations

21 pages

Physical Review E 58, 2518-2825 (1998)

10.1103/PhysRevE.58.2518

solv-int nlin.SI physics.optics

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

solv-int/9812014

Hendry Izaac Elim

Hans J. Wospakrik and Freddy P. Zen

Inhomogeneous Burgers Equation and the Feynman-Kac Path Integral

12 pages

solv-int hep-th nlin.SI

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

solv-int/9812015

Satoru Saito

Satoru Saito

A Realization of Discrete Geometry by String Model

LaTeX, 9 pages

solv-int nlin.SI

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

solv-int/9812016

Igor

I.G. Korepanov and S. Saito

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

LaTeX, 12 pages, 6 eps figures

Theor.Math.Phys. 120 (1999) 862-869; Teor.Mat.Fiz. 120 (1999) 54-63

10.1007/BF02557395

solv-int hep-th math-ph math.MP nlin.SI

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

solv-int/9812017

Ming-Hsien Tu

Ming-Hsien Tu and Jiin-Chang Shaw

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

16 pages, Revtex, no figures

J. Math. Phys. 40 (1999) 3021

10.1063/1.532741

solv-int nlin.SI

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

solv-int/9812018

Antoine Balan

A. Balan, (Ecole Polytechnique)

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

14 pages, nofigure. WITHDRAWN by the author

solv-int nlin.SI

This text has been withdrawn by the author.

solv-int/9812019

David Gomez-Ullate

D. Gomez-Ullate, S. Lafortune and P. Winternitz

Symmetries of Discrete Dynamical Systems Involving Two Species

40 pages, no figures, typed in AMS-LaTeX

J. Math. Phys. 40 (1999) 2782-2804

10.1063/1.532728

CRM-2567

solv-int nlin.SI

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.

solv-int/9812020

V. E. Vekslerchik

V.E. Vekslerchik

Functional representation of the Ablowitz-Ladik hierarchy. II

arxiv version is already official

J. Nonlinear Math. Phys. 9, no. 2 (2002) 157-180

10.2991/jnmp.2002.9.2.3

solv-int nlin.SI

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

solv-int/9812021

Maillet Jean Michel

A.G. Izergin, N. Kitanine, J. M. Maillet, V. Terras

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

18 pages, Latex2e

Nucl. Phys. B 554 (1999) 679-696

10.1016/S0550-3213(99)00273-4

LPENSL-TH-13/98

solv-int nlin.SI

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.

solv-int/9812022

Adrian-Stefan Carstea

A.S.Carstea

Extension of the bilinear formalism to supersymmetric KdV-type equations

11 pages, revtex, no figures, some corrected typos

solv-int nlin.SI

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.

solv-int/9812023

Yavuz Nutku

Y. Nutku

Hamiltonian structure of real Monge-Amp\`ere equations

published in J. Phys. A 29 (1996) 3257

10.1088/0305-4470/29/12/029

solv-int nlin.SI

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

solv-int/9812024

Lafortune

S. Lafortune, B. Grammaticos and A. Ramani

Discrete and Continuous Linearizable Equations

Plain Tex file, 14 pages, no figure

Physica A, 268, 129-141 (1999)

10.1016/S0378-4371(99)00026-6

solv-int nlin.SI

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

solv-int/9812025

Fritz Gesztesy

Fritz Gesztesy and Helge Holden

The Cole-Hopf and Miura transformations revisited

LaTeX, 11 pages

solv-int nlin.SI

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.

solv-int/9812026

Fritz Gesztesy

Fritz Gesztesy and Helge Holden

The classical Boussinesq hierarchy revisited

LaTeX, 17 pages

solv-int nlin.SI

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.

solv-int/9812027

Nugmanova G. N.

F.B.Altynbaeva, A.K.Danlybaeva, G.N.Nugmanova and R.N.Syzdykova

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

18 pages, Latex, no figures

solv-int nlin.SI

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

solv-int/9812028

Nikita A. Slavnov

N. A. Slavnov (Steklov Mathematical Institute, Moscow, Russia)

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

22 pages, Latex, no figures

10.1007/BF02557233

MI-98-91

solv-int nlin.SI

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.

solv-int/9812029

Lin Runliang

Yunbo Zeng, Runliang Lin and Xin Cao (Tsinghua University, Beijing, P.R. China)

The relation between the Toda hierarchy and the KdV hierarchy

11 pages, Tex, no figures, to be published in Physics Letters A

10.1016/S0375-9601(98)00886-X

solv-int nlin.SI

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.

solv-int/9812030

Andrey V. Tsiganov

A.V. Tsiganov

The Lax pairs for the Holt system

7 pages, LaTeX2e, a4.sty

J. Phys. A, Math. Gen. 32, No.45, 7983-7987, (1999)

10.1088/0305-4470/32/45/312

solv-int nlin.SI

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.

solv-int/9812031

O. B. Zaslavskii

O.B.Zaslavskii (Department of Physics, Kharkov State University)

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

7 pages, ReVTeX. Talk given at the 22nd International Colloqium for Group Theoretical Methods in Physics

`Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics', Eds S P Corney, R Delbourgo and P D Jarvis (Cambridge, MA: International Press) 1998, pp.234-238

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

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

solv-int/9901001

Jeremy Schiff

Michael Fisher, Jeremy Schiff

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

8 pages, LaTeX

10.1016/S0375-9601(99)00466-1

solv-int nlin.SI

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.

solv-int/9901002

Marcio J. Martins

M.J. Martins

Unified algebraic Bethe ansatz for two-dimensional lattice models

plain latex, 9 pages

10.1103/PhysRevE.59.7220

UFSCARF-TH-98-33

solv-int nlin.SI

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

solv-int/9901003

Craig A. Tracy

Craig A. Tracy and Harold Widom

Universality of the distribution functions of random matrix theory

11 pages, 3 figures

Statistical Physics on the Eve of the 21st Century: In Honour of J B McGuire on the Occasion of His 65th Birthday, eds. M. T. Batchelor and L. T. Wille, World Scientific Pub., 1999, pgs. 230-239.

solv-int math-ph math.MP nlin.SI

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.

solv-int/9901004

Craig A. Tracy

Craig A. Tracy and Harold Widom

Airy Kernel and Painleve II

14 pages, 1 figure. References updated in second version

in "Isomonodromic Deformations and Applications in Physics," eds. A. Its and J. Harnad, CRM Proceedings & Lecture Notes, Vol. 31, Amer. Math. Soc., Providence, 2002, pp. 85-98.

solv-int math-ph math.MP nlin.SI

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

solv-int/9901005

Sergei Yu. Sakovich

Sergei Yu. Sakovich

Coupled KdV equations of Hirota-Satsuma type

J. Nonlinear Math. Phys. 6 (1999) 255-262

10.2991/jnmp.1999.6.3.2

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

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.

solv-int/9901006

Alfred Ramani

B. Grammaticos (Paris VII) and A. Ramani (Ecole Polytechnique)

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

6 pages, Plain-TeX

solv-int nlin.SI

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

solv-int/9901007

David H. Sattinger

R. Beals, D.H. Sattinger, and J. Szmigielski

Acoustic Scattering and the Extended Korteweg deVries hierarchy

18 pages

Advances in Mathematics, vol 140, (1998), 190-206

solv-int nlin.SI

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

solv-int/9901008

Fritz Gesztesy

Fritz Gesztesy and Helge Holden

Darboux-type transformations and hyperelliptic curves

LaTeX, 27 pages

solv-int nlin.SI

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.

solv-int/9901009

A.N.W.Hone (Roma Tre)

Exact solutions of the associated Camassa-Holm equation

11 pages

solv-int nlin.SI

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

solv-int/9901010

Hendry Izaac Elim

Freddy P. Zen and Hendry I. Elim

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

15 pages, LaTeX2e, PACS 42.65Sf

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

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.

solv-int/9901011

F. Delduc

Fran\c{c}ois Delduc, L. Gallot

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

J. Nonlinear Math. Phys. 6 (1999), no. 3, 332-343

10.2991/jnmp.1999.6.3.8

JNMP 4/2002 (Article)

solv-int nlin.SI

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.

solv-int/9902001

Takeo Kojima

N. Fukushima (Waseda Univ.), T. Kojima (Nihon Univ.)

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

25 pages, LaTEX2e

J.Phys.A:Math.Gen.32,(1999) 6149-6168

10.1088/0305-4470/32/34/304

solv-int hep-th nlin.SI

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

solv-int/9902002

Artur Sergyeyev

Artur G. Sergyeyev (= Arthur G. Sergheyev) (Institute of Mathematics of NAS of Ukraine, Kyiv)

On time-dependent symmetries and formal symmetries of evolution equations

7 pages, Latex, no figures

Symmetry and perturbation theory (Rome, 1998), 303-308, World Sci. Publ., River Edge, NJ, 1999

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

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.

solv-int/9902003

Yuri B. Suris

Yuri B. Suris (TU Berlin)

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

LaTeX, 58 pp

solv-int nlin.SI

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