Datasets:

CCRss

/

arXiv_dataset

like 1

solv-int/9906002

David H. Sattinger

Yi Li and D.H. Sattinger

Soliton Collisions in the Ion Acoustic Plasma Equations

13 pages, 3 figures

Journal of Mathematical Fluid Mechanics, volume 1, (1999), pp. 117-130

10.1007/s000210050006

solv-int nlin.SI

Numerical experiments involving the interaction of two solitary waves of the ion acoustic plasma equations are described. An exact 2-soliton solution of the relevant KdV equation was fitted to the initial data, and good agreement was maintained throughout the entire interaction. The data demonstrates that the soliton interactions are virtually elastic

solv-int/9906003

Antonio Lima Santos

A. Lima-Santos

Reflection K-Matrices for 19-Vertex Models

35 pages, LaTex

Nucl. Phys. B 558 [PM] 637-667

10.1016/S0550-3213(99)00456-3

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

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or $A_{1}^{(1)}$ model, Izergin-Korepin or $A_{2}^{(2)}$ model, sl(2|1) model and osp(2|1) model. We find that there is a general solution for $A_{1}^{(1)}$ and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the $A_{2}^{(2)}$ and osp(2|1) models we find three general solutions, being two complete reflection K-matrices solutions and one incomplete reflection K-matrix solution with some null entries. In both models these solutions have two free parameters. Integrable spin-1 Hamiltonians with general boundary interactions are also presented. Several reduced solutions from these general solutions are presented in the appendices.

solv-int/9906004

John Harnad

J. Harnad (C.R.M., U. de Montreal and Concordia U.)

On the bilinear equations for Fredholm determinants appearing in random matrices

arxiv version is already official

J. Nonlinear Math. Phys., volume 9, no. 4 (2002) 530-550

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

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic $\tau$-functions.

solv-int/9906005

Runliang Lin

Yunbo Zeng (1), Wen-Xiu Ma (2) ((1)Tsinghua University, Beijing, China, (2) City University of Hong Kong, China)

Families of quai-bi-Hamiltonian systems and separability

27 pages, Amstex, no figues, to be published in Journal of Mathematical Physics

10.1063/1.532979

solv-int nlin.SI

It is shown how to construct an infinite number of families of quasi-bi-Hamiltonian (QBH) systems by means of the constrained flows of soliton equations. Three explicit QBH structures are presented for the first three families of the constrained flows. The Nijenhuis coordinates defined by the Nijenhuis tensor for the corresponding families of QBH systems are proved to be exactly the same as the separated variables introduced by means of the Lax matrices for the constrained flows.

solv-int/9906006

F. Nijhoff

F.W. Nijhoff, N. Joshi, A. Hone

On the discrete and continuous Miura Chain associated with the Sixth Painlev\'e Equation

17 pages, LaTeX2e

10.1016/S0375-9601(99)00764-1

solv-int nlin.SI

A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or B\"acklund transformations. We describe such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from \pvi itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-B\"acklund transformation, relating two copies of \pvi with different parameters. We also establish the analogous ordinary difference equations in the discrete counterpart of the chain. Such difference equations govern iterations of solutions of \pvi under B\"acklund transformations. Both discrete and continuous equations constitute a larger system which include partial difference equations, differential-difference equations and partial differential equations, all associated with the lattice Korteweg-de Vries equation subject to similarity constraints.

solv-int/9906007

Sergei M. Sergeev

Sergei M. Sergeev

On exact solution of a classical 3D integrable model

J. Nonlinear Math. Phys. 7 (2000), no. 1, 57-72

10.2991/jnmp.2000.7.1.5

JNMP 4/2002 (Article)

solv-int nlin.SI

We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients of these systems of linear equations. Determinant of any system of linear equations is a polynomial of two numerical quasimomenta of the auxiliary linear variables. For one, this determinant is the generating functions of all integrals of motion for the evolution, and on the other hand it defines a high genus algebraic curve. The dependence of the dynamical variables on the space-time point (exact solution) may be expressed in terms of theta functions on the jacobian of this curve. This is the main result of our paper.

solv-int/9906008

James D. E. Grant

James D.E. Grant

Paraconformal Structures and Integrable Systems

14 pages, Latex 2e, submitted to Nonlinearity

solv-int hep-th nlin.SI

We consider some natural connections which arise between right-flat (p, q) paraconformal structures and integrable systems. We find that such systems may be formulated in Lax form, with a "Lax p-tuple" of linear differential operators, depending a spectral parameter which lives in (q-1)-dimensional complex projective space. Generally, the differential operators contain partial derivatives with respect to the spectral parameter.

solv-int/9906009

G. Tondo

G. Falqui, F. Magri, G. Tondo

Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy

20 pages, LaTeX2e, report to NEEDS in Leeds (1998), to be published in Theor. Math. Phys

10.1007/BF02551195

solv-int nlin.SI

We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.

solv-int/9906010

Yuri B. Suris

Yuri B. Suris

r-matrices for relativistic deformations of integrable systems

J. Nonlinear Math. Phys. 6 (1999), no. 4, 411-447

10.2991/jnmp.1999.6.4.4

JNMP 4/2002 (Article)

solv-int nlin.SI

We include the relativistic lattice KP hierarchy, introduced by Gibbons and Kupershmidt, into the $r$-matrix framework. An $r$-matrix account of the nonrelativistic lattice KP hierarchy is also provided for the reader's convenience. All relativistic constructions are regular one-parameter perturbations of the nonrelativistic ones. We derive in a simple way the linear Hamiltonian structure of the relativistic lattice KP, and find for the first time its quadratic Hamiltonian structure. Amasingly, the latter turns out to coincide with its nonrelativistic counterpart (a phenomenon, known previously only for the simplest case of the relativistic Toda lattice).

solv-int/9906011

Shigeki Matsutani

p-adic Difference-Difference Lotka-Volterra Equation and Ultra-Discrete Limit

AMS-Tex Use. Title changes

solv-int nlin.SI

In this article, we have studied the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We pointed out that the structure of the space given by taking the ultra-discrete limit is the same as that of the $p$-adic valuation space.

solv-int/9906012

S. Yu. Sakovich

S. Yu. Sakovich

Integrability of the higher-order nonlinear Schroedinger equation revisited

6 pages, LaTeX

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

Only the known integrable cases of the Kodama-Hasegawa higher-order nonlinear Schroedinger equation pass the Painleve test. Recent results of Ghosh and Nandy add no new integrable cases of this equation.

solv-int/9906013

Nadja Kutz

Tim Hoffmann, Johannes Kellendonk, Nadja Kutz and Nicolai Reshetikhin

Factorization dynamics and Coxeter-Toda lattices

33 pages, latex, minor corrections

Comm. Math. Phys. 212, Issue 2, 297-321 (2000)

10.1007/s002200000212

solv-int math.QA nlin.SI

It is shown that the factorization relation on simple Lie groups with standard Poisson Lie structure restricted to Coxeter symplectic leaves gives an integrable dynamical system. This system can be regarded as a discretization of the Toda flow. In case of $SL_n$ the integrals of the factorization dynamics are integrals of the relativistic Toda system. A substantial part of the paper is devoted to the study of symplectic leaves in simple complex Lie groups, its Borel subgroups and their doubles.

solv-int/9907001

Peter Forrester

M. Adler, P.J. Forrester, T. Nagao and P. van Moerbeke

Classical skew orthogonal polynomials and random matrices

21 pages, no figures

10.1023/A:1018644606835

solv-int nlin.SI

Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the cases that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre and Jacobi cases.

solv-int/9907002

Tim Hoffmann

Tim Hoffmann

On the equivalence of the discrete nonlinear Schr\"odinger equation and the discrete isotropic Heisenberg magnet

9 pages, LaTeX2e

10.1016/S0375-9601(99)00860-9

sfb288 preprint 381

solv-int nlin.SI

The equivalence of the discrete isotropic Heisenberg magnet (IHM) model and the discrete nonlinear Schr\"odinger equation (NLSE) given by Ablowitz and Ladik is shown. This is used to derive the equivalence of their discretization with the one by Izergin and Korepin. Moreover a doubly discrete IHM is presented that is equivalent to Ablowitz' and Ladiks doubly discrete NLSE.

solv-int/9907003

Sudipta Nandy

Sasanka Ghosh and Sudipta Nandy

A New Class of Optical Solitons

9 pages, no figure

solv-int nlin.SI

Existence of a new class of soliton solutions is shown for higher order nonlinear Schrodinger equation, describing thrid order dispersion, Kerr effect and stimulated Raman scattering. These new solutions have been obtaiened by invoking a group of nonlinear transformations acting on localised stable solutions. Stability of these solutions has been studied for different values of the arbitrary coefficients, involved in the recursion relation and consequently, different values of coefficient lead to different transmission rates for almost same input power. Another series solution containing even powers of localised stable solution is shown to exist for higher order nonlinear Schrodinger equation.

solv-int/9907004

Alexander Sorin

F. Delduc, L. Gallot and A. Sorin

N=2 local and N=4 nonlocal reductions of supersymmetric KP hierarchy in N=2 superspace

26 pages, LaTeX, a few misprints corrected

Nucl.Phys. B558 (1999) 545-572

10.1016/S0550-3213(99)00473-3

LPENSL-TH-14/99

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

A N=4 supersymmetric matrix KP hierarchy is proposed and a wide class of its reductions which are characterized by a finite number of fields are described. This class includes the one-dimensional reduction of the two-dimensional N=(2|2) superconformal Toda lattice hierarchy possessing the N=4 supersymmetry -- the N=4 Toda chain hierarchy -- which may be relevant in the construction of supersymmetric matrix models. The Lax pair representations of the bosonic and fermionic flows, corresponding local and nonlocal Hamiltonians, finite and infinite discrete symmetries, the first two Hamiltonian structures and the recursion operator connecting all evolution equations and the Hamiltonian structures of the N=4 Toda chain hierarchy are constructed in explicit form. Its secondary reduction to the N=2 supersymmetric alpha=-2 KdV hierarchy is discussed.

solv-int/9907005

Atalay Karasu

Atalay Karasu

On A Recently Proposed Relation Between oHS and Ito Systems

Latex, 5 pages, to be published in Physics Letters A

10.1016/S0375-9601(99)00415-6

solv-int nlin.SI

The bi-Hamiltonian structure of original Hirota-Satsuma system proposed by Roy based on a modification of the bi-Hamiltonian structure of Ito system is incorrect.

solv-int/9907006

Marcio J. Martins

M.J. Martins and X.W. Guan

Integrable supersymmetric correlated electron chain with open boundaries

latex, 14 pages

Nucl. Phys. B 562 (1999) 433-444

10.1016/S0550-3213(99)00551-9

UFSCAR-99-20

solv-int nlin.SI

We construct an extended Hubbard model with open boundaries from a $R$-matrix based on the $U_q[Osp(2|2)]$ superalgebra. We study the reflection equation and find two classes of diagonal solutions. The corresponding one-dimensional open Hamiltonians are diagonalized by means of the Bethe ansatz approach.

solv-int/9907007

Hendry Izaac Elim

Hendry I. Elim

New Integrable Coupled Nonlinear Schrodinger Equations

11 pages, LaTeX

solv-int nlin.SI

Two types of integrable coupled nonlinear Schrodinger (NLS) equations are derived by using Zakharov-Shabat (ZS) dressing method.The Lax pairs for the coupled NLS equations are also investigated using the ZS dressing method. These give new types of the integrable coupled NLS equations with certain additional terms. Then, the exact solutions of the new types are obtained. We find that the solution of these new types do not always produce a soliton solution even they are the kind of the integrable NLS equations.

solv-int/9907008

Peter Forrester

P.J. Forrester and E.M. Rains

Inter-relationships between orthogonal, unitary and symplectic matrix ensembles

35 pages. Some results of the replaced preprint `Exact calculation of the distribution of every second eigenvalue in classical random matrix ensembles with orthogonal symmetry' by PJF have been combined with new results of EMR to form the present article

solv-int nlin.SI

We consider the following problem: When do alternate eigenvalues taken from a matrix ensemble themselves form a matrix ensemble? More precisely, we classify all weight functions for which alternate eigenvalues from the corresponding orthogonal ensemble form a symplectic ensemble, and similarly classify those weights for which alternate eigenvalues from a union of two orthogonal ensembles forms a unitary ensemble. Also considered are the $k$-point distributions for the decimated orthogonal ensembles.

solv-int/9907009

Fritz Gesztesy

Fritz Gesztesy

Integrable Systems in the Infinite Genus Limit

LaTeX, 24 pages

solv-int nlin.SI

We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite genus curves. Our approach applies to complex-valued periodic solutions of the KdV hierarchy and naturally identifies the Riemann surface familiar from standard Floquet theoretic considerations with a limit of Burchnall-Chaundy curves.

solv-int/9907010

Jon Links

Jon Links (U. of Queensland)

A construction for R-matrices without difference property in the spectral parameter

LaTeX, 15 pages, no figures

Phys. Lett. A 265 (2000) 194-206

10.1016/S0375-9601(99)00839-7

UQCMP-99-2

solv-int nlin.SI

A new construction is given for obtaining R-matrices which solve the McGuire-Yang-Baxter equation in such a way that the spectral parameters do not possess the difference property. A discussion of the derivation of the supersymmetric U model is given in this context such that applied chemical potential and magnetic field terms can be coupled arbitrarily. As a limiting case the Bariev model is obtained.

solv-int/9907011

Dmitry Pelinovsky

Dmitry E. Pelinovsky and Catherine Sulem

Spectral decomposition for the Dirac system associated to the DSII equation

10.1088/0266-5611/16/1/306

solv-int nlin.SI

A new (scalar) spectral decomposition is found for the Dirac system in two dimensions associated to the focusing Davey--Stewartson II (DSII) equation. Discrete spectrum in the spectral problem corresponds to eigenvalues embedded into a two-dimensional essential spectrum. We show that these embedded eigenvalues are structurally unstable under small variations of the initial data. This instability leads to the decay of localized initial data into continuous wave packets prescribed by the nonlinear dynamics of the DSII equation.

solv-int/9907012

Adam Doliwa

Adam Doliwa and Paolo Maria Santini

The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

48 pages, 6 figures; 1 section added, to appear in J. Geom. & Phys

10.1016/S0393-0440(00)00011-5

solv-int nlin.SI

We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the tau-function of the lattice, defined as the potential connecting these data. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the D-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and D-invariant. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a D-bar formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.

solv-int/9907013

Adam Doliwa

Adam Doliwa

Lattice geometry of the Hirota equation

11 pages, 3 figures, to appear in Proceedings from the Conference "Symmetries and Integrability of Difference Equations III", Sabaudia, 1998

solv-int nlin.SI

Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of the Laplace equation, (ii) the gauge invariant form of the Hirota equation for projective invariants of the Laplace sequence, (iii) the discrete Toda system for the rotation coefficients, (iv) the original form of the Hirota equation for the tau-function of the quadrilateral lattice.

solv-int/9907014

Adam Doliwa

Adam Doliwa and Paolo Maria Santini

Integrable Discrete Geometry: the Quadrilateral Lattice, its Transformations and Reductions

27 pages, 9 figures, to appear in Proceedings from the Conference "Symmetries and Integrability of Difference Equations III", Sabaudia, 1998

solv-int hep-lat nlin.SI

We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized by the planarity of its elementary quadrilaterals. Therefore the linear property of planarity seems to be a basic geometric property underlying integrability. We present the geometric meaning of its tau-function, as the potential connecting its forward and backward data. We present the theory of transformations of the quadrilateral lattice, which is based on the discrete analogue of the theory of rectilinear congruences. In particular, we discuss the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. We also show how the sequence of Laplace transformations of a quadrilateral surface is described by the discrete Toda system. We finally show that these classical transformations are strictly related to the basic operators associated with the quantum field theoretical formulation of the multicomponent Kadomtsev-Petviashvilii hierarchy. We review the properties of quadrilateral hyperplane lattices, which play an interesting role in the reduction theory, when the introduction of additional geometric structures allows to establish a connection between point and hyperplane lattices. We present and fully characterize some geometrically distinguished reductions of the quadrilateral lattice, like the symmetric, circular and Egorov lattices; we review also basic geometric results of the theory of quadrilateral lattices in quadrics, and the corresponding analogue of the Ribaucour reduction of the fundamental transformation.

solv-int/9907015

F. Nijhoff

F.W. Nijhoff (University of Leeds)

Discrete Dubrovin Equations and Separation of Variables for Discrete Systems

Talk presented at the Intl. Conf. on ``Integrability and Chaos in Discrete Systems'', July 2-6, 1997, to appear in: Chaos, Solitons and Fractals, ed. F. Lambert, (Pergamon Press)

10.1016/S0960-0779(98)00264-1

solv-int nlin.SI

A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions in terms of Abelian functions of Kleinian type (i.e. the higher-genus analogues of the Weierstrass elliptic functions) is discussed as well as the connections with the method of separation of variables.

solv-int/9907016

Nobuhiko Shinzawa

Nobuhiko Shinzawa

Symmetric Linear Backlund Transformation for Discrete BKP and DKP equation

18 pages,3 figures

10.1088/0305-4470/33/21/309

solv-int nlin.SI

Proper lattices for the discrete BKP and the discrete DKP equaitons are determined. Linear B\"acklund transformation equations for the discrete BKP and the DKP equations are constructed, which possesses the lattice symmetries and generate auto-B\"acklund transformations

solv-int/9907017

Fritz Gesztesy

F. Gesztesy, C. K. R. T. Jones, Y. Latushkin, and M. Stanislavova

A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr\"odinger Equations

LaTeX, 16 pages

solv-int nlin.SI

A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schr\" odinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions.

solv-int/9907018

Takayuki Tsuchida

Takayuki Tsuchida, Miki Wadati (University of Tokyo)

Multi-Field Integrable Systems Related to WKI-Type Eigenvalue Problems

9 pages, LaTeX209 file, uses jpsj.sty

J. Phys. Soc. Jpn. 68 (1999) 2241-2245

10.1143/JPSJ.68.2241

solv-int nlin.SI

Higher flows of the Heisenberg ferromagnet equation and the Wadati-Konno-Ichikawa equation are generalized into multi-component systems on the basis of the Lax formulation. It is shown that there is a correspondence between the multi-component systems through a gauge transformation. An integrable semi-discretization of the multi-component higher Heisenberg ferromagnet system is obtained.

solv-int/9907019

Liu Qing Ping

Q.P. Liu

Miura Map between Lattice KP and its Modification is Canonical

8 pages, LaTeX

solv-int nlin.SI

We consider the Miura map between the lattice KP hierarchy and the lattice modified KP hierarchy and prove that the map is canonical not only between the first Hamiltonian structures, but also between the second Hamiltonian structures.

solv-int/9907020

Goro Hatayama

Goro Hatayama, Atsuo Kuniba, and Taichiro Takagi

Soliton Cellular Automata Associated With Crystal Bases

29 pages, 1 figure, LaTeX2e

Nuclear Physics B577[PM](2000) 619-645

10.1016/S0550-3213(00)00105-X

solv-int math.QA nlin.SI

We introduce a class of cellular automata associated with crystals of irreducible finite dimensional representations of quantum affine algebras U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for \hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering matrices of two solitons coincide with the combinatorial R matrices of U'_q(C^{(1)}_{n-1})-crystals.

solv-int/9907021

Olaf Lechtenfeld

Olaf Lechtenfeld and Alexander Sorin

Supersymmetric KP hierarchy in N=1 superspace and its N=2 reductions

21 pages, version to be published in Nucl. Phys. B

Nucl.Phys. B566 (2000) 489-510

10.1016/S0550-3213(99)00653-7

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

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

A wide class of N=2 reductions of the supersymmetric KP hierarchy in N=1 superspace is described. This class includes a new N=2 supersymmetric generalization of the Toda chain hierarchy. The Lax pair representations of the bosonic and fermionic flows, local and nonlocal Hamiltonians, finite and infinite discrete symmetries, first two Hamiltonian structures and the recursion operator of this hierarchy are constructed. Its secondary reduction to new N=2 supersymmetric modified KdV hierarchy is discussed.

solv-int/9908001

Ziad Maassarani

Z. Maassarani (University of Virginia)

Non-additive fusion, Hubbard models and non-locality

14 pages, Latex. A remark added in section 2, four typos corrected

J. Phys. A 32 (1999) 8691-8703

10.1088/0305-4470/32/49/310

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

In the framework of quantum groups and additive R-matrices, the fusion procedure allows to construct higher-dimensional solutions of the Yang-Baxter equation. These solutions lead to integrable one-dimensional spin-chain Hamiltonians. Here fusion is shown to generalize naturally to non-additive R-matrices, which therefore do not have a quantum group symmetry. This method is then applied to the generalized Hubbard models. Although the resulting integrable models are not as simple as the starting ones, the general structure is that of two spin-(s times s') sl(2) models coupled at the free-fermion point. An important issue is the probable lack of regular points which give local Hamiltonians. This problem is related to the existence of second order zeroes in the unitarity equation, and arises for the XX models of higher spins, the building blocks of the Hubbard models. A possible connection between some Lax operators L and R-matrices is noted.

solv-int/9908002

V. Kuznetsov

V.B.Kuznetsov, M.Salerno and E.K.Sklyanin

Quantum Backlund transformation for the integrable DST model

24 pages, corrected refs to Sections and a misprint

J.Phys.A33:171-189,2000

10.1088/0305-4470/33/1/311

LPENSL-TH-16/99

solv-int hep-th math.CA math.QA nlin.SI

For the integrable case of the discrete self-trapping (DST) model we construct a Backlund transformation. The dual Lax matrix and the corresponding dual Backlund transformation are also found and studied. The quantum analog of the Backlund transformation (Q-operator) is constructed as the trace of a monodromy matrix with an infinite-dimensional auxiliary space. We present the Q-operator as an explicit integral operator as well as describe its action on the monomial basis. As a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable DST model. These eigenfunctions are special functions of the Heun class which is beyond the hypergeometric class. The found integral equations are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.

solv-int/9908003

Konstantin Selivanov

K.G. Selivanov (ITEP, Moscow)

Classical Solutions Generating Tree Form-Factors in Yang-Mills, Sin(h)-Gordon and Gravity

7 pages, Talk given at the conference Nonlinearity, integrability and all that twenty years after NEEDS '79, Gallipoli, Lecce (Italy), July 1 - July 10, 1999

10.1142/9789812817587_0073

ITEP-TH-99-34

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

Classical solutions generating tree form-factors are defined and constructed in various models.

solv-int/9908004

Alexander Mikhailov

A. V. Mikhailov, V. V. Sokolov

Integrable ODEs on Associative Algebras

19 pages, LaTeX

10.1007/s002200050810

solv-int nlin.SI

In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential equations. We choose existence of hierarchies of first integrals and/or symmetries as a criterion for integrability and justify it by examples. Using our componentless approach we have solved a number of classification problems for integrable equations on free associative algebras. Also, in the simplest case, we have listed all possible Hamiltonian operators of low order.

solv-int/9908005

Wen-Xiu Ma

Wen-Xiu Ma and Ruguang Zhou

A Coupled AKNS-Kaup-Newell Soliton Hierarchy

15 pages, latex

10.1063/1.532976

solv-int nlin.SI

A coupled AKNS-Kaup-Newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from Hamiltonian pairs. Zero curvature representations and tri-Hamiltonian structures are established for all coupled AKNS-Kaup-Newell systems in the hierarchy. Therefore all systems have infinitely many commuting symmetries and conservation laws. Two reductions of the systems lead to the AKNS hierarchy and the Kaup-Newell hierarchy, and thus those two soliton hierarchies also possess tri-Hamiltonian structures.

solv-int/9908006

Takayuki Tsuchida

Takayuki Tsuchida, Miki Wadati (University of Tokyo)

Complete integrability of derivative nonlinear Schr\"{o}dinger-type equations

14 pages, LaTeX2e (IOP style), to appear in Inverse Problems

Inverse Problems 15 (1999) 1363-1373

10.1088/0266-5611/15/5/317

solv-int nlin.SI

We study matrix generalizations of derivative nonlinear Schr\"{o}dinger-type equations, which were shown by Olver and Sokolov to possess a higher symmetry. We prove that two of them are `C-integrable' and the rest of them are `S-integrable' in Calogero's terminology.

solv-int/9908007

Kenji Kajiwara

Kenji Kajiwara(Doshisha Univ.), Tetsu Masuda(Doshisha Univ.), Masatoshi Noumi(Kobe Univ.), Yasuhiro Ohta(Hiroshima Univ.), Yasuhiko Yamada(Kobe Univ.)

Determinant Formulas for the Toda and Discrete Toda Equations

16pages, LaTeX using theorem.sty

solv-int nlin.SI

Determinant formulas for the general solutions of the Toda and discrete Toda equations are presented. Application to the $\tau$ functions for the Painlev\'e equations is also discussed.

solv-int/9908008

Ladislav Hlavaty

L. Hlavaty

Towards the Lax formulation of SU(2) principal models with nonconstant metric

Talk given at the Workshop on Backlund and Darboux Transformations. The geometry of Soliton Theory. June 4-9, 1999 (Halifax, Nova Scotia), 10 pages, Latex2e, no figures

solv-int hep-th nlin.SI

The equations that define the Lax pairs for generalized principal chiral models can be solved for any constant nondegenerate bilinear form on SU(2). Necessary conditions for the nonconstant metric on SU(2) that define the integrable models are given.

solv-int/9908009

Takeo Kojima

H. Furutsu, T. Kojima, and Y.-H. Quano

Type II vertex operators for the $A_{n-1}^{(1)}$ face model

20 pages, LaTEX 2e

Int.J.Mod.Phys. A15 (2000) 1533-1556

10.1142/S0217751X00000690

solv-int hep-th nlin.SI

Presented is a free boson representation of the type II vertex operators for the $A_{n-1}^{(1)}$ face model. Using the bosonization, we derive some properties of the type II vertex operators, such as commutation, inversion and duality relations.

solv-int/9909001

Wen-Xiu Ma

Wen-Xiu Ma and Si-Ming Zhu

Non-symmetry constraints of the AKNS system yielding integrable Hamiltonian systems

latex, 8 pages, to appear in Chaos, Solitons and Fractals

solv-int nlin.SI

This paper aims to show that there exist non-symmetry constraints which yield integrable Hamiltonian systems through nonlinearization of spectral problems of soliton systems, like symmetry constraints. Taking the AKNS spectral problem as an illustrative example, a class of such non-symmetry constraints is introduced for the AKNS system, along with two-dimensional integrable Hamiltonian systems generated from the AKNS spectral problem.

solv-int/9909002

Alexander I. Bobenko

Sergey I. Agafonov, Alexander I. Bobenko

Discrete Z^a and Painleve equations

25 pages, 9 figures

International Math. Research Notices 2000:4 165-193

solv-int math.CV nlin.SI

A discrete analogue of the holomorphic map z^a is studied. It is given by a Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding immersed circle patterns lead to special separatrix solutions of a discrete Painleve equation. Global properties of these solutions, as well as of the discrete $z^a$ are established.

solv-int/9909003

Metin Gurses

Metin Gurses (Bilkent University), Atalay Karasu (METU), and Vladimir Sokolov (Landau Institue)

On Construction of Recursion Operators From Lax Representation

Latex File (AMS format), 23 pages, to be published in Journal of Mathematical Physics

10.1063/1.533102

solv-int nlin.SI

In this work we develop a general procedure for constructing the recursion operators fro non-linear integrable equations admitting Lax representation. Svereal new examples are given. In particular we find the recursion operators for some KdV-type of integrable equations.

solv-int/9909004

Andrey Tsyganov User

Andrey Tsiganov

On integrable deformations of the spherical top

LaTeX file with Amssymb, 9 page

J. Phys. A, Math. Gen. 32, No.47, 8355-8363, (1999)

10.1088/0305-4470/32/47/313

solv-int nlin.SI

The motion on the sphere $S^2$ with the potential $V= (x_1x_2x_3)^{-2/3}$ is considered. The Lax representation and the linearisation procedure for this two-dimensional integrable system are discussed.

solv-int/9909005

Dmitry Demskoy

D.K.Demskoy, A.G.Meshkov

New integrable string-like fields in 1+1 dimensions

solv-int nlin.SI

The symmetry classification method is applied to the string-like scalar fields in two-dimensional space-time. When the configurational space is three-dimensional and reducible we present the complete list of the systems admiting higher polynomial symmetries of the 3rd, 4th and 5th-order.

solv-int/9909006

Andrey V. Tsiganov

Andrey Tsiganov

Canonical transformations of the extended phase space, Toda lattices and Stackel family of integrable systems

LaTeX2e + Amssymb, 22pp

J. Phys. A, Math. Gen. 33, No.22, 4169-4182, (2000)

10.1088/0305-4470/33/22/318

solv-int nlin.SI

We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map completely integrable system into other completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and R-matrices. As an example, we consider canonical transformations of the extended phase space for the Toda lattices and the Stackel systems.

solv-int/9909007

Hans Jacobus Wospakrik

Hans J. Wospakrik and Freddy P. Zen

CPT Symmetries and the Backlund Transformations

12 pages

solv-int nlin.SI

We show that the auto-Backlund transformations of the sine-Gordon, Korteweg-deVries, nonlinear Schrodinger, and Ernst equations are related to their respective CPT symmetries. This is shown by applying the CPT symmetries of these equations to the Riccati equations of the corresponding pseudopotential functions where the fields are allowed to transform into new solutions while the pseudopotential functions and the Backlund parameter are held fixed.

solv-int/9909008

Andrey K. Svinin

A.K. Svinin

Lie point symmetries of integrable evolution equations and invariant solutions

LaTeX, 17 pages

ISDCT-99-4

solv-int nlin.SI

An integrable hierarchies connected with linear stationary Schr\"odinger equation with energy dependent potentials (in general case) are considered. Galilei-like and scaling invariance transformations are constructed. A symmetry method is applied to construct invariant solutions.

solv-int/9909009

Yuri B. Suris

Yuri B. Suris (TU Berlin)

The motion of a rigid body in a quadratic potential: an integrable discretization

LaTeX, 15 pp

Intern. Math. Research Notices, 2000, No 12, p.643-663.

solv-int nlin.SI

The motion of a rigid body in a quadratic potential is an important example of an integrable Hamiltonian system on a dual to a semidirect product Lie algebra so(n) x Symm(n). We give a Lagrangian derivation of the corresponding equations of motion, and introduce a discrete time analog of this system. The construction is based on the discrete time Lagrangian mechanics on Lie groups, accompanied with the discrete time Lagrangian reduction. The resulting multi-valued map (correspondence) on the dual to so(n) x Symm(n) is Poisson with respect to the Lie-Poisson bracket, and is also completely integrable. We find a Lax representation based on matrix factorisations, in the spirit of Veselov-Moser.

solv-int/9909010

Pierre van Moerbeke

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

Pfaff tau-functions

42 pages

solv-int adap-org hep-th nlin.AO nlin.SI

Consider the evolution $$ \frac{\pl m_\iy}{\pl t_n}=\Lb^n m_\iy, \frac{\pl m_\iy}{\pl s_n}=-m_\iy(\Lb^\top)^n, $$ on bi- or semi-infinite matrices $m_\iy=m_\iy(t,s)$, with skew-symmetric initial data $m_{\iy}(0,0)$. Then, $m_\iy(t,-t)$ is skew-symmetric, and so the determinants of the successive "upper-left corners" vanish or are squares of Pfaffians. In this paper, we investigate the rich nature of these Pfaffians, as functions of t. This problem is motivated by questions concerning the spectrum of symmetric and symplectic random matrix ensembles.

solv-int/9909011

David Fairlie

D.B. Fairlie and A.N. Leznov

The Complex Bateman Equation

6 pages latex, no figures

solv-int nlin.SI

The general solution to the Complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special case.

solv-int/9909012

David Fairlie

D.B. Fairlie and A.N. Leznov

The General Solution of the Complex Monge-Amp\`ere Equation in two dimensional space

9 pages, latex, no figures

solv-int nlin.SI

The general solution to the Complex Monge-Amp\`ere equation in a two dimensional space is constructed.

solv-int/9909013

David Fairlie

D.B. Fairlie and A.N. Leznov

The Complex Bateman Equation in a space of arbitrary dimension

13 pages, latex, no figures

solv-int nlin.SI

A general solution to the Complex Bateman equation in a space of arbitrary dimensions is constructed.

solv-int/9909014

David Fairlie

D.B. Fairlie and A.N. Leznov

The General Solution of the Complex Monge-Amp\`ere Equation in a space of arbitrary dimension

13 pages, latex, no figures

10.1088/0305-4470/33/25/307

solv-int nlin.SI

A general solution to the Complex Monge-Amp\`ere equation in a space of arbitrary dimensions is constructed.

solv-int/9909015

Adam Doliwa

Adam Doliwa (Warsaw University)

Discrete asymptotic nets and W-congruences in Plucker line geometry

28 pages, 4 figures; expanded Introduction, new Section, added references

J. Geom. Phys. 39 (2001) 9-29

10.1016/S0393-0440(00)00070-X

solv-int math.DG nlin.SI

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Moutard transformation, it is constructed the discrete analog of the W-congruences, which provide the Darboux-Backlund type transformation of asymptotic lattices.The permutability theorems for the discrete Moutard transformation and for the corresponding transformation of asymptotic lattices are established as well. Moreover, it is proven that the discrete W-congruences are represented by quadrilateral lattices in the quadric of Plucker. These results generalize to a discrete level the classical line-geometric approach to asymptotic nets and W-congruences, and incorporate the theory of asymptotic lattices into more general theory of quadrilateral lattices and their reductions.

solv-int/9909016

Liu Qing Ping

Q. P. Liu, Manuel Manas

Darboux Transformation for Supersymmetric KP Hierarchies

14 pages, LaTeX2e with amsmath,amssymb,amsthm and geometry packages. In this new version we consider both the Manin-Radul and the Jacobian SKP hierachies and we show how the elementary Darboux transformation composed with a reversion of signs in the fermionic times constitute a proper transformation of these hierarchies

Phys.Lett.B485:293-300,2000

10.1016/S0370-2693(00)00663-8

solv-int hep-th nlin.SI

We construct Darboux transformations for the super-symmetric KP hierarchies of Manin--Radul and Jacobian types. We also consider the binary Darboux transformation for the hierarchies. The iterations of both type of Darboux transformations are briefly discussed.

solv-int/9909017

Pilar G. Estevez

P. G. Estevez and G. A. Hernaez (Universidad de Salamanca, Spain)

Darboux transformations for a Bogoyavlenskii equation in 2+1 dimensions

7 pages, latex, to appear in the proceedings of the meeting "Nonlinearity and Integrability" (Gallipoli, Italy, July 1999)

10.1142/9789812817587_0016

solv-int nlin.SI

We use the singular manifold method to obtain the Lax pair, Darboux transformations and soliton solutions for a (2+1) dimensional integrable equation.

solv-int/9909018

Roman Paunov

H. Belich, G. Cuba and R. Paunov

Surfaces of Constant negative Scalar Curvature and the Correpondence between the Liouvulle and the sine-Gordon Equations

latex file, 23 pages, uses ams.tex

IFT-P.070/99, CBPF-NF-045/99

solv-int nlin.SI

By studying the {\it internal} Riemannian geometry of the surfaces of constant negative scalar curvature, we obtain a natural map between the Liouville, and the sine-Gordon equations. First, considering isometric immersions into the Lobachevskian plane, we obtain an uniform expression for the general (locally defined) solution of both the equations. Second, we prove that there is a Lie-B\"acklund transformation interpolating between Liouville and sine-Gordon. Third, we use isometric immersions into the Lobachevskian plane to describe sine-Gordon N-solitons explicitly.

solv-int/9909019

Kazuyasu Shigemoto

M.Horibe and K.Shigemoto

The Structure of the Bazhanov-Baxter Model and a New Solution of the Tetrahedron Equation

23 pages, Latex

Progr. Theor. Phys. 102 (1999), 221-236

10.1143/PTP.102.221

TEZ-99-1

solv-int nlin.SI

We clarify the structure of the Bazhanov-Baxter model of the 3-dim N-state integrable model. There are two essential points, i) the cubic symmetries, and ii) the spherical trigonometry parametrization, to understand the structure of this model. We propose two approaches to find a candidate as a solution of the tetrahedron equation, and we find a new solution.

solv-int/9909020

Vladimir Gerdjikov

V. S. Gerdjikov (Institute for Nuclear Research and Nuclear Energy, Bulg. Acad. of Sci., Sofia,Bulgaria), E. G. Evstatiev (Department of Physics, University of Texas at Austin, Austin, Texas, USA), R. I. Ivanov (Department of Mathematical Physics National University of Ireland - Galway, Galway, Ireland)

The complex Toda chains and the simple Lie algebras - solutions and large time asymptotics -- II

LaTeX, iopart style, 37 pages, no figures

solv-int nlin.SI

We propose a compact and explicit expression for the solutions of the complex Toda chains related to the classical series of simple Lie algebras g. The solutions are parametrized by a minimal set of scattering data for the corresponding Lax matrix. They are expressed as sums over the weight systems of the fundamental representations of g and are explicitly covariant under the corresponding Weyl group action. In deriving these results we start from the Moser formula for the A_r series and obtain the results for the other classical series of Lie algebras by imposing appropriate involutions on the scattering data. Thus we also show how Moser's solution goes into the one of Olshanetsky and Perelomov. The results for the large-time asymptotics of the A_r -CTC solutions are extended to the other classical series B_r - D_r. We exhibit also some `irregular' solutions for the D_{2n+1} algebras whose asymptotic regimes at t ->\pm\infty are qualitatively different. Interesting examples of bounded and periodic solutions are presented and the relations between the solutions for the algebras D_4, B_3 and G_2 $ are analyzed.

solv-int/9909021

Vladimir Marikhin

M. Boiti, V.G. Marikhin, F. Pempinelli, A.B. Shabat

Self-similar solutions of NLS-type dynamical systems

18 pages, AmsTeX

solv-int nlin.SI

We study self-similar solutions of NLS-type dynamical systems. Lagrangian approach is used to show that they can be reduced to three canonical forms, which are related by Miura transformations. The fourth Painleve equation (PIV) is central in our consideration - it connects Heisenberg model, Volterra model and Toda model to each other. The connection between the rational solutions of PIV and Coulomb gas in a parabolic potential is established. We discuss also the possibility to obtain an exact solution for optical soliton i.e. of the NLS equation with time-dependent dispersion.

solv-int/9909022

Igor Loutsenko

I.Loutsenko, V.Spiridonov

Self-similarity in Spectral Problems and q-special Functions

solv-int hep-th nlin.SI

Similarity symmetries of the factorization chains for one-dimensional differential and finite-difference Schr\"odinger equations are discussed. Properties of the potentials defined by self-similar reductions of these chains are reviewed. In particular, their algebraic structure, relations to $q$-special functions, infinite soliton systems, supersymmetry, coherent states, orthogonal polynomials, one-dimensional Ising chains and random matrices are outlined.

solv-int/9909023

Saburo Kakei

Saburo Kakei

Orthogonal and symplectic matrix integrals and coupled KP hierarchy

4 pages, LaTeX, no figures. to appear in J. Phys. Soc. Jpn. Vol. 68, No. 9 (1999)

J.Phys.Soc.Jap. 68 (1999) 2875-2877

10.1143/JPSJ.68.2875

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

Orthogonal and symplectic matrix integrals are investigated. It is shown that the matrix integrals can be considered as a $\tau$-function of the coupled KP hierarchy, whose solution can be expressed in terms of pfaffians.

solv-int/9909024

Saburo Kakei

Saburo Kakei

Dressing method and the coupled KP hierarchy

11 pages, LaTeX, no figures

10.1016/S0375-9601(99)00848-8

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

The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by using the dressing method. It is shown that the coupled KP hierarchy can be reformulated as a reduced case of the 2-component KP hierarchy.

solv-int/9909025

Krzysztof Marciniak

Stefan Rauch-Wojciechowski, Krzysztof Marciniak, Hans Lundmark

Quasi-Lagrangian Systems of Newton Equations

50 pages including 9 figures. Uses epsfig package. To appear in J. Math. Phys

10.1063/1.533098

solv-int nlin.SI

Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$ with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Henon-Heiles type.

solv-int/9909026

Andrew Hone

Frank Nijhoff (University of Leeds), Andrew Hone and Nalini Joshi (University of Adelaide)

On a Schwarzian PDE associated with the KdV Hierarchy

11 pages

10.1016/S0375-9601(00)00063-3

solv-int nlin.SI

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under M\"obius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the {\it full} Painlev\'e VI equation, i.e. with four arbitary parameters.

solv-int/9909027

Sergei Sakovich

Sergei Sakovich

On two aspects of the Painleve analysis

8 pages

Int. J. Analysis 2013 (2013) 172813 (5 pages)

10.1155/2013/172813

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

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Backlund transformation related to the Lax pair.

solv-int/9909028

Johan van de Leur

Johan van de Leur

Matrix integrals and the geometry of spinors

J.Nonlin.Math.Phys. 8 (2001) 288-310

JNMP 4/2002 (Article)

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

We obtain the collection of symmetric and symplectic matrix integrals and the collection of Pfaffian tau-functions, recently described by Peng and Adler and van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector of a fermionic Fock space. This fermionic Fock space is the same space as one constructs to obtain the KP and 1-Toda lattice hierarchy.

solv-int/9910001

Ashok Das

J. Barcelos-Neto, Alin Constandache and Ashok Das

Dispersionless Fermionic KdV

15 pages

Phys.Lett. A268 (2000) 342-351

10.1016/S0375-9601(00)00189-4

solv-int hep-th nlin.SI

We analyze the dispersionless limits of the Kupershmidt equation, the SUSY KdV-B equation and the SUSY KdV equation. We present the Lax description for each of these models and bring out various properties associated with them as well as discuss open questions that need to be addressed in connection with these models.

solv-int/9910002

Ismagil T. Habibullin

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

The KdV equation on a half-line

14 pages. Some revisions are made and more detailes are worked out in comparison with the previous version (which was of 11 pages), pictures are added

solv-int nlin.SI

The initial boundary value problem on a half-line for the KdV equation with the boundary conditions $u|_{x=0}=a\leq0$, $u_{xx}|_{x=0}=3a^2$ is integrated by means of the inverse scattering method. In order to find the time evolution of the scattering matrix it turned out to be sufficient to solve the Riemann problem on a hyperelliptic curve of genus two, where the conjugation matrices are effectively defined by initial data.

solv-int/9910003

Fabian Wagner

A.J. Macfarlane, F. Wagner

Yang-Baxter Algebra for the n-Harmonic Oscillator Realisation of sp(2n,R)

9 pages, Latex, uses amsfonts

Phys. Lett. B468 (3-4) 244-250 (1999)

10.1016/S0370-2693(99)01261-7

DAMTP-1999-141

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

Using a rational R-matrix associated with the 4 x 4 defining matrix representation of c_2=sp(4), the Lie algebra of Sp(4), a one-site operator solution of the associated Yang-Baxter algebra acting in the Fock space of two harmonic oscillators is derived. This is used to define N-site integrable systems, which are soluble by a version of the algebraic Bethe ansatz method without nesting. All essential aspects of the work generalise directly from c_2 to c_n.

solv-int/9910004

Vladimir Konotop

V. V. Konotop (U. of Lisbon)

On integrable discretization of the inhomogeneous Ablowitz-Ladik model

6 pages, 1 figure, LaTeX

Phys. Lett. A 258, 18-24 (1999)

10.1016/S0375-9601(99)00336-9

solv-int nlin.SI

An integrable discretization of the inhomogeneous Ablowitz-Ladik model with a linear force is introduced. Conditions on parameters of the discretization which are necessary for reproducing Bloch oscillations are obtained. In particular, it is shown that the step of the discretization must be comensurable with the period of oscillations imposed by the inhomogeneous force. By proper choice of the step of the discretization the period of oscillations of a soliton in the discrete model can be made equal to an integer number of periods of oscillations in the underline continuous-time lattice.

solv-int/9910005

Pilar G. Estevez

P. G. Estevez and G.A. Hern\'aez (Universidad de Salamanca, SPAIN)

Lax pair, Darboux Transformations and solitonic solutions for a (2+1) dimensional NLSE

LaTeX, 9 figures

solv-int nlin.SI

In this paper the Singular Manifold Method has allowed us to obtain the Lax pair, Darboux transformations and tau functions for a non-linear Schr\"odiger equation in 2+1 dimensions. In this way we can iteratively build different kind of solutions with solitonic behavior.

solv-int/9910006

Anca Visinescu

D. Grecu, Anca Visinescu, A. S. Carstea

Beyond Nonlinear Schr\"odinger Equation Approximation for an Anharmonic Chain with Harmonic Long Range Interaction

6 pages, LaTeX, paper presented at NEEDS'99 Kolymbari, Crete

solv-int nlin.SI

Multi scales method is used to analyze a nonlinear differential-difference equation. In order $\epsilon^3$ the NLS equation is found to determine the space-time evolution of the leading amplitude. In the next order this has to satisfy a complex mKdV equation (the next in the NLS hierarchy) in order to eliminate secular terms. The zero dispersion point case is also analyzed and the relevant equation is a modified NLS equation with a third order derivative term included

solv-int/9910007

Basile Grammaticos

Y. Ohta, K.M. Tamizhmani, B. Grammaticos and A. Ramani

Singularity confinement and algebraic entropy: the case of the discrete Painlev\'e equations

PlainTeX

10.1016/S0375-9601(99)00670-2

solv-int nlin.SI

We examine the validity of the results obtained with the singularity confinement integrability criterion in the case of discrete Painlev\'e equations. The method used is based on the requirement of non-exponential growth of the homogeneous degree of the iterate of the mapping. We show that when we start from an integrable autonomous mapping and deautonomise it using singularity confinement the degrees of growth of the nonautonomous mapping and of the autonomous one are identical. Thus this low-growth based approach is compatible with the integrability of the results obtained through singularity confinement. The origin of the singularity confinement property and its necessary character for integrability are also analysed.

solv-int/9910008

Fabio Musso

F. Musso, O. Ragnisco

Exact Solution of the Quantum Calogero-Gaudin System and of its q-Deformation

20 pages Latex

10.1063/1.1308508

solv-int nlin.SI

A complete set of commuting observables for the Calogero-Gaudin system is diagonalized, and the explicit form of the corresponding eigenvalues and eigenfunctions is derived. We use a purely algebraic procedure exploiting the co-algebra invariance of the model; with the proper technical modifications this procedure can be applied to the $q-$deformed version of the model, which is then also exactly solved.

solv-int/9910009

Fabio Musso

A. Ballesteros, O. Ragnisco

N=2 Hamiltonians with sl(2) coalgebra symmetry and their integrable deformations

14 pages Latex

solv-int nlin.SI

Two dimensional classical integrable systems and different integrable deformations for them are derived from phase space realizations of classical $sl(2)$ Poisson coalgebras and their $q-$deformed analogues. Generalizations of Morse, oscillator and centrifugal potentials are obtained. The N=2 Calogero system is shown to be $sl(2)$ coalgebra invariant and the well-known Jordan-Schwinger realization can be also derived from a (non-coassociative) coproduct on $sl(2)$. The Gaudin Hamiltonian associated to such Jordan-Schwinger construction is presented. Through these examples, it can be clearly appreciated how the coalgebra symmetry of a hamiltonian system allows a straightforward construction of different integrable deformations for it.

solv-int/9910010

Henning Samtleben

D. Korotkin, N. Manojlovic, H. Samtleben

Schlesinger transformations for elliptic isomonodromic deformations

19 pages, LaTeX2e

J.Math.Phys. 41 (2000) 3125-3141

10.1063/1.533296

AEI-1999-31, LPTENS-99/36

solv-int hep-th nlin.SI

Schlesinger transformations are discrete monodromy preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere we construct these transformations for isomonodromic deformations on genus one Riemann surfaces. Their action on the system's tau-function is computed and we obtain an explicit expression for the ratio of the old and the transformed tau-function.

solv-int/9910011

Andrei Maltsev Ya.

Andrei Ya. Maltsev

The averaging of non-local Hamiltonian structures in Whitham's method

Latex, 40 Pages, 1 figure

Intern. Journ. of Math. and Math. Sci. 30:7 (2002) 399-434

solv-int nlin.SI

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

We consider the $m$-phase Whitham's averaging method and propose a procedure of "averaging" of non-local Hamiltonian structures. The procedure is based on the existence of a sufficient number of local commuting integrals of a system and gives a Poisson bracket of Ferapontov type for the Whitham's system. The method can be considered as a generalization of the Dubrovin-Novikov procedure for the local field-theoretical brackets.

solv-int/9910012

Takeo Kojima

H.Furutsu, T.Kojima and Y.-H.Quano

Form factors of the SU(2) invariant massive Thirring model with boundary reflection

LaTEX2e, 15pages

Int.J.Mod.Phys. A15 (2000) 3037-3052

10.1142/S0217751X0000104X

solv-int hep-th nlin.SI

The SU(2) invariant massive Thirring model with a boundary is considered on the basis of the vertex operator approach. The bosonic formulae are presented for the vacuum vector and its dual in the presence of the boundary. The integral representations are also given for form factors of the present model.

solv-int/9911001

Andrey Tsyganov User

Andrey Tsiganov

Canonical transformations of the time for the Toda lattice and the Holt system

LaTeX2e, +amssymb.cls, 8 p

J. Phys. A, Math. Gen. 33, No.26, 4825-4835, (2000)

10.1088/0305-4470/33/26/308

solv-int nlin.SI

For the Toda lattice and the Holt system we consider properties of canonical transformations of the extended phase space, which preserve integrability. The separated variables are invariant under change of the time. On the other hand, mapping of the time induces transformations of the action-angles variables and a shift of the generating function of the B\"{a}cklund transformation.

solv-int/9911002

Korotkin Dmitrii

D.Korotkin

Introduction to the functions on compact Riemann surfaces and theta-functions

31 pages, lectures given at "First non-orthodox school on Non-linearity and geometry", sept 21-28, 1995, Warsaw

"Nonlinearity and Geometry", ed. by D.Wojcik and J.Cieslinski, Polish Scient. Publ. PWN p. p.109-139

solv-int nlin.SI

We collect some classical results related to analysis on the Riemann surfaces. The notes may serve as an introduction to the field: we suppose that the reader is familiar only with the basic facts from topology and complex analysis. the treatment is organized to give a background for further applications to non-linear differential equations.

solv-int/9911003

Cicogna G

Resonant Bifurcations

PlainTeX, no figures

solv-int nlin.SI

We consider dynamical systems depending on one or more real parameters, and assuming that, for some ``critical'' value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existence -- under suitable hypotheses -- of a general class of bifurcating solutions in correspondence to this resonance. These bifurcating solutions include, as particular cases, the usual stationary and Hopf bifurcations. The main idea is to transform the given dynamical system into normal form (in the sense of Poincar\'e-Dulac), and to impose that the normalizing transformation is convergent, using the convergence conditions in the form given by A. Bruno. Some specially interesting situations, including the cases of multiple-periodic solutions, and of degenerate eigenvalues in the presence of symmetry, are also discussed with some detail.

solv-int/9911004

Fis. Teorica. Valladolid.

Angel Ballesteros and Francisco J. Herranz

Integrable deformations of oscillator chains from quantum algebras

15 pages, LaTeX

J. Phys. A, 32 (1999) 8851-8862

10.1088/0305-4470/32/50/306

solv-int math.QA nlin.SI

A family of completely integrable nonlinear deformations of systems of N harmonic oscillators are constructed from the non-standard quantum deformation of the sl(2,R) algebra. Explicit expressions for all the associated integrals of motion are given, and the long-range nature of the interactions introduced by the deformation is shown to be linked to the underlying coalgebra structure. Separability and superintegrability properties of such systems are analysed, and their connection with classical angular momentum chains is used to construct a non-standard integrable deformation of the XXX hyperbolic Gaudin system.

solv-int/9911005

Alexander Sorin

F.Delduc, A.Sorin

A note on real forms of the complex N=4 supersymmetric Toda chain hierarchy in real N=2 and N=4 superspaces

10 pages, latex, new section, one reference and report-no are added

Nucl.Phys. B577 (2000) 461-470

10.1016/S0550-3213(00)00121-8

LPENSL-TH-20/99

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

Three inequivalent real forms of the complex N=4 supersymmetric Toda chain hierarchy (Nucl. Phys. B558 (1999) 545, solv-int/9907004) in the real N=2 superspace with one even and two odd real coordinates are presented. It is demonstrated that the first of them possesses a global N=4 supersymmetry, while the other two admit a twisted N=4 supersymmetry. A new superfield basis in which supersymmetry transformations are local is discussed and a manifest N=4 supersymmetric representation of the N=4 Toda chain in terms of a chiral and an anti-chiral N=4 superfield is constructed. Its relation to the complex N=4 supersymmetric KdV hierarchy is established. Darboux-Backlund symmetries and a new real form of this last hierarchy possessing a twisted N=4 supersymmetry are derived.

solv-int/9911006

Folkert Muller-Hoissen

Aristophanes Dimakis and Folkert Muller-Hoissen

Bicomplexes and finite Toda lattices

5 pages, 1 figure, uses amssymb.sty and diagrams.sty, Proceedings "Quantum Theory and Symmetries" (Goslar, July 1999)

solv-int nlin.SI

We associate bicomplexes with the finite Toda lattice and with a finite Toda field theory in such a way that conserved currents and charges are obtained by a simple iterative construction.

solv-int/9911007

Wen-Xiu Ma

Yunbo Zeng and Wen-Xiu Ma

Separation of variables for soliton equations via their binary constrained flows

39 pages, Amstex

10.1063/1.533105

solv-int nlin.SI

Binary constrained flows of soliton equations admitting $2\times 2$ Lax matrices have 2N degrees of freedom, which is twice as many as degrees of freedom in the case of mono-constrained flows. For their separation of variables only N pairs of canonical separated variables can be introduced via their Lax matrices by using the normal method. A new method to introduce the other N pairs of canonical separated variables and additional separated equations is proposed. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies.

solv-int/9911008

Igor G. Korepanov

I.G. Korepanov

Multidimensional analogs of geometric s<-->t duality

LaTeX2e, pictures using emlines. In this re-submission, an English version of the paper is added (9 pages, file english.tex) to the originally submitted file in Russian (10 pages, russian.tex)

10.1007/BF02551073

solv-int cond-mat gr-qc hep-lat hep-th nlin.SI

The usual propetry of s<-->t duality for scattering amplitudes, e.g. for Veneziano amplitude, is deeply connected with the 2-dimensional geometry. In particular, a simple geometric construction of such amplitudes was proposed in a joint work by this author and S.Saito (solv-int/9812016). Here we propose analogs of one of those amplitudes associated with multidimensional euclidean spaces, paying most attention to the 3-dimensional case. Our results can be regarded as a variant of "Regge calculus" intimately connected with ideas of the theory of integrable models.

solv-int/9911009

Verbus

A. P. Protogenov and V. A. Verbus

Equations and Integrals of Motion in Discrete Integrable $A_{k-1}$ Algebra Models

20 pages

Theor.Math.Phys. 119 (1999) 420-429

solv-int hep-th nlin.SI

We study integrals of motion for Hirota bilinear difference equation that is satisfied by the eigenvalues of the transfer-matrix. The combinations of the eigenvalues of the transfer-matrix are found, which are integrals of motion for integrable discrete models for the $A_{k-1}$ algebra with zero and quasiperiodic boundary conditions. Discrete analogues of the equations of motion for the Bullough-Dodd model and non-Abelian generalization of Liouville model are obtained.

solv-int/9912001

Anjan Kundu

Anjan Kundu

Construction of variable mass sine-Gordon and other novel inhomogeneous quantum integrable models

Latex, 6 pages, no figure; to be published in J. Nonlinear Math. Phys. as Proc. NEEDS'99 (Crete, Greece, June, 1999)

SINP/TNP/99-34

solv-int nlin.SI

The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model, cylindrical NLS, spin chains with impurity, inhomogeneous Toda chain, the Ablowitz-Ladik model etc.

solv-int/9912002

Peter Schupp

Branislav Jurco, Peter Schupp

Quantum Lax scheme for Ruijsenaars models

5 pages, contribution to proceedings of "Quantum Theory and Symmetries" Goslar, 18-22 July 1999

solv-int nlin.SI

We develop a quantum Lax scheme for IRF models and difference versions of Calogero-Moser-Sutherland models introduced by Ruijsenaars. The construction is in the spirit of the Adler-Kostant-Symes method generalized to the case of face Hopf algebras and elliptic quantum groups with dynamical R-matrices.

solv-int/9912003

Francisco Toppan

E. Ivanov (1), S. Krivonos (1), F. Toppan (2) ((1) JINR, Dubna, Russia, (2) CBPF, Rio d.J., Brazil)

N=4 Sugawara construction on affine sl(2|1), sl(3) and mKdV-type superhierarchies

Few references added, misprints corrected

Mod.Phys.Lett. A14 (1999) 2673-2686

10.1142/S0217732399002819

CBPF-NF-046-99, JINR E2-99-302

solv-int hep-th nlin.SI

The local Sugawara constructions of the "small" N=4 SCA in terms of supercurrents of N=2 extensions of the affinization of the sl(2|1) and sl(3) algebras are investigated. The associated super mKdV type hierarchies induced by N=4 SKdV ones are defined. In the sl(3) case the existence of two inequivalent Sugawara constructions is found. The long one involves all the affine sl(3)-valued currents, while the "short" one deals only with those from the subalgebra sl(2)\oplus u(1). As a consequence, the sl(3)-valued affine superfields carry two inequivalent mKdV type super hierarchies induced by the correspondence between "small" N=4 SCA and N=4 SKdV hierarchy. However, only the first hierarchy posseses genuine global N=4 supersymmetry. We discuss peculiarities of the realization of this N=4 supersymmetry on the affine supercurrents.

solv-int/9912004

Wen-Xiu Ma

Wen-Xiu Ma

Integrable Couplings of Soliton Equations by Perturbations I. A General Theory and Application to the KdV Hierarchy

41 pages, latex, to appear in Methods and Applications of Analysis

solv-int nlin.SI

A theory for constructing integrable couplings of soliton equations is developed by using various perturbations around solutions of perturbed soliton equations being analytic with respect to a small perturbation parameter. Multi-scale perturbations can be taken and thus higher dimensional integrable couplings can be presented. The theory is applied to the KdV soliton hierarchy. Infinitely many integrable couplings are constructed for each soliton equation in the KdV hierarchy, which contain integrable couplings possessing quadruple Hamiltonian formulations and two classes of hereditary recursion operators, and integrable couplings possessing local 2+1 dimensional bi-Hamiltonian formulations and consequent 2+1 dimensional hereditary recursion operators.

solv-int/9912005

L. V. Bogdanov

L.V. Bogdanov (L.D. Landau ITP, Moscow) and B.G. Konopelchenko (Universita di Lecce, Italy)

Generalized KP hierarchy: M\"obius Symmetry, Symmetry Constraints and Calogero-Moser System

18 pages, LaTeX, talk at "Solitons, Collapses and Turbulence: Achievements, Developments and Perspectives" (August 1999, Chernogolovka, Russia)

solv-int nlin.SI

Analytic-bilinear approach is used to study continuous and discrete non-isospectral symmetries of the generalized KP hierarchy. It is shown that M\"obius symmetry transformation for the singular manifold equation leads to continuous or discrete non-isospectral symmetry of the basic (scalar or multicomponent KP) hierarchy connected with binary B\"acklund transformation. A more general class of multicomponent M\"obius-type symmetries is studied. It is demonstrated that symmetry constraints of KP hierarchy defined using multicomponent M\"obius-type symmetries give rise to Calogero-Moser system.

solv-int/9912006

Dimitri Kusnezov

Hui Li, Dimitri Kusnezov, Francesco Iachello

Group Theoretical Properties and Band Structure of the Lame Hamiltonian

21 pages Revtex + 6 eps + 2 jpg figures

10.1088/0305-4470/33/36/310

solv-int nlin.SI

We study the group theoretical properties of the Lame equation and its relation to su(1,1) and su(2). We compute the band structure, dispersion relation and transfer matrix and discuss the dynamical symmetry limits.

solv-int/9912007

Dimitri Kusnezov

Hui Li, Dimitri Kusnezov (Yale)

Dynamical Symmetry Approach to Periodic Hamiltonians

20 pages, 7 postscript figures

10.1063/1.533265

solv-int nlin.SI

We show that dynamical symmetry methods can be applied to Hamiltonians with periodic potentials. We construct dynamical symmetry Hamiltonians for the Scarf potential and its extensions using representations of su(1,1) and so(2,2). Energy bands and gaps are readily understood in terms of representation theory. We compute the transfer matrices and dispersion relations for these systems, and find that the complementary series plays a central role as well as non-unitary representations.

solv-int/9912008

Pierre van Moerbeke

M. Adler and P. van Moerbeke

The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff

58 pages

solv-int nlin.SI

We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities, which we show to be equivalent to the Pfaff Lattice. In the semi-infinite case, the tau-functions are Pfaffians; interesting examples are the matrix integrals over symmetric matrices (symmetric matrix integrals) and matrix integrals over self-dual quaternionic Hermitean matrices (symplectic matrix integrals). There is a striking parallel of the Pfaff lattice with the Toda lattice, and more so, there is a map from one to the other. In particular, we exhibit two maps, dual to each other, (i) from the the Hermitean matrix integrals to the symmetric matrix integrals, and (ii) from the Hermitean matrix integrals to the symplectic matrix integrals. The map is given by the skew-Borel decomposition of a skew-symmetric operator. We give explicit examples for the classical weights.

solv-int/9912009

Antonio Lima Santos

A. Lima-Santos

Quantum Lax Pair From Yang-Baxter Equations

11 pages, 4 figures

J. Stat. Mech. 0905-P05008 (2009)

solv-int hep-th nlin.SI

We show explicitly how to construct the quantum Lax pair for systems with open boundary conditions. We demonstrate the method by applying it to the Heisenberg XXZ model with general integrable boundary terms.