Datasets:

Abzu

/

arxiv_stem

like 0

Follow

Abzu 6

solv-int/9707003

Shigeki Matsutani

Statistical Mechanics of Elastica on Plane as a Model of Supercoiled DNA-Origin of the MKdV hierarchy-

AMS-Tex Use

10.1088/0305-4470/31/11/017

solv-int nlin.SI

In this article, I have investigated statistical mechanics of a non-stretched elastica in two dimensional space using path integral method. In the calculation, the MKdV hierarchy naturally appeared as the equations including the temperature fluctuation.I have classified the moduli of the closed elastica in heat bath and summed the Boltzmann weight with the thermalfluctuation over the moduli. Due to the bilinearity of the energy functional,I have obtained its exact partition function.By investigation of the system,I conjectured that an expectation value at a critical point of this system obeys the Painlev\'e equation of the first kind and its related equations extended by the KdV hierarchy.Furthermore I also commented onthe relation between the MKdV hierarchy and BRS transformationin this system.

2009-10-30

solv-int/9707004

Vadim Vereschagin

V.L.Vereschagin

Nonlinear Quasiclassics and the Painlev\'e Equations

5 pp., Latex

solv-int nlin.SI

Problem of asymptotic description for global solutions to the six Painleve equations was investigated. Elliptic anzatzes and appropriate modulation equations were written out.

2008-02-03

solv-int/9707005

Alexander Turbiner

Marcos Rosenbaum, Alexander Turbiner and Antonio Capella

Solvability of the G_2 Integrable System

18 pages, LaTeX, some minor typos corrected

Int.J.Mod.Phys. A13 (1998) 3885-3904

10.1142/S0217751X98001815

Mexico ICN-UNAM 97-05

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

It is shown that the 3-body trigonometric G_2 integrable system is exactly-solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the Hamiltonian can be expressed as a quadratic polynomial in the generators of some Lie algebra of differential operators in a finite-dimensional representation. Four infinite families of eigenstates, represented by polynomials, and the corresponding eigenvalues are described explicitly.

2009-10-30

solv-int/9707006

Shigeki Matsutani

Quantum Coupled Nonlinear Schr\"odinger System with Different Masses

AMS-Tex Use

solv-int nlin.SI

In this letter, I have considered one-dimensional quantum system with different masses $m$ and $M$, which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system should be regarded as a modification of velocity rather than that of momentum. I have also considered the three-body wave function and argued its integrable condition.

2008-02-03

solv-int/9707007

Shigeki Matsutani

On Density of State of Quantized Willmore Surface-A Way to Quantized Extrinsic String in R^3

AMS-Tex Use

10.1088/0305-4470/31/15/021

solv-int nlin.SI

Recently I quantized an elastica with Bernoulli-Euler functional in two-dimensional space using the modified KdV hierarchy. In this article, I will quantize a Willmore surface, or equivalently a surface with the Polyakov extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation. In other words, I show that the density of state of the partition function for the quantized Willmore surface is expressed by volume of a subspace of the moduli of the MNV equation.

2009-10-30

solv-int/9707008

V. E. Vekslerchik

V. E. Vekslerchik (Institute for Radiophysics and Electronics, Kharkov, Ukraine)

Functional representation of the Ablowitz-Ladik hierarchy

15 pages, LaTeX

10.1088/0305-4470/31/3/018

solv-int nlin.SI

The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the inverse scattering approach. After establishing the structure of solutions of the auxiliary linear problems, the ALH, which has been originally introduced as an infinite system of difference-differential equations is presented as a finite system of difference-functional equations. The representation obtained, when rewritten in terms of Hirota's bilinear formalism, is used to demonstrate relations between the ALH and some other integrable systems, the Kadomtsev-Petviashvili hierarchy in particular.

2009-10-30

solv-int/9707009

Fritz Gesztesy

Fritz Gesztesy and Ratnam Ratnaseelan

An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

LaTeX, submitted to Reviews in Mathematical Physics

10.1142/S0129055X98000112

solv-int nlin.SI

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.

2009-10-30

solv-int/9707010

Fritz Gesztesy

Fritz Gesztesy and Helge Holden

A combined sine-Gordon and modified Korteweg-de Vries hierarchy and its algebro-geometric solutions

LaTeX; emphasis put on the mKdV hierarchy

solv-int hep-th nlin.SI

We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectral parameter. We further illustrate our approach by developing the basic algebro-geometric setting for the sGmKdV hierarchy, including Baker-Akhiezer functions, trace formulas, Dubrovin-type equations, and theta function representations for its algebro-geometric solutions. Although we mainly focus on sG-type equations, our formalism also yields the sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type equations combined with the mKdV hierarchy.

2008-02-03

solv-int/9707011

Tamara Grava

T. Grava

Bifurcation diagram of a one-parameter family of dispersive waves

latex2e, 28 pages, 14 figures, revised version to appear in Matematica Contemporanea 2000. Substantial changes and improvements have been added. Sections 2, 3 and 4 have been reduced to one section while sections 5 and 6 have been expanded

solv-int nlin.SI

The Korteweg de Vries (KdV) equation with small dispersion is a model for the formation and propagation of dispersive shock waves in one dimension. Dispersive shock waves in KdV are characterized by the appearance of zones of rapid modulated oscillations in the solution of the Cauchy problem with smooth initial data. The modulation in time and space of the amplitudes, the frequencies and the wave-numbers of these oscillations and their interactions is approximately described by the $g$-phase Whitham equations. We study the initial value problem for the Whitham equations for a one parameter family of monotone decreasing initial data. We obtain the bifurcation diagram of the number $g$ of interacting oscillatory zones.

2008-02-03

solv-int/9707012

Q. P. Liu

The Constrained MKP Hierarchy and the Generalized Kupershmidt-Wilson Theorem

9 pages, LaTex

Lett. Math. Phys., 43 (1997) 65

solv-int hep-th nlin.SI

The constrained Modified KP hierarchy is considered from the viewpoint of modification. It is shown that its second Poisson bracket, which has a rather complicated form, is associated to a vastly simpler bracket via Miura-type map. The similar results are established for a natural reduction of MKP.

2008-02-03

solv-int/9707013

Luis Eduardo Saltini

L.E. Saltini, A. Zadra

Algebra of non-local charges in the O(N) WZNW model at and beyond criticality

10 pages, LaTeX, no figures

solv-int hep-th nlin.SI

We derive the classical algebra of the non-local conserved charges in the O(N) WZNW model and analyze its dependence on the coupling constant of the Wess-Zumino term. As in the non-linear sigma model, we find cubic deformations of the O(N) affine algebra. The surprising result is that the cubic algebra of the WZNW non-local charges does not obey the Jacobi identity, thus opposing our expectations from the known Yangian symmetry of this model.

2008-02-03

solv-int/9707014

Shen-Jane Chang

Jiin-Chang Shaw and Ming-Hsien Tu

The constrained modified KP hierarchy and the generalized Miura transformations

8 pages, latex, no figures

J. Phys. A30 (1997) L725

10.1088/0305-4470/30/21/004

solv-int nlin.SI

In this letter, we consider the second Hamiltonian structure of the constrained modified KP hierarchy. After mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand-Dickey brackets defined by this differential operator. We simplify this Hamiltonian structure by factorizing the Lax operator into linear terms.

2009-10-30

solv-int/9707015

Vsevolod Adler

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

B\"acklund transformation for the Krichever-Novikov equation

3p (8K), LaTeX, submitted to IMRN

Int Math Res Notices 1998, Volume 1998, Issue 1, pp 1-4

10.1155/S1073792898000014

solv-int nlin.SI

The B\"acklund transformation and its nonlinear superposition principle are presented for the Krichever-Novikov equation $u_t= u_{xxx} - {3/(2u_x)} (u^2_{xx} - r(u)) + cu_x, r^{(5)}=0$.

2014-08-27

solv-int/9707016

Adrian-Stefan Carstea

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

Continuum limit of nonlinear discrete systems with long range interaction potentials

11 pages, LaTeX, no figure, submitted to Phys.Rev. E

IFIN-HH, F.T. 430-1997

solv-int nlin.SI

One dimensional nonlinear lattices with harmonic long range interaction potentials (LRIP) having an inverse power kernel type, are studied. For the nearest neighbour nonlinear interaction we consider the anharmonic potential of the Fermi-Pasta-Ulam problem and the \phi^3+\phi^4 potential as well. The continuum limit is obtained following the method used by Ishimori and several Boussinesq and KdV type equations with supplementary Hilbert transform terms are found. These nonlocal terms are introduced by the LRIP. For the \phi^3+\phi^4 nearest neighbour interactions the continuum approximation turns out to admit exact bilinearization in Hirota formalism. Exact rational nonsingular solutions are found. The integrability of these nonlocal equations and the connection with perturbed KdV are also discussed.

2008-02-03

solv-int/9707017

Gregorio Falqui

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)

Bihamiltonian Reductions and W_n Algebras

LaTeX2e, 23 pages, to be published in J. Geom. Phys

10.1016/S0393-0440(97)00060-0

SISSA 72/97/FM

solv-int nlin.SI

We discuss the geometry of the Marsden-Ratiu reduction theorem for a bihamiltonian manifold. We consider the case of the manifolds associated with the Gel'fand-Dickey theory, i.e., loop algebras over sl(n+1). We provide an explicit identification, tailored on the MR reduction, of the Adler-Gel'fand-Dickey brackets with the Poisson brackets on the MR-reduced bihamiltonian manifold N. Such an identification relies on a suitable immersion of the space of sections of the cotangent bundle of N into the algebra of pseudo differential operators connected to geometrical features of the theory of (classical) W_n algebras.

2009-10-30

solv-int/9708001

Atsushi Nagai

A. Nagai, T. Tokihiro, J. Satsuma, R. Willox and K. Kajiwara

Two-dimensional soliton cellular automaton of deautonomized Toda-type

11 pages, LaTeX file

10.1016/S0375-9601(97)00591-4

solv-int nlin.SI

A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed.

2009-10-30

solv-int/9708002

Francois Delduc

F. Delduc, L. Feher, L. Gallot

Nonstandard Drinfeld-Sokolov reduction

19 pages, LaTeX file

10.1088/0305-4470/31/25/006

ENSLAPP-L-658, DIAS-STP-97-12

solv-int hep-th nlin.SI

Subject to some conditions, the input data for the Drinfeld-Sokolov construction of KdV type hierarchies is a quadruplet $(\A,\Lambda, d_1, d_0)$, where the $d_i$ are $\Z$-gradations of a loop algebra $\A$ and $\Lambda\in \A$ is a semisimple element of nonzero $d_1$-grade. A new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented. The proposal relies on splitting the $d_1$-grade zero part of $\A$ into a vector space direct sum of two subalgebras. This permits one to interpret certain Gelfand-Dickey type systems associated with a nonstandard splitting of the algebra of pseudo-differential operators in the Drinfeld-Sokolov framework.

2009-10-30

solv-int/9708003

Sedra Moulay Brahim

E.H. Saidi and M.B. Sedra (UFR-HEP Fac. Sc. Rabat- Morocco / Fac. Sc. Kenitra- Morocco)

Three Graded Modified Classical Yang-Baxter Equations and Integrable Systems

22 pages, Revtex

solv-int hep-th nlin.SI

The $6 = 3\times 2$ huge Lie algebra $\Xi$ of all local and non local differential operators on a circle is applied to the standard Adler-Kostant-Symes (AKS) R-bracket sckeme. It is shown in particular that there exist three additional Lie structures, associated to three graded modified classical Yang-Baxter(GMCYB) equations. As we know from the standard case, these structures can be used to classify in a more consitent way a wide class of integrable systems. Other algebraic properties are also presented.

2008-02-03

solv-int/9708004

Vladimir Gerdjikov

V. S. Gerdjikov (1), E. G. Evstatiev(1), D. J. Kaup(2), G. L. Diankov (3), I. M. Uzunov (4) ((1) Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria, (2) Clarksson Univerity, Potsdam, USA, (3) Institute of Solid State Physics, Sofia, Bulgaria, (4) Institute of Electronics, Sofia, Bulgaria)

Criterion and Regions of Stability for Quasi-Equidistant Soliton Trains

14 pages, LaTeX (revtex style), 5 figures

INRNE-TH-97-4

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

Using the complex Toda chain (CTC) as a model for the propagation of the N-soliton pulse trains of the nonlinear Schrodinger (NLS) equation, we predict the asymptotic behavior of these trains. The following asymptotic regimes are stable: (i)~asymptotically free propagation of all N solitons; (ii)~bound state regime where the N solitons may move quasi-equidistantly (QED); and (iii)~various different combinations of (i) and (ii). For N=2 and 3 we determine analytically the set of initial soliton parameters corresponding to each of these regimes. We find excellent agreement between the solutions of CTC and NLS for all regimes and propose realistic choices for the sets of amplitudes, for which the solitons propagate QED to very large run lengths. This is of importance for optical fiber communication.

2009-09-25

solv-int/9708005

Q. P. Liu

On the Integrable Hierarchies Associated With N=2 Super $W_n$ Algebra

11 pages, AMS-LaTex, to appear in Phys. Lett. A

Phys. Lett. A, 235 (1997) 335

10.1016/S0375-9601(97)00638-5

solv-int hep-th nlin.SI

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with N=2 super $W_n$ algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulted hierarchy contains the N=2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of N=2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of N=2 supersymmetric Boussinesq hierarchies.

2009-10-30

solv-int/9708006

Jarmo Hietarinta

J. Hietarinta

Introduction to the Hirota bilinear method

10 pages in LaTeX. To appear in "Lectures on the Integrability of Nonlinear Systems", Springer Lecture Notes in Physics 495

10.1007/BFb0113694

solv-int nlin.SI

We give an elementary introduction to Hirota's direct method of constructing multisoliton solutions to integrable nonlinear evolution equations. We discuss in detail how this works for equations in the Korteweg-de Vries class. We also show how Hirota's method can be used to search for new integrable evolution equations by testing for the existence of 3- and 4-soliton solutions, and list the results that have been obtained this way for the KdV, mKdV/sG and nlS classes of equations.

2009-10-30

solv-int/9708007

E. Sklyanin

E. K. Sklyanin (Steklov Mathematical Institute at St.Petersburg, Russia)

Generating function of correlators in the sl_2 Gaudin model

16 pages, LaTex 209, macros included

Letters in Mathematical Physics 47 (1999) 275-292

10.1023/A:1007585716273

solv-int nlin.SI

For the sl_2 Gaudin model (degenerated quantum integrable XXX spin chain) an exponential generating function of correlators is calculated explicitely. The calculation relies on the Gauss decomposition for the SL_2 loop group. From the generating function a new explicit expression for the correlators is derived from which the determinant formulas for the norms of Bethe eigenfunctions due to Richardson and Gaudin are obtained.

2015-11-12

solv-int/9708008

Ming-Hsien Tu

Jiin-Chang Shaw and Ming-Hsien Tu

Nonlocal extended conformal algebras associated with multi-constraint KP hierarchy and their free-field realizations

14 pages, RevTex, no figures, typos corrected

Int. J. Mod. Phys. A13 (1998) 2723

10.1142/S0217751X98001384

solv-int nlin.SI

We study the conformal properties of the multi-constraint KP hierarchy and its nonstandard partner by covariantizing their corresponding Lax operators. The associated second Hamiltonian structures turn out to be nonlocal extension of $W_n$ algebra by some integer or half-integer spin fields depending on the order of the Lax operators. In particular, we show that the complicated second Hamiltonian structure of the nonstandard multi-constraint KP hierarchy can be simplified by factorizing its Lax operator to multiplication form. We then diagonalize this simplified Poisson matrix and obtain the free-field realizations of its associated nonlocal algebras.

2009-10-30

solv-int/9708009

Stephane Gourmelen

F. Gieres, S. Gourmelen

d=2, N=2 Superconformally Covariant Operators and Super W-Algebras

29 pages, LaTeX

J.Math.Phys. 39 (1998) 3453-3475

10.1063/1.532446

LYCEN-PUB97-30, MPI-PhT/97-36

solv-int hep-th nlin.SI

We construct and classify superconformally covariant differential operators defined on N=2 super Riemann surfaces. By contrast to the N=1 theory, these operators give rise to partial rather than ordinary differential equations which leads to novel features for their matrix representation. The latter is applied to the derivation of N=2 super W-algebras in terms of N=2 superfields.

2009-10-30

solv-int/9708010

Jarmo Hietarinta

Jarmo Hietarinta

Pure quantum integrability

15 pages in LaTeX2e (uses amsmath), misprints corrected and other small changes

10.1016/S0375-9601(98)00535-0

solv-int nlin.SI

The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not a 1-1, although some close correspondencies exist. If a classical mechanical system is integrable with invariants that are polynomial in momenta one can construct a corresponding commuting set of differential operators. Here we discuss some 2- or 3-dimensional purely quantum integrable systems (the 1-dimensional counterpart is the Lame equation). That is, we have an integrable potential whose amplitude is not free but rather proportional to $\hbar^2$, and in the classical limit the potential vanishes. Furthermore it turns out that some of these systems actually have N+1 commuting differential operators, connected by a nontrivial algebraic relation. Some of them have been discussed recently by A.P. Veselov et. al.} from the point of view of Baker-Akheizer functions.

2009-10-30

solv-int/9709001

Alexander V. Shapovalov

Ya. V. Lisitsyn and A. V. Shapovalov

Separation of variables via integral transformations

14 LaTex pages

solv-int nlin.SI

For a system of linear partial differential equations (LPDEs) we introduce an operator equation for auxiliary operators. These operators are used to construct a kernel of an integral transformation leading the LPDE to the separation of variables (SoV). The auxiliary operators are found for various types of the SoV including conventional SoV in the scalar second order LPDE and the SoV by the functional Bethe anzatz. The operators are shown to relate to separable variables. This approach is similar to the position-momentum transformation to action angle coordinates in the classical mechanics. General statements are illustrated by some examples.

2008-02-03

solv-int/9709002

Park Q.-Han

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

Field Theory for Coherent Optical Pulse Propagation

43 pages, Latex, some comments and references are added. postscript file containing 10 figures can be obtained at http://photon.kyunghee.ac.kr/~qhpark/

10.1103/PhysRevA.57.4621

SNUTP 97-110

solv-int nlin.SI

We introduce a new notion of "matrix potential" to nonlinear optical systems. In terms of a matrix potential $g$, we present a gauge field theoretic formulation of the Maxwell-Bloch equation that provides a semiclassical description of the propagation of optical pulses through resonant multi-level media. We show that the Bloch part of the equation can solved identically through $g$ and the remaining Maxwell equation becomes a second order differential equation with reduced set of variables due to the gauge invariance of the system. Our formulation clarifies the (nonabelian) symmetry structure of the Maxwell-Bloch equations for various multi-level media in association with symmetric spaces $G/H$. In particular, we associate nondegenerate two-level system for self-induced transparency with $G/H=SU(2)/U(1)$ and three-level $\L $- or V-systems with $G/H = SU(3)/U(2)$. We give a detailed analysis for the two-level case in the matrix potential formalism, and address various new properties of the system including soliton numbers, effective potential energy, gauge and discrete symmetries, modified pulse area, conserved topological and nontopological charges. The nontopological charge measures the amount of self-detuning of each pulse. Its conservation law leads to a new type of pulse stability analysis which explains nicely earlier numerical results.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2008-02-03

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.

2008-02-03

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.

2009-10-30

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: lakshman@kaveri.bdu.ernet.in)

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.

2015-06-26

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.

2013-10-15

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.

2009-10-30

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.

2009-10-30

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.

2008-02-03

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''.

2008-02-03

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2007-05-23

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2007-05-23

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.

2007-05-23

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2007-05-23

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.

2009-01-21

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.

2009-10-30

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

2007-05-23

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.

2007-05-23

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.

2007-05-23

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.

2007-05-23

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.

2007-05-23

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2007-05-23

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.

2007-05-23

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.

2009-10-30

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.

2009-10-30

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.

2007-05-23

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 .

2018-01-17

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.

2009-10-30

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.

2007-05-23

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.

2009-10-30

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.

2007-05-23

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.

2007-05-23

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2016-09-08

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.

2009-10-30

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.

2007-05-23

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.

2009-10-30

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.

2009-10-30

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.

2007-05-23

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)]

2007-05-23

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.

2009-10-30

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.

2008-11-26

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.

2007-05-23

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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.

2009-10-30

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. .

2007-05-23

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.

2009-10-30

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.

2007-05-23

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.

2011-08-22