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solv-int/9709003

Park Q.-Han

Q-Han Park, H.J. Shin (Kyunghee Univ.)

Matched Pulse Propagation in a Three-Level System

20 pages, Latex, 12 eps figure files some comments and references are added. postscript file with 12 figures can be obtained at http://photon.kyunghee.ac.kr/~qhpark/

10.1103/PhysRevA.57.4643

SNUTP 97-082

solv-int nlin.SI

The B\"{a}cklund transformation for the three-level Maxwell-Bloch equation is presented in the matrix potential formalism. By applying the B\"{a}cklund transformation to a constant electric field background, we obtain a general solution for matched pulses (a pair of solitary waves) which can emit or absorb a light velocity solitary pulse but otherwise propagate with their shapes invariant. In the special case, this solution describes a steady state pulse without emission or absorption, and becomes the matched pulse solution recently obtained by Hioe and Grobe. A nonlinear superposition rule is derived from the B\"{a}cklund transformation and used for the explicit construction of two solitons as well as nonabelian breathers. Various new features of these solutions are addressed. In particular, we analyze in detail the scattering of "invertons", a specific pair of different wavelength solitons one of which moving with the velocity of light. Unlike the usual case of soliton scattering, the broader inverton changes its sign through the scattering. Surprisingly, the light velocity inverton receives time advance through the scattering thereby moving faster than light, which however does not violate causality.

solv-int/9709004

Henrik Aratyn

H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman

Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy

LaTeX, 13pgs

10.1088/0305-4470/31/47/009

IFT-P.053/97,UICHEP-TH/97-7

solv-int hep-th nlin.SI

The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras leads to a variety of 1+1 soliton equations. By varying the rank of the underlying $sl (n)$ algebra as well as its gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy. The soliton solutions are then obtained as elements of the orbits of the dressing transformations constructed in terms of representations of the vertex operators of the affine $sl (n)$ algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of the constrained KP hierarchy.

solv-int/9709005

Juri Suris

Yuri B. Suris (University of Bremen)

Integrable discretizations for lattice systems: local equations of motion and their Hamiltonian properties

LaTeX, 89pp; section on modified Volterra added

Rev. Math. Phys., 1999, V.11, p. 727-822.

solv-int nlin.SI

We develop the approach to the problem of integrable discretization based on the notion of $r$--matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non--locality. We demonstrate how to overcome this drawback. Namely, we introduce the notion of localizing changes of variables and construct such changes of variables for a large number of examples, including the Toda and the relativistic Toda lattices, the Volterra lattice and its integrable perturbation, the second flows of the Toda and of the Volterra hierarchies, the modified Volterra lattice, the Belov-Chaltikian lattice, the Bogoyavlensky lattices, the Bruschi-Ragnisco lattice. We also introduce a novel class of constrained lattice KP systems, discretize all of them, and find the corresponding localizing change of variables. Pulling back the differential equations of motion under the localizing changes of variables, we find also (sometimes novel) integrable one-parameter perturbations of integrable lattice systems. Poisson properties of the localizing changes of variables are also studied: they produce interesting one-parameter deformations of the known Poisson algebras.

solv-int/9709006

Sergei M. Sergeev

S. M. Sergeev

Solutions of the functional tetrahedron equation connected with the local Yang -- Baxter equation for the ferro-electric

7 pages, LaTeX

solv-int nlin.SI

Local (or modified) Yang -- Baxter equation (LYBE) gives the functional map from the parameters of the weights in the left hand side to the parameters of the correspondent weights in the right hand side of LYBE. Such maps solve the functional tetrahedron equation. In this paper all the maps associated with LYBE of the ferro-electric type with single parameter in each weight matrix are classified.

solv-int/9709007

Manuel Manas

Q. P. Liu, Manuel Manas

Discrete Levy Transformations and Casorati Determinant Solutions of Quadrilateral Lattices

12 pages, LaTeX2e using AMSLaTeX package

10.1016/S0375-9601(97)00933-X

solv-int nlin.SI

Sequences of discrete Levy and adjoint Levy transformations for the multidimensional quadrilateral lattices are studied. After a suitable number of iterations we show how all the relevant geometrical features of the transformed quadrilateral lattice can be expressed in terms of multi-Casorati determinants. As an example we dress the Cartesian lattice.

solv-int/9709008

M. Lakshmanan

M. Lakshmanan

Nonlinear Physics: Integrability, Chaos and Beyond

52 pages, 17 figures, Latex (Lecture given at the IEEE International Workshop on "Visions of Nonlinear Science in the 21st Century" held at Sevilla, Spain on June 26, 1996) To appear in J. Franklin. Inst. (1997) and Int. J. Bifurcation and Chaos (1997), please e-mail Lakshmanan for figures (E-mail: [email protected])

10.1142/S0218127497001187

solv-int nlin.SI

Integrability and chaos are two of the main concepts associated with nonlinear physical systems which have revolutionized our understanding of them. Highly stable exponentially localized solitons are often associated with many of the important integrable nonlinear systems while motions which are sensitively dependent on initial conditions are associated with chaotic systems. Besides dramatically raising our perception of many natural phenomena, these concepts are opening up new vistas of applications and unfolding technologies: Optical soliton based information technology, magnetoelectronics, controlling and synchronization of chaos and secure communications, to name a few. These developments have raised further new interesting questions and potentialities. We present a particular view of some of the challenging problems and payoffs ahead in the next few decades by tracing the early historical events, summarizing the revolutionary era of 1950-70 when many important new ideas including solitons and chaos were realized and reviewing the current status. Important open problems both at the basic and applied levels are discussed.

solv-int/9709009

M. Lakshmanan

M. Lakshmanan, R. Myrzakulov, S. Vijayalakshmi and A. K. Danlybaeva

Motion of Curves and Surfaces and Nonlinear Evolution Equations in (2+1) Dimensions

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

J. Math. Phys. 39 , N7, 3765 (1998)

10.1063/1.532466

solv-int nlin.SI

It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical invariants then define topological conserved quantities. Underlying evolution equations are shown to be associated with a triad of linear equations. Our examples include Ishimori equation and Myrzakulov equations which are shown to be geometrically equivalent to Davey-Stewartson and Zakharov -Strachan (2+1) dimensional nonlinear Schr\"odinger equations respectively.

solv-int/9709010

Dr. Jeremy Schiff

Jeremy Schiff (Bar-Ilan University)

The Camassa-Holm Equation: A Loop Group Approach

19 pages, 7 figures; LaTeX with psfig

10.1016/S0167-2789(98)00099-2

solv-int nlin.SI

A map is presented that associates with each element of a loop group a solution of an equation related by a simple change of coordinates to the Camassa-Holm (CH) Equation. Certain simple automorphisms of the loop group give rise to Backlund transformations of the equation. These are used to find 2-soliton solutions of the CH equation, as well as some novel singular solutions.

solv-int/9709011

Kenji Kajiwara

Kenji Kajiwara and Yasuhiro Ohta

Determinant Structure of the Rational Solutions for the Painlev\'e IV Equation

19 pages, Latex, using theorem.sty

10.1088/0305-4470/31/10/017

solv-int nlin.SI

Rational solutions for the Painlev\'e IV equation are investigated by Hirota bilinear formalism. It is shown that the solutions in one hierarchy are expressed by 3-reduced Schur functions, and those in another two hierarchies by Casorati determinant of the Hermite polynomials, or by special case of the Schur polynomials.

solv-int/9709012

Yuji Kodama

Yuji Kodama

The Whitham Equations for Optical Communications: Mathematical Theory of NRZ

Latex 50 pages with 22 figures (figures are available in epsf)

solv-int nlin.SI

We present a model of optical communication system for high-bit-rate data transmission in the nonreturn-to-zero (NRZ) format over transoceanic distance. The system operates in a small group velocity dispersion regime, and the model equation is given by the Whitham equations describing the slow modulation of multi-phase wavetrains of the (defocusing) nonlinear Schr\"odinger (NLS) equation. The model equation is of hyperbolic type, and certain initial NRZ pulse with phase modulation develops a shock. We then show how one can obtain a global solution by choosing an appropriate Riemann surface on which the Whitham equation is defined. The present analysis may be interpreted as an alternative to the method of inverse scattering transformation for the NLS solitons. We also discuss wavelength-division-multiplexing (WDM) in the NRZ format by using the Whitham equation for a coupled NLS equation, and show that there exists a hydro-dynamic-type instability between channels.

solv-int/9709013

Sergei M. Sergeev

S. M. Sergeev

On a two dimensional system associated with the complex of the solutions of the Tetrahedron equation

14 pages, LaTeX. The references are defined more precisely

solv-int nlin.SI

A sort of two dimensional linear auxiliary problem for the complex of 3D $R$ -- operators associated with the Zamolodchikov -- Bazhanov -- Baxter statistical model is proposed. This problem resembles the problem of the local Yang -- Baxter equation but does not coincide with it. The formulation of the auxiliary problem admits a notion of a ``fusion'', and usual local Yang -- Baxter equation appears among other results of this ``fusion''.

solv-int/9710001

Toppan Francesco

Francesco Toppan (Shizuoka University, Japan)

Susy Hierarchies and Affine Algebras

11 pages, LaTex, uses lamuphys.sty: talk given at the UIC ``Supersymmetry and Integrable Systems Workshop'', Chicago, June 12-14 1997

10.1007/BFb0105325

solv-int hep-th nlin.SI

We review some basic features of the Lie-algebraic classification of W-algebras and related integrable hierarchies in 1+1 dimensions, pointing out the role of affine Lie algebras. We emphasize that the supersymmetric extensions of the above construction possibly lead, though some questions are still opened, to the classification of supersymmetric hierarchies based on ``generic'' supersymmetric affine Lie algebras. Here the word generic is used to make clear that well-known procedures, as those introduced by Inami and Kanno, are too restricted and do not lead to the full spectrum of supersymmetric integrable hierarchies one can construct. A particular attention is devoted to the large-N supersymmetric extensions (here N=4). The attention paid by large-N theories being due to the fact that they arise as dimensional reduction of N=1 models, and moreover that they realize an ``unification'' of known hierarchies.

solv-int/9710002

Nikolai Kitanine

N.M. Bogoliubov, A.G. Izergin, N.A. Kitanine

Correlation functions for a strongly correlated boson system

27 pages LaTeX

10.1016/S0550-3213(98)00038-8

PDMI PREPRINT - 17/1997

solv-int cond-mat hep-th math.QA nlin.SI q-alg

The correlation functions for a strongly correlated exactly solvable one-dimensional boson system on a finite chain as well as in the thermodynamic limit are calculated explicitly. This system which we call the phase model is the strong coupling limit of the integrable q-boson hopping model. The results are presented as determinants.

solv-int/9710003

V.V.Konotop, M.Salerno, S.Takeno

Shock waves in one-dimensional Heisenberg ferromagnets

10 pages, with 3 ps figures

10.1103/PhysRevB.58.14892

solv-int nlin.SI

We use SU(2) coherent state path integral formulation with the stationary phase approximation to investigate, both analytically and numerically, the existence of shock waves in the one- dimensional Heisenberg ferromagnets with anisotropic exchange interaction. As a result we show the existence of shock waves of two types,"bright" and "dark", which can be interpreted as moving magnetic domains.

solv-int/9710004

Alexander Turbiner

Alexander Turbiner

Two-body Elliptic Model in proper variables: Lie-algebraic forms and their discretizations

9 pages, AMSLaTeX, Contribution to the Proceedings of the Workshop on Calogero-Moser-Sutherland models, Montreal, March 10-15, 1997

ICN-UNAM 97-12

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

Two Lie algebraic forms of the 2-body Elliptic Calogero model are presented. Translation-invariant and dilatation-invariant discretizations of the model are obtained.

solv-int/9710005

Shen-Jane Chang

Jiin-Chang Shaw and Ming-Hsien Tu

Binary Darboux-Backlund Transformations for the Manin-Radul Super KdV Hierarchy

14 pages, Revtex, no figures, some typos corrected, two references added

J. Math. Phys. 39 (1998) 4773

10.1063/1.532536

solv-int nlin.SI

We construct the supersymmetric extensions of the Darboux-Backlund transformations (DBTs) for the Manin-Radul super KdV hierarchy using the super-pseudo-differential operators. The elementary DBTs are triggered by the gauge operators constructed from the wave functions and adjoint wave functions of the hierarchy. Iterating these elementary DBTs, we obtain not only Wronskian type but also binary type superdeterminant representations of the solutions.

solv-int/9710006

Peter Schupp

Branislav Jurco, Peter Schupp

AKS scheme for face and Calogero-Moser-Sutherland type models

24 pages, latex

10.1063/1.532453

PUPT-1731, CRM-2507, LMU-TPW 97-24

solv-int hep-th math.QA nlin.SI q-alg

We give the construction of quantum Lax equations for IRF models and difference versions of Calogero-Moser-Sutherland models introduced by Ruijsenaars. We solve the equations using factorization properties of the underlying face Hopf algebras/elliptic quantum groups. This construction is in the spirit of the Adler-Kostant-Symes method and generalizes our previous work to the case of face Hopf algebras/elliptic quantum groups with dynamical R-matrices.

solv-int/9710007

E. Alfinito, V. Grassi, R. A. Leo, G. Profilo and G. Soliani

Equations of the reaction-diffusion type with a loop algebra structure

16 pages, LaTex. submitted to Inverse Problems

Inv. Prob. 14, 1387-1401 (1998)

10.1088/0266-5611/14/6/003

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

A system of equations of the reaction-diffusion type is studied in the framework of both the direct and the inverse prolongation structure. We find that this system allows an incomplete prolongation Lie algebra, which is used to find the spectral problem and a whole class of nonlinear field equations containing the original ones as a special case.

solv-int/9710008

B.G. Konopelchenko, G. Landolfi

On classical string configurations

10 pages, Latex, no figures, trivial corrections, submitted to Modern Physics Letters A

10.1142/S0217732397003289

solv-int nlin.SI

Equations which define classical configurations of strings in $R^3$ are presented in a simple form. General properties as well as particular classes of solutions of these equations are considered.

solv-int/9710009

Ivan Avramidi

Ivan G. Avramidi and Rainer Schimming (University of Greifswald)

A new explicit expression for the Korteweg-De Vries hierarchy

17 pages, LaTeX, 37 KB, no figures

Math.Nachr. 219 (2000) 45-64

University of Greifswald (Oct. 1997)

solv-int hep-th nlin.SI

We derive an improved fully explicit expression for the right-hand sides of the matrix KdV hierarchy using the relation to the heat kernel of the one-dimensional Schr\"odinger operator. Our method of "matrix elements" produces, moreover, an explicit expression for the powers of a Schr\"odinger-like differential operator of any order.

solv-int/9710010

R. P. Malik

R.P.Malik

On Fifth Order KdV-Type Equation

12 pages, latex, (no figures)

solv-int hep-th nlin.SI

The dynamics of the highly nonlinear fifth order $KdV$-type equation is discussed in the framework of the Lagrangian and Hamiltonian formalisms. The symmetries of the Lagrangian produce three commuting conserved quantities that are found to be recursively related to one-another for a certain specific value of the power of nonlinearity. The above cited recursion relations are obeyed with a second Poisson bracket which sheds light on the integrability properties of the above nonlinear equation. It is shown that a Miura-type transformation can be made to obtain the fifth order $mKdV$-type equation from the fifth order $KdV$-type equation. The spatial dependence of the fields involved is, however, not physically interesting from the point of view of the solitonic solutions. As a consequence, it seems that the fifth order $KdV$- and $mKdV$-type equations are completely independent nonlinear evolution equations in their own right.

solv-int/9710011

Harry Braden

H. W. Braden

A Conjectured R-Matrix

12 pages Latex

10.1088/0305-4470/31/7/008

MS-97-013

solv-int nlin.SI

A new spectral parameter independent R-matrix (that depends on all of the dynamical variables) is proposed for the elliptic Calogero-Moser models. Necessary and sufficient conditions for this R-matrix to exist reduce to an equality between determinants of matrices involving elliptic functions. The needed identity appears new and is still unproven in full generality: we present it as a conjecture.

solv-int/9710012

John Harnad

J. Harnad

Hamiltonian Dynamics, Classical R-matrices and Isomonodromic Deformations

LaTeX 13pgs (requires lamuphys.sty). Text of talk given at workshop: Supersymmetric and Integrable Systems, University of Illinois, Chicago Circle, June 12-14, 1997. To appear in: Springer Lecture notes in Physics

Lect.Notes Phys.502:63-75,1998

10.1007/BFb0105314

CRM 2511 (1997)

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

The Hamiltonian approach to the theory of dual isomonodromic deformations is developed within the framework of rational classical R-matrix structures on loop algebras. Particular solutions to the isomonodromic deformation equations appearing in the computation of correlation functions in integrable quantum field theory models are constructed through the Riemann-Hilbert problem method. The corresponding $\tau$-functions are shown to be given by the Fredholm determinant of a special class of integral operators.

solv-int/9710013

Manna Miguel

M. A. Manna and V. Merle

Asymptotic dynamics of short-waves in nonlinear dispersive models

to appears in Physical Review E. 4 pages, revtex files

10.1103/PhysRevE.57.6206

solv-int nlin.SI

The multiple-scale perturbation theory, well known for long-waves, is extended to the study of the far-field behaviour of short-waves, commonly called ripples. It is proved that the Benjamin-Bona-Mahony- Peregrine equation can propagates short-waves. This result contradict the Benjamin hypothesis that short-waves tends not to propagate in this model and close a part of the old controversy between Korteweg-de Vries and Benjamin-Bona-Mahony-Peregrine equations. We shown that a nonlinear (quadratic) Klein-Gordon type equation substitutes in a short-wave analysis the ubiquitous Korteweg-de Vries equation of long-wave approach. Moreover the kink solutions of phi-4 and sine-Gordon equations are understood as an all orders asymptotic behaviour of short-waves. It is proved that the antikink solution of phi-4 model which was never obtained perturbatively can be obtained by perturbation expansion in the wave-number k in the short-wave limit.

solv-int/9710014

Manuel Manas

Q. P. Liu and Manuel Manas

Vectorial Ribaucour Transformations for the Lame Equations

12 pages. LaTeX2e with AMSLaTeX packages

J. Phys. A: Math. & Gen. 31 (1998) L193

10.1088/0305-4470/31/10/003

solv-int nlin.SI

The vectorial extension of the Ribaucour transformation for the Lame equations of orthogonal conjugates nets in multidimensions is given. We show that the composition of two vectorial Ribaucour transformations with appropriate transformation data is again a vectorial Ribaucour transformation, from which it follows the permutability of the vectorial Ribaucour transformations. Finally, as an example we apply the vectorial Ribaucour transformation to the Cartesian background.

solv-int/9710015

Anton Zabrodin

A.Zabrodin

Hidden quantum R-matrix in discrete time classical Heisenberg magnet

23 pages, latex, typos corrected

ITEP-TH-45/97

solv-int hep-th nlin.SI

We construct local M-operators for an integrable discrete time version of the classical Heisenberg magnet by convolution of the twisted quantum trigonometric 4$\times$4 R-matrix with certain vectors in its "quantum" space. Components of the vectors are identified with $\tau$-functions of the model. Hirota's bilinear formalism is extensively used. The construction generalizes the known representation of M-operators in continuous time models in terms of Lax operators and classical $r$-matrix.

solv-int/9710016

John Harnad

J. Harnad

Bispectral Operators, Dual Isomonodromic Deformations and the Riemann-Hilbert Dressing Method

AMSTeX 13pgs. Text of talk presented at the workshop on the Bispectral Problem, Centre de recherches mathematiques, Universite de Montreal, March 17--21, 1997. To appear in: CRM Proceedings and Lecture Notes series (1997/98)

CRM Proc. Lecture Notes 14, 67-79, (Amer. Math. Soc., Providence, RI, 1998)

CRM 2512 (1997)

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

A comparison is made between bispectral systems and dual isomonodromic deformation equations. A number of examples are given, showing how bispectral systems may be embedded into isomonodromic ones. Sufficiency conditions are given for the construction of rational solutions of isomonodromic deformation equations through the Riemann-Hilbert problem dressing method, and these are shown, in certain cases, to reduce to bispectral systems.

solv-int/9710017

Dr S. Chaturvedi

S. Chaturvedi

Jack polynomials, generalized binomial coefficients and polynomial solutions of the generalized Laplace's equation

19 pages, latex, no figures, 12 tables Minor typographical errors in some of the equations and the tables have been corrected

10.1142/S0217732398000772

solv-int nlin.SI

We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace's equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials and the constraints on the coefficients of expansion are derived. These constraints involve generalized binomial coefficients defined through Jack polynomials. Generalized binomial coefficients for partitions of $k$ upto $k=6$ are tabulated.

solv-int/9710018

Anjan Kundu

Anjan Kundu

Unifying structures in quantum integrable systems

Latex, 18 pages, no figure (Invited review article by Indian J.Phys.)

Indian J. Phys. 72B (1998) 283-299

SINP/TNP/97-16

solv-int nlin.SI

Basic concepts of quantum integrable systems (QIS) are presented stressing on the unifying structures underlying such diverse models. Variety of ultralocal and nonultralocal models is shown to be described by a few basic relations defining novel algebraic entries. Such properties can generate and classify integrable models systematically and also help to solve exactly their eigenvalue problem in an almost model-independent way. The unifying thread stretches also beyond the QIS to establish its deep connections with statistical models, conformal field theory etc. as well as with abstract mathematical objects like quantum group, braided or quadratic algebra

solv-int/9710019

Alex Kasman

Yu. Berest and A. Kasman

D-modules and Darboux transformations

to appear Lett. Math. Phys

CRM-2499

solv-int math.QA nlin.SI q-alg

A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of D-modules, leads to a new class of examples of commutative rings of partial differential operators with rational spectral varieties. As an application, we briefly discuss their link to the bispectral problem and to the theory of lacunas.

solv-int/9710020

Robert Conte

R. Conte (CEA Saclay)

The Painlev\'e approach to nonlinear ordinary differential equations

113 pages, no figure, standard Latex, to appear in The Painlev\'e property, one century later, ed. R. Conte, CRM series in mathematical physics (Springer--Verlag, Berlin, 1998) (Carg\`ese school, 3-22 June 1996)

S97/103

solv-int nlin.SI

The ``Painlev\'e analysis'' is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory of the (explicit) integration of nonlinear differential equations. To achieve our goal, we will not start the exposition with a more or less precise ``Painlev\'e test''. On the contrary, we will finish with it, after a gradual introduction to the rich world of singularities of nonlinear differential equations, so as to remove any cooking recipe. The emphasis is put on embedding each method of the test into the well known theorem of perturbations of Poincar\'e. A summary can be found at the beginning of each chapter.

solv-int/9710021

Nalini Joshi

Clio Cresswell and Nalini Joshi

The Discrete Painlev\'e I Hierarchy

9 pages in LaTeX. To appear in Proceedings of SIDEII, Kent, UK 1996, (eds) P.A.Clarkson and F.Nijhoff

solv-int nlin.SI

The discrete Painlev\'e I equation (dP$\rm_I$) is an integrable difference equation which has the classical first Painlev\'e equation (P$\rm_I$) as a continuum limit. dP$\rm_I$ is believed to be integrable because it is the discrete isomonodromy condition for an associated (single-valued) linear problem. In this paper, we derive higher-order difference equations as isomonodromy conditions that are associated to the same linear deformation problem. These form a hierarchy that may be compared to hierarchies of integrable ordinary differential equations (ODEs). We strengthen this comparison by continuum limit calculations that lead to equations in the P$\rm_I$ hierarchy. We propose that our difference equations are discrete versions of higher-order Painlev\'e equations.

solv-int/9710022

Nalini Joshi

Nalini Joshi

The Second Painlev\'e Equation in the Large-Parameter Limit I: Local Asymptotic Analysis

30 pages in LaTeX2e. Submitted

solv-int nlin.SI

In this paper, we find all possible asymptotic behaviours of the solutions of the second Painlev\'e equation $y''=2y^3+xy +\alpha$ as the parameter $\alpha\to\infty$ in the local region $x\ll\alpha^{2/3}$. We prove that these are asymptotic behaviours by finding explicit error bounds. Moreover, we show that they are connected and complete in the sense that they correspond to all possible values of initial data given at a point in the local region.

solv-int/9710023

Nalini Joshi

Martin D.Kruskal, Nalini Joshi, and Rod Halburd

Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property

40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B. Grammaticos and K. Tamizhmani

10.1007/BFb0113696

solv-int nlin.SI

The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable equations have the Painlev\'e property, that is, all solutions are single-valued around all movable singularities. In this expository article, we review methods for analysing such singularity structure. In particular, we describe well known techniques of nonlinear regular-singular-type analysis, i.e. the Painlev\'e tests for ordinary and partial differential equations. Then we discuss methods of obtaining sufficiency conditions for the Painlev\'e property. Recently, extensions of \textit{irregular} singularity analysis to nonlinear equations have been achieved. Also, new asymptotic limits of differential equations preserving the Painlev\'e property have been found. We discuss these also.

solv-int/9710024

Choong-Ki You

C. Ahn and C.K. You

Complete Nondiagonal Reflection Matrices of RSOS/SOS and Hard Hexagon Models

18pages,Latex

10.1088/0305-4470/31/9/003

solv-int hep-th nlin.SI

In this paper we compute the most general nondiagonal reflection matrices of the RSOS/SOS models and hard hexagon model using the boundary Yang-Baxter equations. We find new one-parameter family of reflection matrices for the RSOS model in addition to the previous result without any parameter. We also find three classes of reflection matrices for the SOS model, which has one or two parameters. For the hard hexagon model which can be mapped to RSOS(5) model by folding four RSOS heights into two, the solutions can be obtained similarly with a main difference in the boundary unitarity conditions. Due to this, the reflection matrices can have two free parameters. We show that these extra terms can be identified with the `decorated' solutions. We also generalize the hard hexagon model by `folding' the RSOS heights of the general RSOS(p) model and show that they satisfy the integrability conditions such as the Yang- Baxter and boundary Yang-Baxter equations. These models can be solved using the results for the RSOS models.

solv-int/9710025

Krzysztof Gawedzki

Krzysztof Gawedzki and Pascal Tran-Ngoc-Bich

Self-duality of the SL_2 Hitchin integrable system at genus two

32 pages, latex, no figures, references and a discussion inspired by one of them added

10.1007/s002200050438

IHES/P/97/80

solv-int alg-geom hep-th math.AG nlin.SI

We revisit the Hitchin integrable system whose phase space is the bundle cotangent to the moduli space $N$ of holomorphic $SL_2$-bundles over a smooth complex curve of genus two. $N$ may be identified with the 3-dimensional projective space of theta functions of the second order, We prove that the Hitchin system on $T^*N$ possesses a remarkable symmetry: it is invariant under the interchange of positions and momenta. This property allows to complete the work of van Geemen-Previato which, basing on the classical results on geometry of the Kummer quartic surfaces, specified the explicit form of the Hamiltonians of the Hitchin system. The resulting integrable system resembles the classic Neumann systems which are also self-dual. Its quantization produces a commuting family of differential operators of the second order acting on homogeneous polynomials in four complex variables. As recently shown by van Geemen-de Jong, these operators realize the Knizhnik-Zamolodchikov-Bernard-Hitchin connection for group SU(2) and genus 2 curves.

solv-int/9710026

Henrik Aratyn

Henrik Aratyn and Ashok Das

The sAKNS Hierarchy

LaTeX, 16 pgs

10.1142/S0217732398001261

solv-int hep-th nlin.SI

We study, systematically, the properties of the supersymmetric AKNS (sAKNS) hierarchy. In particular, we discuss the Lax representation in terms of a bosonic Lax operator and some special features of the equations and construct the bosonic local charges as well as the fermionic nonlocal charges associated with the system starting from the Lax operator. We obtain the Hamiltonian structures of the system and check the Jacobi identity through the method of prolongation. We also show that this hierarchy of equations can equivalently be described in terms of a fermionic Lax operator. We obtain the zero curvature formulation as well as the conserved charges of the system starting from this fermionic Lax operator which suggests a connection between the two. Finally, starting from the fermionic description of the system, we construct the soliton solutions for this system of equations through Darboux-Backlund transformations and describe some open problems.

solv-int/9710027

Roland Beutler

R. Beutler, B.G. Konopelchenko

Surfaces of Revolution via the Schroedinger Equation : Construction, Integrable Dynamics and Visualization

29 pages, 27 figures

solv-int nlin.SI

Surfaces of revolution in three-dimensional Euclidean space are considered. Several new examples of surfaces of revolution associated with well-known solvable cases of the Schoedinger equation (infinite well, harmonic oscillator, Coulomb potential, Bargmann potential, etc.) are analyzed and visualized. The properties of such surfaces are discussed. Two types of deformations (evolutions), namely 1) preserving the Gaussian curvature and 2) via the dynamics of the Korteweg-de-Vries equation are discussed.

solv-int/9710028

Nikolai Kitanine

A.G. Izergin, V.S. Kapitonov, N.A. Kitanine

Equal-time temperature correlators of the one-dimensional Heisenberg XY chain

25 pages, LaTeX

Zap. Nauchn. Semin. POMI 245 (1997) 173-206 (in russian)

solv-int nlin.SI

Representations as determinants of $M\times M$ dimensional matrices are obtained for equal-time temperature correlators of the anisotropic Heisenberg XY chain. These representations are simple deformations of the answers for the isotropic XX0 chain. In the thermodynamic limit, the correlators are expressed in terms of the Fredholm determinants of linear integral operators.

solv-int/9711001

Jose Carlos Brunelli

J. C. Brunelli and Ashok Das

Integrable Models and the Higher Dimensional Representations of Graded Lie Algebras

13 pages, latex

Mod.Phys.Lett. A13 (1998) 133-144

10.1142/S0217732398000176

solv-int hep-th nlin.SI

We construct a zero curvature formulation, in superspace, for the sTB-B hierarchy which naturally reduces to the zero curvature condition in terms of components, thus solving one of the puzzling features of this model. This analysis, further, suggests a systematic method of constructing higher dimensional representations for the zero curvature condition starting with the fundamental representation. We illustrate this with the examples of the sTB hierarchy and the sKdV hierarchy. This would be particularly useful in constructing explicit higher dimensional representations of graded Lie algebras.

solv-int/9711002

Manuel Manas

Q. P. Liu and Manuel Manas

Darboux Transformations for SUSY Integrable Systems

13 pages. LaTeX209 with LamuPhys and EPSF packages, 3 figures. Contribution to the proceedings of the "Integrable Models and Supersymmetry" meeting held at Chicago on July'97

10.1007/BFb0105324

solv-int nlin.SI

Several types of Darboux transformations for supersymmetric integrable systems such as the Manin-Radul KdV, Mathieu KdV and SUSY sine-Gordon equations are considered. We also present solutions such as supersolitons and superkinks.

solv-int/9711003

Leonid Dickey

L.A.Dickey

Additional symmetries of the Zakharov-Shabat hierarchy, String equation and Isomonodromy

11 pages, LaTeX

solv-int nlin.SI

Isomonodromic deformations are nothing but symmetries of the Zakharov-Shabat (isospectral) hierarchy, both the basic ones (belonging to the hierarchy) and additional, restricted to the submanifold of solutions to the string equation.

solv-int/9711004

Ju Guo-xing

Guo-xing Ju, Chi Xiong

On the Integrability of the One-Dimensional Open XYZ Spin Chain

6 pages,latex,no figures

10.1088/0253-6102/30/3/337

solv-int hep-th nlin.SI

The Lax pair for the one-dimensional open XYZ spin chain is constructed, this shows that the system is completely integrable .

solv-int/9711005

Ming-Hsien Tu

Jiin-Chang Shaw and Ming-Hsien Tu

A Note on the Gauge Equivalence between the Manin-Radul and Laberge-Mathieu Super KdV Hierarchies

8 pages, revtex, 1 figure

J. Phys. A31 (1998) 4805

10.1088/0305-4470/31/20/017

solv-int nlin.SI

The gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies is revisited. Apart from the Inami-Kanno transformation, we show that there is another gauge transformation which also possess the canonical property. We explore the relationship of these two gauge transformations from the Kupershmidt-Wilson theorem viewpoint and, as a by-product, obtain the Darboux-Backlund transformation for the Manin-Radul super KdV hierarchy. The geometrical intepretation of these transformations is also briefly discussed.

solv-int/9711006

Kirill N. Ilinski

Kirill Ilinski, Alexander Stepanenko (University of Birmingham)

Comment on ``Equal-time temperature correlators of the one-dimensional Heisenberg XY chain'', preprint solv-int/9710028

1 page, Latex

solv-int nlin.SI

In the comment we give references to our papers where the problem was solved for more general case of time-dependent finite temperature correlators.

solv-int/9711007

Tomaz Prosen

Tomaz Prosen (Physics Dept., Faculty of Math.&Phys., University of Ljubljana, Ljubljana, Slovenia)

A new class of completely integrable quantum spin chains

4 pages in RevTex

10.1088/0305-4470/31/21/002

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

A large (infinitely-dimensional) class of completely integrable (possibly non-autonomous) spin chains is discovered associated to an infinite-dimensional Lie Algebra of infinite rank. The complete set of integrals of motion is constructed explicitly, as well as their eigenstates and spectra. As an example we outline kicked Ising model: Ising chain periodically kicked with transversal magnetic field.

solv-int/9711008

Yang Wenli

Bo-yu Hou and Wen-li Yang

The nondynamical r-matrix structure of the elliptic Calogero-Moser model

7 pages, Latex file 17k

IMPNWU-960810

solv-int hep-th nlin.SI

In this paper, we construct a new Lax operator for the elliptic Calogero-Moser model with N=2. The nondynamical r-matrix structure of this Lax operator is also studied . The relation between our Lax operator and the Lax operator given by Krichever is also obtained.

solv-int/9711009

Loriano Bonora

L.Bonora, S.Krivonos and A.Sorin

The N=2 supersymmetric matrix GNLS hierarchies

13 pages, Latex, one reference added

Lett.Math.Phys. 45 (1998) 63-79

SISSA 142/97/EP

solv-int hep-th nlin.SI

We construct the matrix generalization of the N=2 supersymmetric GNLS hierarchies. This is done by exhibiting the corresponding matrix super Lax operators in terms of N=2 superfields in two different superfield bases. We present the second Hamiltonian structure and discrete symmetries. We then extend our discussion by conjecturing the Lax operators of different reductions of the N=2 supersymmetric matrix KP hierarchy and discuss the simplest examples.

solv-int/9711010

V. Kuznetsov

V.B. Kuznetsov and E.K. Sklyanin

Few remarks on Baecklund transformations for many-body systems

14 pages, latex v.2.09, no figures

J.Phys.A 31 (1998) 2241-2251

10.1088/0305-4470/31/9/012

solv-int nlin.SI

Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Baecklund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable mu for the parameter lambda of the BT B_lambda such that mu belongs to the spectrum of the Lax operator L(lambda). As a consequence, the generating function of the composition of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by mu is introduced.

solv-int/9711011

Luiz Agostinho Ferreira

H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman

Vertex Operators and Solitons of Constrained KP Hierarchies

13 pages, needs lamuphys.tex and lamuphys.sty, talk presented at the 1997 UIC Workshop on Supersymmetry and Integrable Models, Chicago, USA, June/97. To be published in Lecture Notes in Physics, Springer-Verlag

10.1007/BFb0105320

IFT-P.071/97

solv-int hep-th nlin.SI

We construct the vertex operator representation for the Affine Kac-Moody $SL(M+K+1)$ algebra, which is relevant for the construction of the soliton solutions of the constrained KP hierarchies. The oscillators involved in the vertex operator construction are provided by the Heisenberg subalgebras of $SL(M+K+1)$ realized in the unconventional gradations. The well-known limiting cases are the homogeneous Heisenberg subalgebra of $SL(M+1)$ and the principal Heisenberg subalgebra of ${\hat{sl}}(K+1)$. The explicit example of $M=K=1$ is discussed in detail and the corresponding soliton solutions and tau-functions are given.

solv-int/9711012

Loriano Bonora

L.Bonora, S.Krivonos and A.Sorin

Coset approach to the N=2 supersymmetric matrix GNLS hierarchies

13 pages, Latex, a few misprints have been corrected

10.1016/S0375-9601(98)00112-1

SISSA 143/97/EP

solv-int hep-th nlin.SI

We discuss a large class of coset constructions of the N=2 sl(n|n-1) affine superalgebra. We select admissible subalgebras, i.e. subalgebras that induce linear chiral/antichiral constraints on the coset supercurrents. We show that all the corresponding coset constructions lead to N=2 matrix GNLS hierarchies. We develop an algorithm to compute the relative Hamiltonians and flows. We spell out completely the case of the N=2 affine sl(3|2), which possesses four admissible subalgebras. The non-local second Hamiltonian structure of the N=2 matrix GNLS hierarchies is obtained via Dirac procedure from the local N=2 sl(n|n-1) affine superalgebra. We observe that to any second Hamiltonian structure with pure bosonic or pure fermionic superfield content there correspond two different N=2 matrix GNLS hierarchies.

solv-int/9711013

Andrew J. Bordner

Andrew J. Bordner

Commuting Charges of the Quantum Korteweg-deVries and Boussinesq Theories from the Reduction of W(infinity) and W(1+infinity) Algebras

11 pages, RevTeX

Mod. Phys. Lett. A 13, (1998) 541.

10.1142/S0217732398000607

YITP-97-58

solv-int nlin.SI

Integrability of the quantum Boussinesq equation for c=-2 is demonstrated by giving a recursive algorithm for generating explicit expressions for the infinite number of commuting charges based on a reduction of the W(infinity) algebra. These charges exist for all spins $s \geq 2$. Likewise, reductions of the W(infinity/2) and W((1+infinity)/2) algebras yield the commuting quantum charges for the quantum KdV equation at c=-2 and c=1/2, respectively.

solv-int/9711014

Jarmo Hietarinta

Jarmo Hietarinta and Claude Viallet

Singularity confinement and chaos in discrete systems

4 pages, revtex, 2 PostScript-figures

Phys. Rev. Lett., 81 (1998) 325

10.1103/PhysRevLett.81.325

solv-int nlin.SI

We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the analysis of the complexity (``algebraic entropy'') of the map using the growth of the degree of its iterates: integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems.

solv-int/9711015

Metin Gurses

Metin Gurses and Atalay Karasu

Integrable Coupled KdV Systems

17pp, LateX, to be published in J.Math.Phys

10.1063/1.532278

solv-int nlin.SI

We give the conditions for a system of N- coupled Korteweg de Vries(KdV) type of equations to be integrable. Recursion operators of each subclasses are also given. All examples for N=2 are explicitly given.

solv-int/9712001

R. A. Sharipov

O. N. Mikhailov and R. A. Sharipov

On the geometry of point-expansions for certain class of differential equations of the second order

AmS-TeX, version 2.1, 8 pages, amsppt style

solv-int nlin.SI

Second order ordinary differential equations of the form $y'' = P(x,y) + 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered and their point-expansions are constructed. Geometrical structures connected with these expansions are described.

solv-int/9712002

Alexander Sorin

S. Krivonos and A. Sorin

Extended N=2 supersymmetric matrix (1,s)-KdV hierarchies

LaTeX, 8 pages

Phys.Lett. A251 (1999) 109-114

10.1016/S0375-9601(98)00863-9

JINR E2-97-365

solv-int hep-th nlin.SI

We propose the Lax operators for N=2 supersymmetric matrix generalization of the bosonic (1,s)-KdV hierarchies. The simplest examples - the N=2 supersymmetric a=4 KdV and a=5/2 Boussinesq hierarchies - are discussed in detail.

solv-int/9712003

A. V. Tsiganov

A.V. Tsiganov

The Stackel systems and algebraic curves

21 pages, LaTeX, no figures

solv-int nlin.SI

We show how the Abel-Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel-Jacobi map is just the St\"{a}ckel matrix, which determines $n$-orthogonal curvilinear coordinate systems in a flat space. The Lax pairs, $r$-matrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space.

solv-int/9712004

Rossen Ivanov

V. S. Gerdjikov, E. G. Evstatiev, R. I. Ivanov (Institute for Nuclear Energy and Nuclear Research, Bulg. Acad. of Sci., Sofia, Bulgaria)

The Complex Toda Chains and the Simple Lie Algebras - Solutions and Large Time Asymptotics

LaTeX, article style, 16 pages; corrections of formulas and text improvements

10.1088/0305-4470/31/40/014

INRNE preprint, TH-97-13

solv-int nlin.PS nlin.SI patt-sol

The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends related to the classical series of simple Lie algebras are classified. It is shown that the CTC models have much richer variety of asymptotic regimes than the real Toda chain (RTC). Besides asymptotically free propagation (the only possible regime for the RTC), CTC allow bound state regimes, various intermediate regimes when one (or several) group(s) of particles form bound state(s), singular and degenerate solutions. These results can be used e.g., in describing the soliton interactions of the nonlinear Schroedinger equation. Explicit expressions for the solutions in terms of minimal sets of scattering data are proposed for all classical series B_r - D_r.

solv-int/9712005

Nikita A. Slavnov

V. E. Korepin (State University of New York, Stony Brook, USA) and N. A. Slavnov (Steklov Mathematical Institute, Moscow, Russia)

The New Identity for the Scattering Matrx of Exactly Solvable Models

7 pages, Latex, no figures

10.1007/s100510050477

ITP-SUNY-SB-97-72

solv-int cond-mat hep-th math.QA nlin.SI q-alg

We discovered a simple quadratic equation, which relates scattering phases of particles on Fermi surface. We consider one dimensional Bose gas and XXZ Heisenberg spin chain.

solv-int/9712006

Igor G. Korepanov

I.G. Korepanov

Integrability in 3+1 Dimensions: Relaxing a Tetrahedron Relation

LaTeX, 3 pages

solv-int alg-geom math.AG nlin.SI

I propose a scheme of constructing classical integrable models in 3+1 discrete dimensions, based on a relaxed version of the problem of factorizing a matrix into the product of four matrices of a special form.

solv-int/9712007

Kojima Takeo

T. Kojima (RIMS Kyoto University)

Dynamical Correlation Functions for an Impenetrable Bose gas with open boundary conditions

LaTEX, 15 pages

solv-int hep-th math.QA nlin.SI q-alg

We study the time and temperature dependent correlation functions for an impenetrable bose gas with open boundary conditions. We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. In the case of time independent ground state, our Fredholm determinant formulae degenerate to the one which have been obtained by the help of fermions [T. Kojima, J.Stat.Phys.Vol.88,713-(1997)]

solv-int/9712008

Ziad Maassarani

Z. Maassarani (Laval university)

The XXC Models

6 pages, LaTeX

Phys. Lett. A 244 (1998) 160-164

10.1016/S0375-9601(98)00322-3

LAVAL-PHY-27/97

solv-int cond-mat math.QA nlin.SI q-alg

A class of recently introduced multi-states XX models is generalized to include a deformation parameter. This corresponds to an additional nearest-neighbor CC interaction in the defining quadratic hamiltonian. Complete integrability of the one-dimensional models is shown in the context of the quantum inverse scattering method. The new R-matrix is derived. The diagonalization of the XXC models is carried out using the algebraic Bethe Ansatz.

solv-int/9712009

Ming-Hsien Tu

Jiin-Chang Shaw and Ming-Hsien Tu

Canonical gauge equivalences of the sAKNS and sTB hierarchies

10 pages, Revtex, no figures

J.Phys.A31:6517,1998

10.1088/0305-4470/31/30/016

solv-int nlin.SI

We study the gauge transformations between the supersymmetric AKNS (sAKNS) and supersymmetric two-boson (sTB) hierarchies. The Hamiltonian nature of these gauge transformations is investigated, which turns out to be canonical. We also obtain the Darboux-Backlund transformations for the sAKNS hierarchy from these gauge transformations.

solv-int/9712010

Andrei Mal'tsev

A.Ya.Maltsev (Landau Institute for Theoretical Physics)

The averaging of Hamiltonian structures in discrete variant of Whitham method

Latex, 4 Pages

Russian Math. Surveys 53:1 (1998), 214-216

solv-int nlin.SI

Paper is devoted to the construction of averaging procedure of Hamiltonian structures in discrete Whitham method. The procedure is analogous to Dubrovin-Novikov procedure of averaging of local field-theoretical Poisson brackets and gives the Poisson bracket of Hydrodynamic Type starting from Poisson bracket for a discrete chain.

solv-int/9712011

Ju Guo-xing

Guo-xing Ju, Shi-kun Wang, Ke Wu, Chi Xiong

Boundary K-matrices and the Lax pair for 1D open XYZ spin-chain

LaTeX, 17 pages, errors in references corrected

10.1142/S0217751X98002006

solv-int hep-th nlin.SI

We analysis the symmetries of the reflection equation for open $XYZ$ model and find their solutions $K^{\pm}$ case by case. In the general open boundary conditions, the Lax pair for open one-dimensional $XYZ$ spin-chain is given.

solv-int/9712012

Henrik Aratyn

H. Aratyn, E. Nissimov and S. Pacheva

A New ``Dual'' Symmetry Structure of the KP Hierarchy

Added one reference, LaTeX, 8 pgs

10.1016/S0375-9601(98)00340-5

BGU-97/21/Dec-PH, UICHEP-TH/97-16

solv-int hep-th nlin.SI

A new infinite set of commuting additional (``ghost'') symmetries is proposed for the KP-type integrable hierarchy. These symmetries allow for a Lax representation in which they are realized as standard isospectral flows. This gives rise to a new double-KP hierarchy embedding ``ghost'' and original KP-type Lax hierarchies connected to each other via a ``duality'' mapping exchanging the isospectral and ``ghost'' ``times''. A new representation of 2D Toda lattice hierarchy as a special Darboux-Backlund orbit of the double-KP hierarchy is found and parametrized entirely in terms of (adjoint) eigenfunctions of the original KP subsystem.

solv-int/9712013

Szmigielski

David H. Sattinger and Jacek Szmigielski

A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator

Inverse Problems 12 (1996) 1003-1025

10.1088/0266-5611/12/6/014

solv-int nlin.SI

\We consider an inverse scattering problem for Schr\"odinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for two distinct symmetry classes. As an application we prove global existence theorems for the two distinct systems of partial differential equations $u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted, complementary classes of initial data.

solv-int/9712014

Marcio J. Martins

M.J. Martins and P.B. Ramos

The Quantum Inverse Scattering Method for Hubbard-like Models

latex file, 71 pages

10.1016/S0550-3213(98)00199-0

IFTA-97-35/UFSCARTH-97-19

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

This work is concerned with various aspects of the formulation of the quantum inverse scattering method for the one-dimensional Hubbard model. We first establish the essential tools to solve the eigenvalue problem for the transfer matrix of the classical ``covering'' Hubbard model within the algebraic Bethe Ansatz framework. The fundamental commutation rules exhibit a hidden 6-vertex symmetry which plays a crucial role in the whole algebraic construction. Next we apply this formalism to study the SU(2) highest weights properties of the eigenvectors and the solution of a related coupled spin model with twisted boundary conditions. The machinery developed in this paper is applicable to many other models, and as an example we present the algebraic solution of the Bariev XY coupled model.

solv-int/9712015

Pierre Vandergheynst

M. Adler, E. Horozov, P. van Moerbeke

The solution to the q-KdV equation

18 pages, LaTeX

10.1016/S0375-9601(98)00082-6

Math-97

solv-int nlin.SI

Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this paper is to show that any KdV solution leads effectively to a solution of the q-approximation of KdV. Two different q-KdV approximations were proposed, one by Frenkel and a variation by Khesin et al. We show there is a dictionary between the solutions of q-KP and the 1-Toda lattice equations, obeying some special requirement; this is based on an algebra isomorphism between difference operators and D-operators, where $Df(x)=f(qx)$. Therefore, every notion about the 1-Toda lattice can be transcribed into q-language.

solv-int/9712016

Pierre Vandergheynst

M. Adler and P. van Moerbeke

Toda-Darboux maps and vertex operators

23 pages, LaTeX

Math-97

solv-int nlin.SI

The purpose of this paper is to study Toda-Darboux transforms, i.e., Darboux transforms for operators L(t) flowing according to the Toda lattice. Each element of the null-space $L(t)-z$ specifies a factorization for all t and thus a Toda-Darboux transform on $L(t)$. The Toda-Darboux map induces a transformation on the tau-vectors, given by a certain vertex operator, and on eigenfunctions, given by a Wronskian. .

solv-int/9712017

Adam Doliwa

A. Doliwa, P. M. Santini and M. Manas

Transformations of Quadrilateral Lattices

50 pages, 15 figures; minor corrections, added references

J. Math. Phys. 41 (2000) 944-990

10.1063/1.533175

solv-int nlin.SI

Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogues: the multidimensional quadrilateral lattices, i.e. lattices x: Z^N -> R^M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analogue of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between "transformations" and "discretizations" is also investigated for quadrilateral lattices. We finally interpret these results within the D-bar formalism.

solv-int/9712018

Metin Gurses

Metin Gurses

Sigma Models and Minimal Surfaces

Latex, 13pp, to be published in Letters in Mathematical Physics

solv-int nlin.SI

The correspondance is established between the sigma models, the minimal surfaces and the Monge-Ampere equation. The Lax -Pairs of the minimality condition of the minimal surfaces and the Monge-Ampere equations are given. Existance of infinitely many nonlocal conservation laws is shown and some Backlund transformations are also given.

solv-int/9712019

Matveev V. S.

V.S. Matveev (Bremen University)

Quadratically integrable geodesic flows on the torus and on the Klein bottle

10 pages, latex2e

Regular and Chaotic Dynamics, vol 2 no 1 (1997), 96-103

solv-int math.DG nlin.SI

In the present paper we prove, that if the geodesic flow of a metric G on the torus T is quadratically integrable, then the torus T isometrically covers a torus with a Liouville metric on it, and describe the set of quadratically integrable geodesic flows on the Klein bottle.

solv-int/9712020

Valery Shchesnovich

V.S. Shchesnovich

Polarization scattering by soliton-soliton collisions

Second formula in Eq. (7) is corrected; 5 pages, Latex

solv-int nlin.SI

Collision of two solitons of the Manakov system is analytically studied. Existence of a complete polarization mode switching regime is proved and the parameters of solitons prepared for polarization switching are found.

solv-int/9801001

Arthur Vartanian

A. V. Kitaev, A. H. Vartanian

Asymptotics of Solutions to the Modified Nonlinear Schr\"{o}dinger Equation: Solitons on a Non-Vanishing Continuous Background

38 pages, 1 figure, LaTeX

solv-int nlin.SI

Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leading-order asymptotics as $t \to \pm \infty$ of the solution to the Cauchy problem for the modified nonlinear Schr\"{o}dinger equation, $i \partial_{t} u + {1/2} \partial_{x}^{2} u + | u |^{2} u + i s \partial_{x} (| u |^{2} u) = 0$, $s \in \Bbb R_{>0}$, which is a model for nonlinear pulse propagation in optical fibers in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent nonlinear evolution equations; in particular, the derivative nonlinear Schr\"{o}dinger equation, $i \partial_{t} q + \partial_{x}^{2} q - i \partial_{x} (| q |^{2} q) = 0$. As an application of these asymptotic results, explicit expressions for position and phase shifts of solitons in the presence of the continuous spectrum are calculated.

solv-int/9801002

Nobuhiko Shinzawa

Nobuhiko Shinzawa and Satoru Saito

A Symmetric Generalization of Linear B\"acklund Transformation associated with the Hirota Bilinear Difference Equation

Latex, 12 pages, 1 figure

10.1088/0305-4470/31/19/016

solv-int nlin.SI

The Hirota bilinear difference equation is generalized to discrete space of arbitrary dimension. Solutions to the nonlinear difference equations can be obtained via B\"acklund transformation of the corresponding linear problems.

solv-int/9801003

YU-Song Ju

Yu. S.J(1), K. Toda(1),N. Sasa(2) and T. Fukuyama(1)((1)Ritsumeikan Univ., (2)Japan Atomic Energy Research Institute)

N Soliton Solutions to The Bogoyavlenskii-Schiff Equation and A Quest for The Soliton Solution in (3 + 1) Dimensions

14 pages, 8 figures([email protected]), uses ioplppt.sty

solv-int nlin.SI

We study the integrable systems in higher dimensions which can be written not by the Hirota's bilinear form but by the trilinear form. We explicitly discuss about the Bogoyavlenskii-Schiff(BS) equation in (2 + 1) dimensions. Its analytical proof of multi soliton solution and a new feature are given. Being guided by the strong symmetry, we also propose a new equation in (3 + 1) dimensions.

solv-int/9801004

Andrei Pronko

A. G. Izergin, A. G. Pronko

Temperature correlators in the two-component one-dimensional gas

40 pages, LaTeX, a4.sty

10.1016/S0550-3213(98)00182-5

PDMI PREPRINT - 19/1997

solv-int nlin.SI

The quantum nonrelativistic two-component Bose and Fermi gases with the infinitely strong point-like coupling between particles in one space dimension are considered. Time and temperature dependent correlation functions are represented in the thermodynamic limit as Fredholm determinants of integrable linear integral operators.

solv-int/9801005

Shigeki Matsutani

Statistical Mechanics of Non-stretching Elastica in Three Dimensional Space

AMS-Tex Use

10.1016/S0393-0440(98)00042-4

solv-int nlin.SI

Recently I proposed a new calculation scheme of a partition function of an immersion object using path integral method and theory of soliton (to appear in J.Phys.A). I applied the scheme to problem of elastica in two-dimensional space and Willmore surface in three dimensional space. In this article, I will apply the scheme to elastica in three dimensional space as a more physical model in polymer science. Then orbit space of the nonlinear Schrodinger and complex modified Korteweg-de Vries equations can be regarded as the functional space of the partition function. By investigation of the partition function, I gives a conjecture of the relation of these soliton equations.

solv-int/9801006

Shigeki Matsutani

Dirac Operator of a Conformal Surface Immersed in R^4: Further Generalized Weierstrass Relation

AMS-Tex Use

solv-int nlin.SI

In the previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac operator defined over a conformal surface immersed in R^3 is identified with the Dirac operator which is generalized the Weierstrass- Enneper equation and Lax operator of the modified Novikov-Veselov (MNV) equation. In this article, I determine the Dirac operator defined over a conformal surface immersed in R^4, which is reduced to the Lax operators of the nonlinear Schrodinger and the MNV equations by taking appropriate limits. Thus the Dirac operator might be the Lax operator of (2+1)- dimensional soliton equation.

solv-int/9801007

Hasan Gumral

Hasan Gumral

General vorticity conservation

Latex, 20 pages

solv-int nlin.SI

The motion of an incompressible fluid in Lagrangian coordinates involves infinitely many symmetries generated by the left Lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid. Utilizing a 1+3-dimensional Hamiltonian setting an explicit realization of this symmetry algebra and the related Lagrangian and Eulerian conservation laws are constructed recursively. Their Lie algebraic structures are inherited from the same construction. The laws of general vorticity and helicity conservations are formulated globally in terms of invariant differential forms of the velocity field.

solv-int/9801008

Harold Widom

Craig A. Tracy, Harold Widom

Asymptotics of a class of Fredholm determinants

8 pages, LaTeX file

"Spectral Problems in Geometry and Arithmetic," ed. T. Branson, Amer. Math. Soc., Providence, 1999, pgs 167-174

solv-int math.FA nlin.SI

In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptotics, are also discussed.

solv-int/9801009

Szmigielski

J. Dorfmeister, H. Gradl and J. Szmigielski

Systems of PDEs obtained from factorization in loop groups

1 figure

solv-int nlin.SI

We propose a generalization of a Drinfeld-Sokolov scheme of attaching integrable systems of PDEs to affine Kac-Moody algebras. With every affine Kac-Moody algebra $\gg$ and a parabolic subalgebra $\gp$, we associate two hierarchies of PDEs. One, called positive, is a generalization of the KdV hierarchy, the other, called negative, generalizes the Toda hierarchy. We prove a coordinatization theorem, which establishes that the number of functions needed to express all PDEs of the the total hierarchy equals the rank of $\gg$. The choice of functions, however, is shown to depend in a noncanonical way on $\gp$. We employ a version of the Birkhoff decomposition and a ``2-loop'' formulation which allows us to incorporate geometrically meaningful solutions to those hierarchies. We illustrate our formalism for positive hierarchies with a generalization of the Boussinesq system and for the negative hierarchies with the stationary Bogoyavlenskii equation.

solv-int/9801010

David H. Sattinger

Richard Beals and D. H. Sattinger

Integrable Systems and Isomonodromy Deformations

Physica D, vol 65, (1993), 17-47

10.1016/0167-2789(93)90003-J

solv-int nlin.SI

We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the $n\times n$ AKNS hierarchies and the Gel'fand-Dikii hierarchies. The third arises in string theory as the representation of the Heisenberg group by $[(L^{k/n})_+,L]=I$ where $L$ is an $n^{th}$ order scalar differential operator. The monodromy data is constructed in each case; the inverse monodromy problem is solved as a Riemann-Hilbert problem; and a simple proof of the Painlev\'e property is given for the general case

solv-int/9801011

David H. Sattinger

D.H. Sattinger and J.S. Szmigielski

Factorization and the Dressing Method for the Gel'fand-Dikii Hierarch

Physica D, vol 64, (1993), 1-34

10.1016/0167-2789(93)90247-X

solv-int nlin.SI

The isospectral flows of an $n^{th}$ order linear scalar differential operator $L$ under the hypothesis that it possess a Baker-Akhiezer function were originally investigated by Segal and Wilson from the point of view of infinite dimensional Grassmanians, and the reduction of the KP hierarchy to the Gel'fand-Dikii hierarchy. The associated first order systems and their formal asymptotic solutions have a rich Lie algebraic structure which was investigated by Drinfeld and Sokolov. We investigate the matrix Riemann-Hilbert factorizations for these systems, and show that different factorizations lead respectively to the potential, modified, and ordinary Gel'fand-Dikii flows. Lie algebra decompositions (the Adler-Kostant-Symes method) are obtained for the modified and potential flows. For $n>3$ the appropriate factorization for the Gel'fand-Dikii flows is not a group factorization, as would be expected; yet a modification of the dressing method still works. A direct proof, based on a Fredholm determinant associated with the factorization problem, is given that the potentials are meromorphic in $x$ and in the time variables. Potentials with Baker-Akhiezer functions include the multisoliton and rational solutions, as well as potentials in the scattering class with compactly supported scattering data. The latter are dense in the scattering class.

solv-int/9801012

Andrey V. Tsiganov

Andrey Tsiganov

Dynamical boundary conditions for integrable lattices

LaTeX, 12pages

J. Phys. A, Math. Gen. 31, No.39, 8049-8061, (1998)

10.1088/0305-4470/31/39/017

solv-int nlin.SI

Some special solutions to the reflection equation are considered. These boundary matrices are defined on the common quantum space with the other operators in the chain. The relations with the Drinfeld twist are discussed.

solv-int/9801013

Igor G. Korepanov

Igor G. Korepanov

Particles and strings in a (2+1)-D integrable quantum model

J. Nonlinear Math. Phys. 7 (2000), no. 1, 94-119

10.2991/jnmp.2000.7.1.7

JNMP 4/2002 (Review Article)

solv-int nlin.SI

We give a review of some recent work on generalization of the Bethe ansatz in the case of $2+1$-dimensional models of quantum field theory. As such a model, we consider one associated with the tetrahedron equation, i.e. the $2+1$-dimensional generalization of the famous Yang--Baxter equation. We construct some eigenstates of the transfer matrix of that model. There arise, together with states composed of point-like particles analogous to those in the usual $1+1$-dimensional Bethe ansatz, new string-like states and string-particle hybrids.

solv-int/9801014

Oleg M. Kiselev

R.R. Gadyl'shin, O.M. Kiselev (Institute of Mathematics, Ufa Science Centre, Russian Acad. of Sciences)

Asymptotics of perturbed soliton for Davey--Stewartson II equation

In this replaced version the formula for the perturbed parameter of the soliton is corrected. Amstex, 13 pages

solv-int nlin.SI

It is shown that, under a small perturbation of lump (soliton) for Davey--Stewartson (DS-II) equation, the scattering data gain the nonsoliton structure. As a result, the solution has the form of Fourier type integral. Asymptotic analysis shows that, in spite of dispertion, the principal term of the asymptotic expansion for the solution has the solitary wave form up to large time.

solv-int/9801015

Igor G. Korepanov

R.M. Kashaev, I.G. Korepanov, S.M. Sergeev

Functional Tetrahedron Equation

LaTeX, 16 pages

Theor. Math. Phys. 117:3 (1998) 1402 - 1413; Teor. Mat. Fiz. 117:3 (1998) 370 - 384

10.1007/BF02557179

solv-int nlin.SI

We describe a scheme of constructing classical integrable models in 2+1-dimensional discrete space-time, based on the functional tetrahedron equation - equation that makes manifest the symmetries of a model in local form. We construct a very general "block-matrix model" together with its algebro-geometric solutions, study its various particular cases, and also present a remarkably simple scheme of quantization for one of those cases.

solv-int/9801016

Evgeny Doktorov

V.S. Shchesnovich and E.V. Doktorov

Perturbation theory for the modified nonlinear Schr{\"o}dinger solitons

22 pages, Latex, no figures. Submitted to Physica D

10.1016/S0167-2789(98)00209-7

solv-int nlin.PS nlin.SI patt-sol

The perturbation theory based on the Riemann-Hilbert problem is developed for the modified nonlinear Schr{\"o}dinger equation which describes the propagation of femtosecond optical pulses in nonlinear single-mode optical fibers. A detailed analysis of the adiabatic approximation to perturbation-induced evolution of the soliton parameters is given. The linear perturbation and the Raman gain are considered as examples.

solv-int/9801017

Kjell Rosquist

Kjell Rosquist

The classical r-matrix in a geometric framework

LaTeX2e file; requires amsmath,eufrak and eucal packages

10.1016/S0375-9601(99)00177-2

USITP 98-01

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

We use a Riemannian (or pseudo-Riemannian) geometric framework to formulate the theory of the classical r-matrix for integrable systems. In this picture the r-matrix is related to a fourth rank tensor, named the r-tensor, on the configuration space. The r-matrix itself carries one connection type index and three tensorial indices. Being defined on the configuration space it has no momentum dependence but is dynamical in the sense of depending on the configuration variables. The tensorial nature of the r-matrix is used to derive its transformation properties. The resulting transformation formula turns out to be valid for a general r-matrix structure independently of the geometric framework. Moreover, the entire structure of the r-matrix equation follows directly from a simple covariant expression involving the Lax matrix and its covariant derivative. Therefore it is argued that the geometric formulation proposed here helps to improve the understanding of general r-matrix structures. It is also shown how the Jacobi identity gives rise to a generalized dynamical classical Yang-Baxter equation involving the Riemannian curvature.

solv-int/9801018

Krzysztof Kowalski

Krzysztof Kowalski

Nonlinear dynamical systems and classical orthogonal polynomials

21 pages latex, uses revtex

J. Math. Phys. 38 (1997) 2483-2505

10.1063/1.531990

kft-97-45

solv-int nlin.SI quant-ph

It is demonstrated that nonlinear dynamical systems with analytic nonlinearities can be brought down to the abstract Schr\"odinger equation in Hilbert space with boson Hamiltonian. The Fourier coefficients of the expansion of solutions to the Schr\"odinger equation in the particular occupation number representation are expressed by means of the classical orthogonal polynomials. The introduced formalism amounts a generalization of the classical methods for linearization of nonlinear differential equations such as the Carleman embedding technique and Koopman approach.

solv-int/9801019

Ming-Hsien Tu

Wen-Jui Huang, Jiin-Chang Shaw and Ming-Hsien Tu

Matrix Formulation of Hamiltonian Structures of Constrained KP Hierarchy

19 pages, Revtex, no figures. Minor changes, reference corrected

J. Math. Phys. 39 (1998) 3738

10.1063/1.532464

solv-int nlin.SI

We give a matrix formulation of the Hamiltonian structures of constrained KP hierarchy. First, we derive from the matrix formulation the Hamiltonian structure of the one-constraint KP hierarchy, which was originally obtained by Oevel and Strampp. We then generalize the derivation to the multi-constraint case and show that the resulting bracket is actually the second Gelfand-Dickey bracket associated with the corresponding Lax operator. The matrix formulation of the Hamiltonian structure of the one-constraint KP hierarchy in the form introduced in the study of matrix model is also discussed

solv-int/9801020

Krzysztof Kowalski

Krzysztof Kowalski

Universal formats for nonlinear dynamical systems

9 pages LaTeX

Chem. Phys. Lett. 209 (1993) 167 -170

DB-93-17

solv-int chao-dyn nlin.CD nlin.SI

It is demonstrated that very general nonlinear dynamical systems covering all cases arising in practice can be brought down to rate equations of chemical kinetics

solv-int/9801021

Henrik Aratyn

H. Aratyn, E. Nissimov and S. Pacheva

Supersymmetric KP Hierarchy: ``Ghost'' Symmetry Structure, Reductions and Darboux-Backlund Solutions

Minor corrections in few equations. LaTeX, 12 pgs

10.1063/1.532736

BGU-98/01/Jan-PH, UICHEP-TH/98-1

solv-int hep-th nlin.SI

This paper studies Manin-Radul supersymmetric KP hierarchy (MR-SKP) in three related aspects: (i) We find an infinite set of additional (``ghost'') symmetry flows spanning the same (anti-)commutation algebra as the ordinary MR-SKP flows; (ii) The latter are used to construct consistent reductions of the initial unconstrained MR-SKP hierarchy which involves a nontrivial modification for the fermionic flows; (iii) For the simplest constrained MR-SKP hierarchy we show that the orbit of Darboux-Backlund transformations lies on a supersymmetric Toda lattice being a square-root of the standard one-dimensional Toda lattice, and also we find explicit Wronskian-ratio solutions for the super-tau function.

solv-int/9801022

Anton Zabrodin

I.Krichever and A.Zabrodin

Vacuum curves of elliptic L-operators and representations of Sklyanin algebra

27 pages, latex, typos corrected

ITEP-TH-76/97

solv-int hep-th nlin.SI

An algebro-geometric approach to representations of Sklyanin algebra is proposed. To each 2 \times 2 quantum L-operator an algebraic curve parametrizing its possible vacuum states is associated. This curve is called the vacuum curve of the L-operator. An explicit description of the vacuum curve for quantum L-operators of the integrable spin chain of XYZ type with arbitrary spin $\ell$ is given. The curve is highly reducible. For half-integer $\ell$ it splits into $\ell +{1/2}$ components isomorphic to an elliptic curve. For integer $\ell$ it splits into $\ell$ elliptic components and one rational component. The action of elements of the L-operator to functions on the vacuum curve leads to a new realization of the Sklyanin algebra by difference operators in two variables restricted to an invariant functional subspace.

solv-int/9801023

Unal Goktas (1), Willy Hereman (1) ((1) Colorado School of Mines)

Computation of conservation laws for nonlinear lattices

To appear in Physica D, 17 pages, Latex, uses the style files elsart.sty and elsart12.sty

10.1016/S0167-2789(98)00140-7

solv-int nlin.SI

An algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented. The algorithm is implemented in Mathematica and can be used as an automated integrability test. With the code diffdens.m, conserved densities are obtained for several well-known lattice equations. For systems with parameters, the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist.

solv-int/9801024

Unal Goktas (1), Willy Hereman (1) ((1) Colorado School of Mines)

Invariants and Symmetries for Partial Differential Equations and Lattices

To appear in Proceedings of Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation (June 1-5, 1998, Golden, CO), 5 pages, Latex, uses the style file proc209.sty

solv-int nlin.SI

Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as Mathematica. Invariants and symmetries are shown for several well-known equations. Our Mathematica package allows one to automatically compute invariants and symmetries. Applied to systems with parameters, the package determines the conditions on these parameters so that a sequence of invariants or symmetries exists. The software can thus be used to test the integrability of model equations for wave phenomena.

solv-int/9801025

Myrzakulov Ratbay

G.N.Nugmanova (Centre for Nonlinear Problems, Alma-Ata-35, Kazakstan)

On the Lakshmanan and gauge equivalent counterpart of the Myrzakulov-VIII equation

5 pages, Latex, no figures; [email protected]

solv-int nlin.SI

The Lakshmanan equivalent counterparts of the some Myrzakulov equations are found.

solv-int/9801026

Yasuhiro Fujii

Yasuhiro Fujii and Miki Wadati

Correlation Functions of Finite XXZ model with Boundaries

16pages, LaTeX2e file, errors corrected

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

The finite XXZ model with boundaries is considered. We use the Matrix Product Ansatz (MPA), which was originally developed in the studies on the asymmetric simple exclusion process and the quantum antiferromagnetic spin chain. The MPA tells that the eigenstate of the Hamiltonian is constructed by the Zamolodchikov-Faddeev algebra (ZF-algebra) and the boundary states. We adopt the type I vertex operator of $U_q(\hat{sl}_2)$ as the ZF-algebra and realize the boundary states in the bosonic $U_q(\hat{sl}_2)$ form. The correlation functions are given by the product of the vertex operators and the bosonic boundary states. We express them in the integration forms.