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solv-int/9604004

Wen-Xiu Ma

W. X. Ma and B. Fuchssteiner

Integrable Theory of the Perturbation Equations

27 pages, latex, to appear in Chaos, Soliton & Fractals

10.1016/0960-0779(95)00104-2

solv-int hep-th nlin.SI

An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations: hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures etc. and provides us a method to generate hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.

solv-int/9604005

Holger Frahm

Fabian H. L. Essler, Holger Frahm, Alexander R. Its and Vladimir E. Korepin

Painlev\'e Transcendent Describes Quantum Correlation Function of the XXZ Antiferromagnet away from the free-fermion point

10 pages, LaTeX2e

J.Phys.A29:5619-5626,1996

10.1088/0305-4470/29/17/032

OUTP-96-16S, ITP-UH-06/96

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

We consider quantum correlation functions of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg XXZ spin chain in a magnetic field. We show that for a magnetic field close to the critical field $h_c$ (for the critical magnetic field the ground state is ferromagnetic) certain correlation functions can be expressed in terms of the solution of the Painlev\'e V transcendent. This establishes a relation between solutions of Painlev\'e differential equations and quantum correlation functions in models of {\sl interacting} fermions. Painlev\'e transcendents were known to describe correlation functions in models with free fermionic spectra.

solv-int/9605001

Juri Suris

O.Ragnisco (Rome), Yu.B.Suris (Bremen)

Integrable discretizations of the spin Ruijsenaars-Schneider models

LaTeX file

J.Math.Phys. 38 (1997) 4680-4691

10.1063/1.532114

solv-int hep-th nlin.SI

Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars--Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous--time counterparts. Explicit solutions are obtained for arbitrary flows of the hierarchies, including the discrete time ones.

solv-int/9605002

Wen-Xiu Ma

Wen-Xiu Ma

Darboux Transformations for a Lax Integrable System in $2n$-Dimensions

Latex, 14 pages, to be published in Lett. Math. Phys

10.1007/s11005-997-3049-3

solv-int nlin.SI

A $2n$-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux transformations is established for this Lax integrable system. The Vandermonde and generalized Cauchy determinant formulas lead to a description for deriving explicit solutions and thus some rational and analytic solutions are obtained.

solv-int/9605003

Yu Song-Ju

Yu.Song-Ju, T.Fukuyama

The Painlev\'e Test of Higher Dimensional KdV Equation

7 pages, LaTeX

solv-int nlin.SI

We argue the integrability of the generalized KdV(GKdV) equation using the Painlev\'e test. For $d( \le 2)$ dimensional space, GKdV equation passes the Painlev\'e test but does not for $d \geq 3$ dimensional space. We also apply the Ablowitz-Ramani-Segur's conjecture to the GKdV equation in order to complement the Painlev\'e test.

solv-int/9605004

Ayse Bilge

Ayse Humeyra Bilge

Classification of Integrable Evolution Equations of the Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$

10 pages, no figures

solv-int nlin.SI

We obtain the classification of integrable equations of the form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ using the formal symmetry method of Mikhailov et al [A.V.Mikhailov, A.B.Shabat and V.V.Sokolov, in {\it What is Integrability} edited by V.E. Zakharov (Springer-Verlag, Berlin 1991)]. We show that all such equations can be transformed to an integrable equation of the form $v_t=v_{xxx}+f(v,v_x,v_{xx})$ using transformations $\Phi(x,t,u,v,u_x,v_x)=0$, and the $u_{xx}$ dependence can be eliminated except for two equations.

solv-int/9605005

Harm Dorren

H.J.S. Dorren and R.K. Snieder

A stability analysis for the Korteweg-de Vries equation

15 pages LaTeX. The figures are available upon request ([email protected])

solv-int nlin.SI

In this paper the stability of the Korteweg-de Vries (KdV) equation is investigated. It is shown analytically and numerically that small perturbations of solutions of the KdV-equation introduce effects of dispersion, hence the perturbation propagates with a different velocity then the unperturbed solution. This effect is investigated analytically by formulating a differential equation for perturbations of solutions of the KdV-equation. This differential equation is solved generally using an Inverse Scattering Technique (IST) using the continuous part of the spectrum of the Schr\"{o}dinger equation. It is shown explicitly that the perturbation consist of two parts. The first part represents the time-evolution of the perturbation only. The second part represents the interaction between the perturbation and the unperturbed solution. It is shown explicitly that singular non-dispersive solutions of the KdV-equation are unstable.

solv-int/9605006

Juri Suris

Yuri B. Suris (Bremen)

New integrable systems related to the relativistic Toda lattice

LaTeX, 22 pp. Substantially extended version: several new systems added!

J. Phys. A: Math. and Gen., 1997, V. 30, p. 1745-1761.

10.1088/0305-4470/30/5/035

solv-int nlin.SI

New integrable lattice systems are introduced, their different integrable discretization are obtained. B\"acklund transformations between these new systems and the relativistic Toda lattice (in the both continuous and discrete time formulations) are established.

solv-int/9605007

A. A. Andrianov, M. V. Ioffe and D. N. Nishnianidze

Higher Order SUSY in Quantum Mechanics and Integrability of Two-dimensional Hamiltonians

11 pages, LaTeX

SPbU-IP-96-12

solv-int hep-th nlin.SI

The new method based on the SUSY algebra with supercharges of higher order in derivatives is proposed to search for dynamical symmetry operators in 2-dim quantum and classical systems. These symmetry operators arise when closing the SUSY algebra for a wide set of potentials. In some cases they are of 2-nd order in derivatives. The particular solutions are obtained also for potentials accepting symmetry operators of 4-th order. The investigation of quasiclassical limit of the SUSY algebra yields new classical integrals of motion for a certain type of systems which are polynomials of 4-th order in momenta. The general SUSY-inspired algorithm to construct classical systems with additional integrals of motion is outlined.

solv-int/9605008

Adrian-Stefan Carstea

A. S. C\^arstea and D. Grecu

On a class of rational and mixed soliton-rational solutions of Toda lattice

10 pages, Latex

10.1143/PTP.96.29

FT-413-1996, Inst.Atomic Physics, Bucharest, Romania

solv-int nlin.SI

A class of rational solutions of Toda lattice satisfying certain Backlund transformations and a class of mixed rational-soliton solutions (quasisolitons) in wronskian formare obtained using the method of Ablowitz and Satsuma. Also an extended class of rational solutions are found using an appropriate recursion relation. They are also solutions of Boussinesq equation and it is conjectured that there is a larger class of common solutions of both equations.

solv-int/9605009

Wen-Xiu Ma

Wen-Xiu Ma and Benno Fuchssteiner

Binary Nonlinearization of Lax Pairs

8 pages, latex, to appear in the Proceedings of Nonlinear Physics, Italy

solv-int nlin.SI

A kind of Bargmann symmetry constraints involving Lax pairs and adjoint Lax pairs is proposed for soliton hierarchy. The Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative finite dimensional integrable Hamiltonian systems and explicit integrals of motion may also be generated. The corresponding binary nonlinearization procedure leads to a sort of involutive solutions to every system in soliton hierarchy which are all of finite gap. An illustrative example is given in the case of AKNS soliton hierarchy.

solv-int/9605010

Juri Suris

Yuri B. Suris (Bremen)

A new integrable system related to the Toda lattice

LaTeX, 14 pp

J. Phys. A: Math. and Gen., 1997, V. 30, p. 2235-2249.

10.1088/0305-4470/30/6/041

solv-int nlin.SI

A new integrable lattice system is introduced, and its integrable discretizations are obtained. A B\"acklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established.

solv-int/9606001

Dr. Elena I. Ganzha

E. I. Ganzha (Krasnoyarsk State Pedagogical University)

On completeness of the Moutard transformations

6 pages (17 Kbytes), standard LaTeX 2.09, run twice to get the right cross-references. Resubmitted with the only correction: acknowledgment of grant RBFR 96-01-00050 support

solv-int nlin.SI

In this paper we solve positively the problem of (local) density of the "potentials" M(x,y) of the Moutard equation, u_{xy} = M(x,y) u, u=u(x,y), (used in many papers for construction of exact solutions of (2+1)-dimensional integrable systems) obtainable from a given initial potential with consecutive Moutard transformations.

solv-int/9606002

Dr. Elena I. Ganzha

E. I. Ganzha (Krasnoyarsk State Pedagogical University)

On completeness of the Ribaucour transformations for triply orthogonal curvilinear coordinate systems in R^3

7 pages (26 Kbytes), standard LaTeX 2.09, run twice to get the right cross-references. Resubmitted with the only correction: acknowledgment of grant RBFR 96-01-00050 support

solv-int nlin.SI

In this paper we solve positively the problem of (local) density of solutions of the (2+1)-dimentional integrable system describing triply orthogonal curvilinear coordinates in R^3 (a (2+1)-dimensional generalization of the 3-wave system) obtainable from a given initial solution with consecutive B\"acklund transformations (called Ribaucour transformations in classical differential geometry) in the space of all solutions of the system in question.

solv-int/9606003

S. P. Tsarev

E. T. Ganzha, S. P. Tsarev (Krasnoyarsk State Pedagogical University)

On superposition of the autoBaecklund transformations for (2+1)-dimensional integrable systems

11 pages (90 Kbytes), standard LaTeX 2.09, run twice to get the right cross-references. Resubmitted with the only correction: acknowledgment of grant RBFR 96-01-00050 support

solv-int nlin.SI

The usual superposition formulas for Baecklund transformations of (2+1)-dimensional integrable systems include quadratures unlike the well known case of (1+1)-dimensional inegrable systems where the fourth solution is found with algebraic operations. In the present paper we show how in the case of (2+1)-dimensional integrable systems one can find an extended formula of nonlinear superposition such that the resulting solution will be found uniquely from the given previous solution with algebraic operations.

solv-int/9606004

Dr. Jeremy Schiff

Jeremy Schiff (Dept of Math and Comp Sci, Bar Ilan University)

Symmetries of KdV and Loop Groups

36 pages (sorry), LaTeX using a4 documentstyle

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

A simple version of the Segal-Wilson map from the SL(2,C) loop group to a class of solutions of the KdV hierarchy is given, clarifying certain aspects of this map. It is explained how the known symmetries, including Backlund transformations, of KdV arise from simple, field independent, actions on the loop group. A variety of issues in understanding the algebraic structure of Backlund transformations are thus resolved.

solv-int/9606005

Robert Carroll

R. Carroll (Mathematics Dept., University of Illinois, Urbana, IL)

Some survey remarks on Whitham theory and EM duality

Latex, 12 pages

solv-int hep-th nlin.SI

The nature of the BA function and its adjoint for KP-Toda is traced through the averaging method in generating the Whitham equations, differentials, and symplectic forms, with connections to EM duality.

solv-int/9606006

Barbara Shipman

Barbara Shipman

On the geometry of certain isospectral sets in the full Kostant-Toda lattice

22 pages, LaTeX, 7 figures in PicTeX available on request

solv-int nlin.SI

We use momentum mappings on generalized flag manifolds and their momentum polytopes to study the geometry of the level sets of the 1-chop integrals of the full Kostant-Toda lattice in certain isospectral submanifolds of the phase space. We derive expressions for these integrals in terms of Pl\"ucker coordinates on the flag manifold in the case that all eigenvalues are zero and compare the geometry of the base locus of their level set varieties with the corresponding geometry for distinct eigenvalues. Finally, we illustrate and extend our results in the context of the full sl(3,C) and sl(4,C) Kostant-Toda lattices.

solv-int/9606007

Leonid Bogdanov

L.V. Bogdanov (IINS, Landau Institute, Moscow) and B.G. Konopelchenko (Universita' degli Studi di Lecce)

Generalized integrable hierarchies and Combescure symmetry transformations

17 pages, LaTeX

10.1088/0305-4470/30/5/022

solv-int nlin.SI

Unifying hierarchies of integrable equations are discussed. They are constructed via generalized Hirota identity. It is shown that the Combescure transformations, known for a long time for the Darboux system and having a simple geometrical meaning, are in fact the symmetry transformations of generalized integrable hierarchies. Generalized equation written in terms of invariants of Combescure transformations are the usual integrable equations and their modified partners. The KP-mKP, DS-mDS hierarchies and Darboux system are considered.

solv-int/9606008

Stephanie F. Singer

M. Quinn and S.F. Singer

Loop algebras, gauge invariants and a new completely integrable system

solv-int nlin.SI

One fruitful motivating principle of much research on the family of integrable systems known as ``Toda lattices'' has been the heuristic assumption that the periodic Toda lattice in an affine Lie algebra is directly analogous to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper shows that the analogy is not perfect. A discrepancy arises because the natural generalization of the structure theory of finite-dimensional simple Lie algebras is not the structure theory of loop algebras but the structure theory of affine Kac-Moody algebras. In this paper we use this natural generalization to construct the natural analog of the nonperiodic Toda lattice. Surprisingly, the result is not the periodic Toda lattice but a new completely integrable system on the periodic Toda lattice phase space. This integrable system is prescribed purely in terms of Lie-theoretic data. The commuting functions are precisely the gauge-invariant functions one obtains by viewing elements of the loop algebra as connections on a bundle over $S^1$.

solv-int/9606009

Leonid Dickey

L.A.Dickey and W.Strampp

On a generalization of the Fay-Sato identity for KP Baker functions and its application to constrained hierarchies

LaTeX, 11 pages

solv-int nlin.SI

Some new formulas for the KP hierarchy are derived from the differential Fay identity. They proved to be useful for the $k$-constrained hierarchies providing a series of determinant identities for them. A differential equation is introduced which is called ``universal" since it plays an important role for all the $k$-constrained hierarchies. In the cases $k=1,2$ and 3 explicit formulas are presented, in all the others recurrence relations are given which enable one to obtain the identities.

solv-int/9606010

Paolo Casati (Dip. di Matematica, Univ. di Milano, Italy), Gregorio Falqui (SISSA, Trieste, Italy), Franco Magri (Dip. di Matematica, Univ. di Milano, Italy), and Marco Pedroni (Dip. di Matematica, Univ. di Genova, Italy)

A Note on Fractional KdV Hierarchies

Final version to appear in J. Math. Phys. Some changes in the order of presentation, with more emphasis on the geometrical picture. One figure added (using epsf.sty). 30 pages, Latex

10.1063/1.532110

solv-int nlin.SI

We introduce a hierarchy of mutually commuting dynamical systems on a finite number of Laurent series. This hierarchy can be seen as a prolongation of the KP hierarchy, or a ``reduction'' in which the space coordinate is identified with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV hierarchies are gotten by means of further reductions, obtained by constraining the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are discussed in detail.

solv-int/9607001

Hsien-chung Kao

Hsien-chung Kao, Shih-Chang Lee, and Wen-Jer Tzeng

Exact Solution of Frenkel-Kontorova Models with a Complete Devil's Staircase in Higher Dimensions

14 pages, uses revtex. Four figures included

solv-int nlin.SI

We solve exactly a class of Frenkel-Kontorova models with piecewise parabolic potential, which has $d$ sub-wells in a period. With careful analysis, we show that the phase diagram of the minimum enthalpy configurations exhibits the structure of a complete $d$-dimensional devil's staircase. The winding number of a minimum enthalpy configuration is locked to rational values, while the fraction of atoms in each sub-well is locked to values which are sub-commensurable with the winding number.

solv-int/9607002

Kenji Kajiwara

Kenji Kajiwara (Dept. of Elect. Engin., Doshisha Univ.) and Yasuhiro Ohta (Dept. Appl. Math., Hiroshima Univ.)

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

16 pages LaTeX. To appear in J. Math. Phys.(1996)

10.1063/1.531648

solv-int nlin.SI

Two types of determinant representations of the rational solutions for the Painlev\'e II equation are discussed by using the bilinear formalism. One of them is a representation by the Devisme polynomials, and another one is a Hankel determinant representation. They are derived from the determinant solutions of the KP hierarchy and Toda lattice, respectively.

solv-int/9607003

Wen-Xiu Ma

Wen-Xiu MA (Paderborn Univ.) and Zi-Xiang ZHOU (Fudan Univ.)

Coupled Integrable Systems Associated with a Polynomial Spectral Problem and their Virasoro Symmetry Algebras

8 pages, Plain-tex, to appear in Prog. Theor. Phys

10.1143/PTP.96.449

solv-int nlin.SI

An isospectral hierarchy of commutative integrable systems associated with a polynomial spectral problem is proposed. The resulting hierarchy possesses a recursion structure controlled by a hereditary operator. The nonisospectral flows generate the time first order dependent symmetries of the isospectral hierarchy, which constitute Virasoro symmetry algebras together with commutative symmetries.

solv-int/9607004

Kirill L. Vaninsky

K.L. Vaninsky

A Convexity Theorem in the Scattering Theory for the Dirac Operator

20 pages, AMS-TEX

Trans. AMS, vol 350, no. 5, pp. 1895--1911.

solv-int nlin.SI

The Dirac operator enters into zero curvature representation for the cubic nonlinear Schr\"{o}dinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The action variables turn out to be limiting boundary values of the imaginary part of this map. We describe the image of the momentum map (convexity theorem) in the simplest case of a potential from the Schwartz class. We apply this description to the invariant manifolds for the nonlinear Schr\"{o}dinger equation.

solv-int/9607005

Juri Suris

Yuri B. Suris (University of Bremen)

Some further curiosities from the world of integrable lattice systems and their discretizations

14 pp., LaTeX

Symmetries and Integrability of Difference Equations, Eds. P.Clarkson, F.Nijhoff, Cambridge Univ. Press, 1999, p. 79-94.

solv-int nlin.SI

Unexpected relations are found between the Toda lattice, the relativistic Toda lattice and the Bruschi--Ragnisco lattice, as well as between their integrable discretizations.

solv-int/9607006

Nalini Joshi

Nalini Joshi

A Local Asymptotic Analysis of the First Discrete Painlev\'e Equation as the Discrete Independent Variable Approaches Infinity

21 pages in LaTeX2e, to appear in \textit{Methods and Applications of Analysis}

solv-int nlin.SI

The first discrete Painlev\'e equation (dPI), which appears in a model of quantum gravity, is an integrable nonlinear nonautonomous difference equation which yields the well known first Painlev\'e equation (PI) in a continuum limit. The asymptotic study of its solutions as the discrete time-step $n\to\infty$ is important both for physical application and for checking the accuracy of its role as a numerical discretization of PI. Here we show that the asymptotic analysis carried out by Boutroux (1913) for PI as its independent variable approaches infinity can also be achieved for dPI as its discrete independent variable approaches the same limit.

solv-int/9607007

Nalini Joshi

Nalini Joshi and Gopala K. Srinivasan

The Radius of Convergence and the Well-Posedness of the Painlev\'e Expansions of the Korteweg-deVries equation

9 pages in AMSTeX, to appear \textit{Nonlinearity}

10.1088/0951-7715/10/1/005

solv-int nlin.SI

In this paper we obtain explicit lower bounds for the radius of convergence of the Painlev\'e expansions of the Korteweg-de-Vries equation around a movable singularity manifold ${\Cal S}$ in terms of the sup norms of the arbitrary functions involved. We use this estimate to prove the well-posedness of the singular Cauchy problem on ${\Cal S}$ in the form of continuous dependence of the meromorphic solution on the arbitrary data.

solv-int/9607008

Kirill L. Vaninsky

K.L. Vaninsky

Gibbs' States for Moser-Calogero Potentials

8 pages, LATEX

Intern J. Mod. Phys. B, vol. 11, no. 1-2, pp. 203-211 (1997)

10.1142/S0217979297000277

solv-int nlin.SI

We present two independent approaches for computing the thermodynamics for classical particles interacting via the Moser--Calogero potential. Combining the results we propose the form of equation of state or, what is equivalent, the asymptotics of the Jacobian between volume elements corresponding two symplectic structures on the phase space.

solv-int/9608001

Adrian-Stefan Carstea

A. S. C\^arstea

On the dynamics of rational solutions for 1-D generalized Volterra system

9 pages, Latex

10.1016/S0375-9601(97)00481-7

FT-419-1996

solv-int nlin.SI

The Hirota bilinear formalism and soliton solutions for a generalized Volterra system is presented. Also, starting from the soliton solutions, we obtain a class of nonsingular rational solutions using the "long wave" limit procedure of Ablowitz and Satsuma, and appropriate "gauge" transformations. Their properties are also discussed and it is shown that these solutions interact elastically with no phase shift.

solv-int/9608002

David Fairlie

D.B. Fairlie

Equations with an infinite number of explicit Conservation Laws

12 pages, latex, no figures

DTP/94/41 to appear in Proc.Roy.Soc.Edin

solv-int nlin.SI

A large class of first order partial nonlinear differential equations in two independent variables which possess an infinite set of polynomial conservation laws derived from an explicit generating function is constructed. The conserved charge densities are all homogeneous polynomials in the unknown functions which satisfy the differential equations in question. The simplest member of the class of equations is related to the Born-Infeld equation in two dimensions. It is observed that some members of this class possess identical charge densities. This enables the construction of a set of multivariable equations with an infinite number of conservation laws.

solv-int/9608003

Wen-Xiu Ma

Wen-Xiu Ma, Qing Ding, Wei-Guo Zhang and Bao-Qun Lu

Binary Nonlinearization of Lax pairs of Kaup-Newell Soliton Hierarchy

15 pages, plain+ams tex, to be published in Il Nuovo Cimento B

10.1007/BF02743224

solv-int nlin.SI

Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different from the original ones. Binary nonlinearization procedure corresponding to the Bargmann symmetry constraint is carried out for those Lax pairs. The proposed Lax pairs together with adjoint Lax pairs are constrained as a hierarchy of commutative, finite dimensional integrable Hamiltonian systems in the Liouville sense, which also provides us with new examples of finite dimensional integrable Hamiltonian systems. A sort of involutive solutions to the Kaup-Newell hierarchy are exhibited through the obtained finite dimensional integrable systems and the general involutive system engendered by binary nonlinearization is reduced to a specific involutive system generated by mono-nonlinearization.

solv-int/9608004

Tim Baker

T. H. Baker (Uni. of Melbourne) and P. J. Forrester (RIMS, Kyoto University)

The Calogero-Sutherland Model and Generalized Classical Polynomials

LaTeX 2.09, 41 pages, uses subeqnarray.sty

Commun.Math.Phys. 188 (1997) 175-216

10.1007/s002200050161

RIMS-1094

solv-int hep-th nlin.SI

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schr\"odinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case.

solv-int/9608005

YuKui. Zhou

Y-K Zhou (ANU)

Fusion Hierarchies with Open Boundaries and Exactly Solvable Models

8 pages, no figures, talk given in Tianjin, August 1995. To appear in "Statistical Models, Yang-Baxter Equation and Related Topics", M. L. Ge and F. Y. Wu eds, World Scientific, Singapore (1996)

MRR 079-95

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

The formulation of integrable models with open boundary conditions and the functional relations of fused transfer matrices are discussed. It is shown that finite-size corrections to the transfer matrices and unitarity relations of free energies can be obtained from the functional relations. Unitarity relations of surface free energies presented in previous papers are also reviewed.

solv-int/9608006

Ctirad Klimcik

P. Severa

On Simplest Hamiltonian Systems

2 pages, LaTeX

solv-int nlin.SI

Simple Hamiltonian systems, such as mathematical pendulum or Euler equations for rigid body, are solved without computation. It is nothing but a joke but maybe you will find it nice.

solv-int/9608007

Simon Labrunie

Simon Labrunie and Robert Conte (Service de physique de l'\'etat condens\'e, CEA Saclay, Gif-sur-Yvette, France)

A geometrical method towards first integrals for dynamical systems

15 pages, RevTeX with aps and prb styles

10.1063/1.531772

preprint SPEC s96/017

solv-int nlin.SI

We develop a method, based on Darboux' and Liouville's works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements' form. We apply it to three dynamical systems: Lotka--Volterra, Lorenz and Rikitake.

solv-int/9608008

Simon Labrunie

Simon Labrunie and Robert Conte (Service de physique de l'\'etat condens\'e, CEA Saclay, Gif-sur-Yvette, France)

Discrete version of the Chazy class III equation

8 pages, LaTeX

preprint SPEC s96/039

solv-int nlin.SI

We study the discretisation of the Chazy class III equation by two means: a discrete Painlev\'e test, and the preservation of a two-parameter solution to the continuous equation. We get that way a best discretisation scheme.

solv-int/9608009

Kakei Saburo

Saburo Kakei

Common Algebraic Structure for the Calogero-Sutherland Models

7 pages, LaTeX, no figures, some text and references added, minor misprints corrected

J. Phys. A29 (1996) L619-L624

10.1088/0305-4470/29/24/002

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

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case.

solv-int/9608010

Jarmo Hietarinta

Jarmo Hietarinta

Nambu tensors and commuting vector fields

9 pages in LaTeX2e

solv-int hep-th nlin.SI

Takhtajan has recently studied the consistency conditions for Nambu brackets, and suggested that they have to be skew-symmetric, and satisfy Leibnitz rule and the Fundamental Identity (FI, it is a generalization of the Jacobi identity). If the n'th order Nambu brackets in dimension N is written in terms of the Nambu tensor \eta, the FI implies two conditions on it, one algebraic and one differential. The algebraic part of FI implies decomposability of \eta and in this letter we show that the Nambu bracket can then be written in terms of the usual totally antisymmetric n-dimensional tensor and n vector fields D. Our main result is that the differential part of the FI is satisfied iff the vector fields D commute. Examples are provided by integrable Hamiltonian systems. It turns out that then the Nambu bracket itself guarantees that the motions stays on the manifold defined by the constants of motion of the integrable system, while the n-1 Nambu Hamiltonians determine the (possibly non-integrable) motion on this manifold.

solv-int/9609001

Yuji Kodama

Yuji Kodama and Jian Ye

Toda lattices with indefinite metric II: Topology of the iso-spectral manifolds

LaTex 20 pages with 4 figures

solv-int hep-th nlin.SI

We consider the iso-spectral real manifolds of tridiagonal Hessenberg matrices with real eigenvalues. The manifolds are described by the iso-spectral flows of indefinite Toda lattice equations introduced by the authors [Physica, 91D (1996), 321-339]. These Toda lattices consist of $2^{N-1}$ different systems with hamiltonians $H = (1/2) \sum_{k=1}^{N} y_k^2 + \sum_{k=1}^{N-1} s_ks_{k+1} \exp(x_k-x_{k+1})$, where $s_i=\pm 1$. We compactify the manifolds by adding infinities according to the Toda flows which blow up in finite time except the case with all $s_is_{i+1}=1$. The resulting manifolds are shown to be nonorientable for $N>2$, and the symmetric group is the semi-direct product of $(\ZZ_2)^{N-1}$ and the permutation group $S_N$. These properties identify themselves with ``small covers'' introduced by Davis and Januszkiewicz [Duke Mathematical Journal, 62 (1991), 417-451]. As a corollary of our construction, we give a formula on the total numbers of zeroes for a system of exponential polynomials generated as Hankel determinant.

solv-int/9609002

Leon Jerome

M. Boiti, J. Leon, F. Pempinelli

Nonlinear Spectral Characterization of Discrete Data

RevTex file, to appear in Physical Review E

10.1103/PhysRevE.54.5739

solv-int nlin.SI

The explicit analytical expression of the Nonlinear Fourier Transform (NFT) of a finite set of data is provided. Then a simple recursion relation for the NFT is constructed as a function of the spectral parameter. These tools provide a complete characterization of the nonlinear coherent structures (solitons, breathers, ...) present in numerical or experimental data representing the solution, at a given value of time, of a nonlinear evolution equation (e.g. of the nonlinear Schroedinger family).

solv-int/9609003

Andrey V. Tsiganov

S. Rauch-Wojciechowski and A.V. Tsiganov

Quasi-point separation of variables for the Henon-Heiles system and a system with quartic potential

11 pages, Latex

10.1088/0305-4470/29/23/032

solv-int hep-th nlin.SI

We examine the problem of integrability of two-dimensional Hamiltonian systems by means of separation of variables. The systematic approach to construction of the special non-pure coordinate separation of variables for certain natural two-dimensional Hamiltonians is presented. The relations with SUSY quantum mechanics are discussed.

solv-int/9609004

Pac3

P.A. Clarkson, E.L. Mansfield and T.J. Priestley (University of Kent, Canterbury, UK)

Symmetries of a class of Nonlinear Third Order Partial Differential Equations

22 pages, tex, Mathematical and Computer Modelling (to appear)

UKC/IMS/95/27

solv-int nlin.SI

In this paper we study symmetry reductions of a class of nonlinear third order partial differential equations $u_t -\epsilon u_{xxt} +2\kappa u_x= u u_{xxx} +\alpha u u_x +\beta u_x u_{xx}$ where $\epsilon$, $\kappa$, $\alpha$ and $\beta$ are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters $\epsilon=1$, $\alpha=-1$, $\beta=3$ and $\kappa=\tfr12$, admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters $\epsilon=0$, $\alpha=1$, $\beta=3$ and $\kappa=0$, admits a ``compacton'' solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters $\epsilon=1$, $\alpha=-3$ and $\beta=2$, has a ``peakon'' solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.

solv-int/9609005

Pac3

A.P. Bassom (University of Exeter, UK), P.A. Clarkson (University of Kent, Canterbury, UK), C.K. Law (National Sun Yat-sen University, Taiwan) and J.B. McLeod (University of Pittsburgh, USA)

Application of Uniform Asymptotics to the Second Painlev{\'e} Transcendent

28 pages, amstex, no figures

UKC/IMS/96/45

solv-int nlin.SI

In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlev{\'e} equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort. Our method is reliant on finding uniform approximations of differential equations of the generic form ${d^2\phi}/{d\eta^2} = - \xi^2F(\eta,\xi)\phi$ as the complex-valued parameter $\xi \to \infty.$ The details of the treatment rely heavily on the locations of the zeros of the function $F$ in this limit. If they are isolated then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of $F$ coalesce as $|\xi| \to \infty$ then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the ``classical'' connection problem associated with the second Painlev{\'e} transcendent. Future papers will show how the technique can be applied with very little change to the other Painlev{\'e} equations, and to the wider problem of the asymptotic behaviour of the general solution to any of these equations.

solv-int/9609006

Wen-Xiu Ma

Wen-Xiu Ma and Kam-Shun Li

Virasoro Symmetry Algebra of Dirac Soliton Hierarchy

8 pages, latex, to appear in Inverse Problems

solv-int nlin.SI

A hierarchy of first-degree time-dependent symmetries is proposed for Dirac soliton hierarchy and their commutator relations with time-dependent symmetries are exhibited. Meantime, a hereditary structure of Dirac soliton hierarchy is elucidated and a Lax operator algebra associated with Virasoro symmetry algebra is given.

solv-int/9609007

Andrey V. Tsiganov

A.V. Tsiganov (St.Petersburg University)

The Kowalewski top: a new Lax representation

17 pages, Latex

10.1063/1.531850

ISRN-LiTH-MAT-R-95-27, 1995

solv-int nlin.SI

The 2x2 monodromy matrices for the Kowalewski top on the Lie algebras e(3), so(4) and so(3,1) are presented. The corresponding quadratic R-matrix structure is the dynamical deformation of the standard R-matrix algebras. Some tops and Toda lattices related to the Kowalewski top are discussed.

solv-int/9609008

Francois Delduc

F. Delduc, L. Gallot

N=2 KP and KdV hierarchies in extended superspace

18 pages, LaTeX file, important reference added

Commun.Math.Phys. 190 (1997) 395-410

10.1007/s002200050246

ENSLAPP-L-617

solv-int hep-th nlin.SI

We give the formulation in extended superspace of an $N=2$ supersymmetric KP hierarchy using chirality preserving pseudo-differential operators. We obtain two quadratic hamiltonian structures, which lead to different reductions of the KP hierarchy. In particular we find two different hierarchies with the $N=2$ classical super-${\cal W}_n$ algebra as a hamiltonian structure. The relation with the formulation in $N=1$ superspace is carried out.

solv-int/9609009

Leonid Bogdanov

L.V. Bogdanov (IINS, Landau Institute, Moscow) and B.G. Konopelchenko (Universit\'a degli Studi di Lecce)

Analytic-bilinear approach to integrable hierarchies. I.Generalized KP hierarchy

25 pages, LaTeX

solv-int nlin.SI

Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Resolution of these functional equations leads to the $\tau$-function and addition formulae to it. General discrete transformations of the $\tau$-function are presented in the determinant form. Closed one-form and other formulae also arise naturally within the approach proposed. Generalized KP hierarchy written in terms of different invariants of Combescure symmetry transformations coincides with the usual KP hierarchy and the mKP hierarchy.

solv-int/9609010

Tim Baker

T. H. Baker and P. J. Forrester (Uni. of Melbourne)

The Calogero-Sutherland Model and Polynomials with Prescribed Symmetry

LaTeX 2.09, 31 pages

10.1016/S0550-3213(97)00112-0

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

The Schr\"odinger operators with exchange terms for certain Calogero-Sutherland quantum many body systems have eigenfunctions which factor into the symmetric ground state and a multivariable polynomial. The polynomial can be chosen to have a prescribed symmetry (i.e. be symmetric or antisymmetric) with respect to the interchange of some specified variables. For four particular Calogero-Sutherland systems we construct an eigenoperator for these polynomials which separates the eigenvalues and establishes orthogonality. In two of the cases this involves identifying new operators which commute with the corresponding Schr\"odinger operators. In each case we express a particular class of the polynomials with prescribed symmetry in a factored form involving the corresponding symmetric polynomials.

solv-int/9610001

Juri Suris

Yuri B. Suris (University of Bremen)

Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices

35 pages, LaTeX

J. Math. Phys., 1997, V. 38, p. 4179-4201.

10.1063/1.532090

solv-int nlin.SI

A new Lax representation for the Bogoyavlensky lattice is found, its $r$--matrix interpretation is elaborated. The $r$--matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of associative algebras. The theory of such nonlocal structures is developed, the Poisson property of the monodromy map is worked out in the most general situation. Some problems concerning the duality of Lax representations are raised.

solv-int/9610002

Paolo Casati (Dip. di Matematica, Univ. di Milano, Italy), Gregorio Falqui (SISSA/ISAS, Trieste, Italy), Franco Magri (Dip. di Matematica, Univ. di Milano, Italy), and Marco Pedroni (Dip. di Matematica, Univ. di Genova, Italy)

Darboux Coverings and Rational Reductions of the KP Hierarchy

16 pages, LaTeX

SISSA 131/96/FM

solv-int nlin.SI

We use the method of Darboux coverings to discuss the invariant submanifolds of the KP equations, presented as conservation laws in the space of monic Laurent series in the spectral parameter (the space of the Hamiltonian densities). We identify a special class of these submanifolds with the rational invariant submanifolds entering matrix models of $2D$--gravity, recently characterized by Dickey and Krichever. Four examples of the general procedure are provided.

solv-int/9610003

Andrey V. Tsiganov

A.V. Tsiganov

Automorphisms of sl(2) and dynamical r-matrices

14 pages, Latex

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

Two outer automorphisms of infinite-dimensional representations of $sl(2)$ algebra are considered. The similar constructions for the loop algebras and yangians are presented. The corresponding linear and quadratic $R$-brackets include the dynamical $r$-matrices.

solv-int/9610004

Parma

N.A. Gromov, I.V. Kostyakov and V.V. Kuratov

Contractions of Integrable Equations

6 pages, LaTeX, submitted to Proceedings of ' II International Workshop on Classical and Quantum Integrible Systems' (Dubna, 8-12 July,1996), to be published in Int.J.Mod.Phys

10.1142/S0217751X97000256

solv-int nlin.SI

The contraction is applied to obtaining of integrable systems associated with nonsemisimple algebras. The effect of contraction is splitting off some components from initial system without loss of integrability.

solv-int/9610005

Kenichi Maruno

K. Maruno, K. Kajiwara, S. Nakao, M. Oikawa

Bilinearization of Discrete Soliton Equations and Singularity Confinement

14 pages, LaTex

10.1016/S0375-9601(97)00171-0

solv-int nlin.SI

Bilinear forms for some nonlinear partial difference equations(discrete soliton equations) are derived based on the results of singularity confinement. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed.

solv-int/9610006

Mts

Andrey Yu. Boldin and Ruslan A. Sharipov (Baskir State University, Math. Department, Russia)

On the solution of normality equations for the dimension $n\geq 3$

AmS-TeX, amsppt style, 18 pages

solv-int nlin.SI

The normality equations for the Newtonian dynamical systems on an arbitrary Riemannian manifold of the dimension $n \geq 3$ are considered. Locally the solution of such equations reduces to three possible cases: in two of them the solution is written out explicitly, and in the third case the equations of normality are reduced to an ordinary differential equation of the second order. Some new examples of explicit solutions of normality equations are constructed.

solv-int/9610007

Robert Conte

R. Conte (CEA Saclay), M. Musette (VUB Brussels)

A new method to test discrete Painlev\'e equations

12 pages, no figure, standard Latex, to appear in Physics Letters A

10.1016/S0375-9601(96)00783-9

S96/032

solv-int nlin.SI

Necessary discretization rules to preserve the Painlev\'e property are stated. A new method is added to the discrete Painlev\'e test, which perturbs the continuous limit and generates infinitely many no-log conditions.

solv-int/9610008

Wenxiu Ma

Wen-Xiu Ma and Fu-Kui Guo

Lax Representations and Zero Curvature Representations by Kronecker Product

9 pages, Latex, to appear in Intern. J. Theoret. Phys

10.1007/BF02435889

solv-int nlin.SI

It is showed that Kronecker product can be applied to construct not only new Lax representations but also new zero curvature representations of integrable models. Meantime a different characteristic between continuous and discrete zero curvature equations is pointed out.

solv-int/9610009

Jan de Gier

Jan de Gier and Bernard Nienhuis (University of Amsterdam, The Netherlands)

The exact solution of an octagonal rectangle triangle random tiling

26 pages, LaTeX, including 5 figures, to appear in J. Stat. Phys

J. Stat. Phys. 87 (1997) 415-437

10.1007/BF02181494

ITFA 96-35

solv-int cond-mat nlin.SI

We present a detailed calculation of the recently published exact solution of a random tiling model possessing an eight-fold symmetric phase. The solution is obtained using Bethe Ansatz and provides closed expressions for the entropy and phason elastic constants. Qualitatively, this model has the same features as the square-triangle random tiling model. We use the method of P. Kalugin, who solved the Bethe Ansatz equations for the square-triangle tiling, which were found by M. Widom.

solv-int/9610010

Kakei Saburo

Saburo Kakei

An orthogonal basis for the $B_N$-type Calogero model

9 pages, LaTeX, no figures, several errors are corrected, Appendix is added

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

We investigate algebraic structure for the $B_N$-type Calogero model by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis.

solv-int/9610011

Atsuo Kuniba

Zengo Tsuboi (Univ. of Tokyo, Komaba)

Solutions of Discretized Affine Toda Field Equations for A_{n}^{(1)}, B_{n}^{(1)}, C_{n}^{(1)}, D_{n}^{(1)}, A_{n}^{(2)} and D_{n+1}^{(2)}

22 pages, no figure, LaTeX: Introduction, Summary and Discussion are revised. (e-mail: [email protected])

J.Phys.Soc.Jap. 66 (1997) 3391-3398

10.1143/JPSJ.66.3391

solv-int hep-th nlin.SI

It is known that a family of transfer matrix functional equations, the T-system, can be compactly written in terms of the Cartan matrix of a simple Lie algebra. We formally replace this Cartan matrix of a simple Lie algebra with that of an affine Lie algebra, and then we obtain a system of functional equations different from the T-system. It may be viewed as an X_{n}^{(a)} type affine Toda field equation on discrete space time. We present, for A_{n}^{(1)}, B_{n}^{(1)}, C_{n}^{(1)}, D_{n}^{(1)}, A_{n}^{(2)} and D_{n+1}^{(2)}, its solutions in terms of determinants or Pfaffians.

solv-int/9610012

J. T. Liu and D. F. Wang

Integrabilities of the long range t-J model of twisted boundary condition

preprint of Rockefeller-ETH, submitted to PRB rapid communication

10.1103/PhysRevB.55.R3344

solv-int cond-mat nlin.SI

The integrability of the one-dimensional long range supersymmetric t-J model has previously been established for both open systems and those closed by periodic boundary conditions through explicit construction of its integrals of motion. Recently the system has been extended to include the effect of magnetic flux, which gives rise to a closed chain with twisted boundary conditions. While the t-J model with twisted boundary conditions has been solved for the ground state and full energy spectrum, proof of its integrability has so far been lacking. In this letter we extend the proof of integrability of the long range supersymmetric t-J model and its SU(m|n) generalization to include the case of twisted boundary conditions.

solv-int/9610013

Nalini Joshi

Nalini Joshi and Johannes A. Petersen

Complex Blow-Up in Burgers' Equation: an Iterative Approach

11 pages in LaTeX. To appear in Bull. Aust. Math. Soc

solv-int nlin.SI

We show that for a given holomorphic noncharacteristic surface S in two-dimensional complex space, and a given holomorphic function on S, there exists a unique meromorphic solution of Burgers' equation which blows up on S. This proves the convergence of the formal Laurent series expansion found by the Painlev\'e test. The method used is an adaptation of Nirenberg's iterative proof of the abstract Cauchy-Kowalevski theorem.

solv-int/9611001

Nalini Joshi

Rod Halburd and Nalini Joshi

The Coalescence Limit of the Second Painlev\'E Equation

16 pages in LaTeX (1 figure included)

Stud. Appl. Math. 97 (1996) 1--15

solv-int nlin.SI

In this paper, we study a well known asymptotic limit in which the second Painlev\'e equation (P_II) becomes the first Painlev\'e equation (P_I). The limit preserves the Painlev\'e property (i.e. that all movable singularities of all solutions are poles). Indeed it has been commonly accepted that the movable simple poles of opposite residue of the generic solution of P_{II} must coalesce in the limit to become movable double poles of the solutions of P_I, even though the limit naively carried out on the Laurent expansion of any solution of P_{II} makes no sense. Here we show rigorously that a coalescence of poles occurs. Moreover we show that locally all analytic solutions of P_I arise as limits of solutions of P_{II}.

solv-int/9611002

V.V.Konotop (University of Madeira, Portugal), M. Salerno (University of Salerno, Italy)

Small-amplitude excitations in a deformable discrete nonlinear Schroedinger equation

18 pages (RevTex), 13 figures available upon request

10.1103/PhysRevE.55.4706

solv-int nlin.SI

A detailed analysis of the small-amplitude solutions of a deformed discrete nonlinear Schr\"{o}dinger equation is performed. For generic deformations the system possesses "singular" points which split the infinite chain in a number of independent segments. We show that small-amplitude dark solitons in the vicinity of the singular points are described by the Toda-lattice equation while away from the singular points are described by the Korteweg-de Vries equation. Depending on the value of the deformation parameter and of the background level several kinds of solutions are possible. In particular we delimit the regions in the parameter space in which dark solitons are stable in contrast with regions in which bright pulses on nonzero background are possible. On the boundaries of these regions we find that shock waves and rapidly spreading solutions may exist.

solv-int/9611003

Alexander V. Razumov

A. V. Razumov, M. V. Saveliev

Some explicit solutions of the Lam\'e and Bourlet type equations

12 pages, LaTeX file

LPTENS-96/61

solv-int dg-ga hep-th math.DG nlin.SI

Some special solutions to the multidimensional Lam\'e and Bourlet type equations are constructed in an explicit form.

solv-int/9611004

D. F. Wang

Spinless Calogero-Sutherland model with twisted boundary condition

preprint of ETH-L, appearing in recent PRE

10.1103/PhysRevE.54.4586

solv-int nlin.SI

In this work, the spinless Calogero-Sutherland model with twisted boundary condition is studied. The ground state wavefunctions, the ground state energies, the full energy spectrum are provided in details.

solv-int/9611005

Jan de Gier

Jan de Gier, Bernard Nienhuis (University of Amsterdam)

On the integrability of the square-triangle random tiling model

11 pages, LaTeX, inluding 2 postscript figures

Phys. Rev. E 55, 3926 (1997)

10.1103/PhysRevE.55.3926

IFTA 96-47

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

It is shown that the square-triangle random tiling model is equivalent to an asymmetric limit of the three colouring model on the honeycomb lattice. The latter model is known to be the O(n) model at T=0 and corresponds to the integrable model connected to the affine $A_2^{(1)}$ Lie algebra. Thus it is shown that the weights of the square-triangle random tiling satisfy the Yang-Baxter equation, albeit in a singular limit of a more general model. The three colouring model for general vertex weights is solved by algebraic Bethe Ansatz.

solv-int/9611006

Denis Uglov

Kouichi Takemura and Denis Uglov

The orthogonal eigenbasis and norms of eigenvectors in the Spin Calogero-Sutherland Model

35 pages, AMSLaTeX

10.1088/0305-4470/30/10/039

RIMS-1114

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

Using a technique based on the Yangian Gelfand-Zetlin algebra and the associated Yangian Gelfand-Zetlin bases we construct an orthogonal basis of eigenvectors in the Calogero-Sutherland Model with spin, and derive product-type formulas for norms of these eigenvectors.

solv-int/9611007

YuKui. Zhou

Y.-K. Zhou and K. D. Schotte

The L-Matrix for the Massive Thirring Model

10 pages, no ps figures, Tex file

Phys Rev D 47 (1993) R1281-R1284

10.1103/PhysRevD.47.R1281

solv-int nlin.SI

As the new results for the massive Thirring model the L-matrix and the algebraic relations for its action angle variables are given. So it is shown most directly that this model which describes self-interacting relativistic Fermions in one-dimensional space is a quantum integrable system.

solv-int/9611008

Andrei Mal'tsev

A.Ya. Maltsev

The conservation of the Hamiltonian structures in Whitham's method of averaging

39 pages, some improvement, corrected misprints

Izvestiya, Mathematics 63:6 (1999), 1171-1201

solv-int hep-th nlin.SI

The work is devoted to the proof of the conservation of local field-theoretical Hamiltonian structures in Whitham's method of averaging. The consideration is based on the procedure of averaging of local Poisson bracket, proposed by B.A.Dubrovin and S.P.Novikov. Using the Dirac procedure of restriction of the Poisson bracket on the submanifold in the functional space, it is shown in the generic case that the Poisson bracket, constructed by method of Dubrovin and Novikov, satisfies the Jacobi identity. Besides that, the invariance of this bracket with respect to the choice of the set of local conservation laws, used in this procedure, is proved.

solv-int/9612001

Uwe Grimm

Uwe Grimm

Representations of Two-Colour BWM Algebras and Solvable Lattice Models

6 pages, LaTeX, heron2e.sty (included), Poster presented at GROUP21

Proceedings of the Quantum Group Symposium at the XXI International Colloquium on Group Theoretical Methods in Physics, edited by H.-D. Doebner and V.K. Dobrev, Heron Press, Sofia (1997), pp. 114-119

solv-int nlin.SI

Many of the known solutions of the Yang-Baxter equation, which are related to solvable lattice models of vertex- and IRF-type, yield representations of the Birman-Wenzl-Murakami algebra. From these, representations of a two-colour generalization of the Birman-Wenzl-Murakami algebra can be constructed, which in turn are used to derive trigonometric solutions to the Yang-Baxter equation. In spirit, this construction resembles the fusion procedure, in the sense that starting from known solutions of the Yang-Baxter equation new solutions can be obtained.

solv-int/9612002

Nicolai Kitanine

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

Correlators of the phase model

LaTeX, 7 pages, One reference has been changed

10.1016/S0375-9601(97)00326-5

ENSLAPP-L-622/96, HU-TFT-96-41

solv-int nlin.SI

We introduce the phase model on a lattice and solve it using the algebraic Bethe ansatz. Time-dependent temperature correlation functions of phase operators and the "darkness formation probability" are calculated in the thermodynamical limit. These results can be used to construct integrable equations for the correlation functions and to calculate there asymptotics.

solv-int/9612003

Zora Thomova

Z. Thomova, P. Winternitz, W.J. Zakrzewski

Solutions of (2+1)-dimensional spin systems

TeX and phyzzx, 33 pages

J.Math.Phys. 39, 3927-3944 (1998)

10.1063/1.532476

CRM-2373

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

We use the methods of group theory to reduce the equations of motion of two spin systems in (2+1) dimensions to sets of coupled ordinary differential equations. We present solutions of some classes of these sets and discuss their physical significance.

solv-int/9612004

Alexander V. Razumov

A. V. Razumov, M. V. Saveliev

Riemannian Manifolds with Diagonal Metric. The Lam\'e and Bourlet Systems

LaTeX file, 22 pages, to appear in the proceedings of the international conference "Selected Topics of Theoretical and Modern Mathematical Physics (SIMI-96)", September 22-29, 1996, Tbilisi, Georgia

solv-int dg-ga hep-th math.DG nlin.SI

We discuss a Lie algebraic and differential geometry construction of solutions to some multidimensional nonlinear integrable systems describing diagonal metrics on Riemannian manifolds, in particular those of zero and constant curvature. Here some special solutions to the Lam\'e and Bourlet type equations, determining by n arbitrary functions of one variable are obtained in an explicit form. For the case when the sum of the diagonal elements of the metric is a constant, these solutions are expressed as a product of the Jacobi elliptic functions and are determined by 2n arbitrary constants.

solv-int/9612005

Ryuji Kemmoku 0426-77-1111x3377

Ryuji Kemmoku

Difference Operator Approach to the Moyal Quantization and Its Application to Integrable Systems

19 pages, to appear in J. Phys. Soc. Jpn

10.1143/JPSJ.66.51

TMUP-HEL-9609

solv-int nlin.SI

Inspired by the fact that the Moyal quantization is related with nonlocal operation, I define a difference analogue of vector fields and rephrase quantum description on the phase space. Applying this prescription to the theory of the KP-hierarchy, I show that their integrability follows to the nature of their Wigner distribution. Furthermore the definition of the ``expectation value'' clarifies the relation between our approach and the Hamiltonian structure of the KP-hierarchy. A trial of the explicit construction of the Moyal bracket structure in the integrable system is also made.

solv-int/9612006

Adam Doliwa

A. Doliwa

Geometric Discretisation of the Toda System

12 pages, LaTeX, 2 Postscript figures

Phys. Lett. A 234 (1997) 187.

10.1016/S0375-9601(97)00477-5

ROME1-1160/96

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

The Laplace sequence of the discrete conjugate nets is constructed. The invariants of the nets satisfy, in full analogy to the continuous case, the system of difference equations equivalent to the discrete version of the generalized Toda equation.

solv-int/9612007

Adam Doliwa

A. Doliwa and P. M. Santini

Multidimensional Quadrilateral Lattices are Integrable

18 pages, LaTeX, 6 Postscript figures

Phys. Lett. A 233 (1997) 365.

10.1016/S0375-9601(97)00456-8

ROME1-1162/96

solv-int hep-lat nlin.SI

The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in particular, it is clarified the number of initial--boundary data which define the lattice uniquely.

solv-int/9612008

Jaroslav Dittrich

J. Dittrich, V. I. Inozemtsev

On the two-magnon bound states for the quantum Heisenberg chain with variable range exchange

8 pages, latex, no figures

10.1142/S0217984997000554

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

The spectrum of finite-difference two-magnon operator is investigated for quantum S=1/2 chain with variable range exchange of the form $h(j-k)\propto \sinh^{-2}a(j-k)$. It is found that usual bound state appears for some values of the total pseudomomentum of two magnons as for the Heisenberg chain with nearest-neighbor spin interaction. Besides this state, a new type of bound state with oscillating wave function appears at larger values of the total pseudomomentum.

solv-int/9612009

Yves Brihaye

Y. Brihaye, S. Giller, P. Kosinski, J. Nuyts

Irreducible Representations of an Algebra underlying Hidden Symmetries of a class of Quasi Exactly Solvable Systems of Equations

38 pages, latex

10.1007/s002200050133

solv-int nlin.SI

The set of linear, differential operators preserving the vector space of couples of polynomials of degrees n and n-2 in one real variable leads to an abstract associative graded algebra A(2). The irreducible, finite dimensional representations of this algebra are classified into five infinite discrete sets and one exceptional case. Their matrix elements are given explicitely. The results are related to the theory of quasi exactly solvable equations.

solv-int/9612010

Robert Carroll

R. Carroll and J. Chang (Mathematics Dept., University of Illinois, Urbana, IL)

The Whitham equations revisited

Latex 40 pages

solv-int nlin.SI

We survey some topics involving the Whitham equations, concentrating on the role of the Baker Akhiezer function in averaging. Some connections to symplectic geometry and Seiberg-Witten theory are indicated.

solv-int/9612011

Robert Carroll

R. Carroll (Mathematics Dept., University of Illinois, Urbana, IL)

Some kernels on a Riemann surface

Latex 22 pages

solv-int nlin.SI

We discuss certain kernels on a Riemann surface, constructed mainly via Baker Akhiezer functions, and indicate relations to dispersionless theory.

solv-int/9612012

John Harnad

J. Harnad

Bispectral Operators of Rank 1 and Dual Isomonodromic Deformations

16pgs, AMSTeX

CRM Proc. Lecture Notes 11, 155-167 (Amer. Math. Soc., Providence RI, 1997)

CRM 2443 (1996)

solv-int hep-th nlin.SI

A comparison is made between bispectral operator pairs and dual pairs of isomonodromic deformation equations. Through examples, it is shown how operators belonging to rank one bispectral algebras may be viewed equivalently as defining 1-parameter families of rational first order differential operators with matricial coefficients on the Riemann sphere, whose monodromy is trivial. By interchanging the r\^oles of the two variables entering in the bispectral pair, a second 1-parameter family of operators with trivial monodromy is obtained, which may be viewed as the dual isomonodromic deformation system.

solv-int/9701001

Arthur Vartanian

A. V. Kitaev and A. H. Vartanian

Leading Order Temporal Asymptotics of the Modified Non-Linear Schrodinger Equation: Solitonless Sector

29 pages, 5 figures, LaTeX, revised version of the original submission, to be published in Inverse Problems

10.1088/0266-5611/13/5/014

solv-int nlin.PS nlin.SI patt-sol physics.plasm-ph

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution equations (NLEEs) integrable in the sense of the inverse scattering method, we obtain, in the solitonless sector, the leading-order asymptotics as $t$ tends to plus and minus infinity of the solution to the Cauchy initial-value problem for the modified non-linear Schrodinger equation: also obtained are analogous results for two gauge-equivalent NLEEs; in particular, the derivative non-linear Schrodinger equation.

solv-int/9701002

Jarmo Hietarinta

Jarmo Hietarinta

Painleve equations in terms of entire functions

22 pages, lectures given at the summer school ``The Painleve property, one century later'', Cargese, 3-22 June, 1996

solv-int nlin.SI

In these lectures we discuss how the Painleve equations can be written in terms of entire functions, and then in the Hirota bilinear (or multilinear) form. Hirota's method, which has been so useful in soliton theory, is reviewed and connections from soliton equations to Painleve equations through similarity reductions are discussed from this point of view. In the main part we discuss how singularity structure of the solutions and formal integration of the Painleve equations can be used to find a representation in terms of entire functions. Sometimes the final result is a pair of Hirota bilinear equations, but for $P_{VI}$ we need also a quadrilinear expression. The use of discrete versions of Painleve equations is also discussed briefly. It turns out that with discrete equations one gets better information on the singularities, which can then be represented in terms of functions with a simple zero.

solv-int/9701003

Craig A. Tracy

C. A. Tracy and H. Widom

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

29 pages, no figures, LaTeX file

Commun. Math. Phys 190 (1998) 697-721

10.1007/s002200050257

solv-int funct-an math.FA nlin.SI

The small t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, q_k(t), have the representation q_k(t) = log det(I-lambda K_k) - log det(I-lambda K_{k-1}) where K_k are integral operators. This class includes the n-periodic cylindrical Toda equations. For n=2 our results reduce to the previously computed asymptotics of the 2D radial sinh-Gordon equation and for n=3 (and with an additional symmetry contraint) they reduce to earlier results for the radial Bullough-Dodd equation.

solv-int/9701004

V. Kuznetsov

V.B. Kuznetsov, F.W. Nijhoff and E.K. Sklyanin

Separation of variables for the Ruijsenaars system

26 pages, LaTex, no figures

Commun.Math.Phys. 189(1997) 855-877

10.1007/s002200050231

CNLS-Leeds preprint, 10 January 1997

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

We construct a separation of variables for the classical n-particle Ruijsenaars system (the relativistic analog of the elliptic Calogero-Moser system). The separated coordinates appear as the poles of the properly normalised eigenvector (Baker-Akhiezer function) of the corresponding Lax matrix. Two different normalisations of the BA functions are analysed. The canonicity of the separated variables is verified with the use of r-matrix technique. The explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n=2 and n=3. Taking nonrelativistic limit we also construct a separation of variables for the elliptic Calogero-Moser system.

solv-int/9701005

Manuel

Q. P. Liu, M. Manas

Crum Transformations and Wronskian Type Solutions for Supersymmetric KdV equation

13 pp, AMS-LaTeX

Phys.Lett.B396:133,1997

10.1016/S0370-2693(97)00134-2

solv-int hep-th nlin.SI

Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard soliton of the KdV equation. The KdV soliton appears as the body of the super soliton.

solv-int/9701006

Luiz Agostinho Ferreira

Luiz A. Ferreira and Joaquin Sanchez Guillen

Aspects of Solitons in Affine Integrable Hierarchies

39 pages, LaTeX, Two talks presented by the authors at the ``International Workshop on Selected Topics of Theoretical and Modern Mathematical Physics - SIMI/96'', Tbilisi, Georgia, September/96; Some misprints corrected in this replaced version

US-FT/49-96, IFT-P.005/97

solv-int hep-th nlin.SI

We argue that one of the basic ingredients for the appearance of soliton solutions in integrable hierarchies, is the existence of ``vacuum solutions'' corresponding to Lax operators lying in some abelian subalgebra of the associated affine Kac-Moody algebra. Using the dressing transformation we construct the solutions in the orbit of those vacuum solutions, and conjecture that the solitons correspond to some special points in those orbits. The generalized tau-function for those hierarchies are defined for integrable highest weight representations. It applies for any level of the representation. We illustrate our methods with the recently proposed non abelian Toda models coupled to matter fields. A very special class of such theories possess a U(1) Noether charge that is proportional to a topological charge. That leads to a mechanism that confines the matter fields inside the solitons.

solv-int/9701007

Manuel

Q. P. Liu, M. Manas

Darboux Transformation for the Manin-Radul Supersymmetric KdV equation

14 pp, 2 figures, AMS-LaTeX, to appear in Phys. Lett. B

Phys. Lett. B, 394 (1997) 337

10.1016/S0370-2693(97)00026-9

solv-int hep-th nlin.SI

In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots.

solv-int/9701008

Sergei Kharchev

A. Yu. Orlov and P. Winternitz

$P_\infty$ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators

21 pages, Latex, no figures (some references added and misprints are corrected)

10.1142/S0217979297001532

solv-int nlin.SI

The symmetry algebra $P_\infty = W_\infty \oplus H \oplus I_\infty$ of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equations are obtained. It is shown that one (``positive'') half of the point symmetries belongs to the $W_\infty$ symmetries while the other (``negative'') part belongs to the $I_\infty$ ones. The corresponing action on the tau-function is obtained for the positive part of the symmetries. The negative part can not be obtained from the free fermion algebra. A new embedding of the Virasoro algebra into $gl(\infty )$n describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.

solv-int/9701009

V. Kuznetsov

Vadim B. Kuznetsov

Separation of variables for the Dn type periodic Toda lattice

15 pages, LaTex, no figures

J Phys A 30 (1997) 2127-2138

10.1088/0305-4470/30/6/033

preprint, University of Leeds, October 1996

solv-int hep-th nlin.SI

We prove separation of variables for the most general (Dn type) periodic Toda lattice with 2x2 Lax matrix. It is achieved by finding proper normalisation for the corresponding Baker-Akhiezer function. Separation of variables for all other periodic Toda lattices associated with infinite series of root systems follows by taking appropriate limits.

solv-int/9701010

Juri Suris

Yuri B. Suris (University of Bremen)

A note on the integrable discretization of the nonlinear Schr\"odinger equation

24 pages, LaTeX, revised and extended version

Inverse Problems, 1997, V. 13, p. 1121-1136.

10.1088/0266-5611/13/4/016

solv-int nlin.SI

We revisit integrable discretizations for the nonlinear Schr\"odinger equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the non-locality, can be overcome. Namely, we factorize the non-local difference scheme into the product of local ones. This must improve the performance of the scheme in the numerical computations dramatically. Using the equivalence of the Ablowitz--Ladik and the relativistic Toda hierarchies, we find the interpolating Hamiltonians for the local schemes and show how to solve them in terms of matrix factorizations.

solv-int/9701011

Peter Schupp

Branislav Jurco, Peter Schupp

Twisted Quantum Lax Equations

23 pages, latex

Int.J.Mod.Phys. A12 (1997) 5735-5752

10.1142/S0217751X97003005

LMU-TPW 97-1, CRM-2448

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

We give the construction of twisted quantum Lax equations associated with quantum groups. We solve these equations using factorization properties of the corresponding quantum groups. Our construction generalizes in many respects the Adler-Kostant-Symes construction for Lie groups and the construction of M. A. Semenov Tian-Shansky for the Lie-Poisson case.

solv-int/9701012

Luiz Agostinho Ferreira

H.S. Blas Achic, L.A. Ferreira, J.F. Gomes and A.H. Zimerman

Some comments on the bi(tri)-Hamiltonian structure of Generalized AKNS and DNLS hierarchies

10 pages, LaTex

Phys.Lett. A237 (1998) 225-233

10.1016/S0375-9601(97)00865-7

IFT-P/006/97

solv-int nlin.SI

We give the correct prescriptions for the terms involving the inverse of the derivative of the delta function, in the Hamiltonian structures of the AKNS and DNLS systems, in order for the Jacobi identities to hold. We also establish that the sl(2) AKNS and DNLS systems are tri-Hamiltonians and construct two compatible Hamiltonian structures for the sl(3) AKNS system. We also give a derivation of the recursion operator for the sl(n+1) DNLS system.

solv-int/9701013

Luiz Agostinho Ferreira

Luiz A. Ferreira

The structures underlying soliton solutions in integrable hierarchies

Talk given at the I Latin American Symposium on High Energy Physics, I SILAFAE, Merida, Mexico, November/96, 5 pages, LaTeX, needs aipproc.tex, aipproc.sty, aipproc.cls, available from ftp://ftp.aip.org/ems/tex/macros/proceedings/6x9/

10.1063/1.53232

IFT-P/009/97

solv-int hep-th nlin.SI

We point out that a common feature of integrable hierarchies presenting soliton solutions is the existence of some special ``vacuum solutions'' such that the Lax operators evaluated on them, lie in some abelian subalgebra of the associated Kac-Moody algebra. The soliton solutions are constructed out of those ``vacuum solitons'' by the dressing transformation procedure.

solv-int/9701014

Oleg Kiselev

O.M. Kiselev (Ufa Institute of Mathematics, Russian Acad. of Sci.)

The Fourier method for the linearized Davey-Stewartson I equation

4 pages, LaTex

Complex analysis, differential equations, numerical methods and applications. vol.III, Ufa, 1996, p. 93-97, (in russian)

solv-int nlin.SI

The linearized Davey-Stewartson equation with varing coefficients is solved by Fourier method. The approach uses the inverse scattering transform for the Davey-Stewartson equation.

solv-int/9701015

Koichi Takemura

Kouichi Takemura

The Yangian Symmetry in the Spin Calogero Model and its Applications

18 pages, AMSLaTeX

10.1088/0305-4470/30/17/025

RIMS-1126

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

By using the non-symmetric Hermite polynomials and a technique based on the Yangian Gelfand-Zetlin bases, we decompose the space of states of the Calogero model with spin into irreducible Yangian modules, construct an orthogonal basis of eigenvectors and derive product-type formulas for norms of these eigenvectors.

solv-int/9701016

Galina A. Korepanova

I.G. Korepanov

Some eigenstates for a model associated with solutions of tetrahedron equation

7 pages, LaTeX

solv-int nlin.SI

Here we present some eigenstates for a 2+1-dimensional model associated with a solution of the tetrahedron equation. The eigenstates include those "particle-like" (namely one-particle and two-particle ones), constructed in analogy with the usual 1+1-dimensional Bethe ansatz, and some simple "string-like" ones.

solv-int/9701017

Henrik Aratyn

H. Aratyn, E. Nissimov and S. Pacheva

Method of Squared Eigenfunction Potentials in Integrable Hierarchies of KP Type

LaTeX, 30pgs, appendix added on binary Darboux-Backlund transformations

10.1007/s002200050338

BGU-97/01/Jan-PH, UICHEP-TH/97-1

solv-int hep-th nlin.SI

The method of squared eigenfunction potentials (SEP) is developed systematically to describe and gain new information about Kadomtsev-Petviashvili (KP) hierarchy and its reductions. Interrelation to the tau-function method is discussed in detail. The principal result, which forms the basis of our SEP method, is the proof that any eigenfunction of the general KP hierarchy can be represented as a spectral integral over the Baker-Akhiezer (BA) wave function with a spectral density expressed in terms of SEP. In fact, the spectral representations of the (adjoint) BA functions can, in turn, be considered as defining equations for the KP hierarchy. The SEP method is subsequently used to show how the reduction of the full KP hierarchy to the constrained KP hierarchies can be given entirely in terms of linear constraint equations on the pertinent tau-functions. The concept of SEP turns out to be crucial in providing a description of constrained KP hierarchies in the language of universal Sato Grassmannian and finding the non-isospectral Virasoro symmetry generators acting on the underlying tau-functions. The SEP method is used to write down generalized binary Darboux-Backlund transformations for constrained KP hierarchies whose orbits are shown to correspond to a new Toda model on a square lattice. As a result, we obtain a series of new determinant solutions for the tau-functions generalizing the known Wronskian (multi-soliton) solutions. Finally, applications to random matrix models in condensed matter physics are briefly discussed.