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solv-int/9802001

Akira Takamura

Akira Takamura, Ken'ichi Takano

Braid Structure and Raising-Lowering Operator Formalism in Sutherland Model

11 pages, Latex, no figures

10.1088/0305-4470/31/25/002

DPNU-98-03

solv-int cond-mat nlin.SI

We algebraically construct the Fock space of the Sutherland model in terms of the eigenstates of the pseudomomenta as basis vectors. For this purpose, we derive the raising and lowering operators which increase and decrease eigenvalues of pseudomomenta. The operators exchanging eigenvalues of two pseudomomenta have been known. All the eigenstates are systematically produced by starting from the ground state and multiplying these operators to it.

solv-int/9802002

Dionisio Bazeia

D. Bazeia and F. Moraes

Chiral Solitons in Generalized Korteweg-de Vries Equations

9 pages, latex, no figures. References added, typos corrected

Phys. Lett. A, 249 (1998) 450

10.1016/S0375-9601(98)00727-0

MIT-CTP-2713

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

Generalizations of the Korteweg-de Vries equation are considered, and some explicit solutions are presented. There are situations where solutions engender the interesting property of being chiral, that is, of having velocity determined in terms of the parameters that define the generalized equation, with a definite sign.

solv-int/9802003

Ziemowit Popowicz

S.Krivonos, A.Pashnev and Z.Popowicz

Lax pairs for N=2,3 Supersymmetric KdV Equations and their Extensions

8 pages, LaTex

10.1142/S0217732398001510

IFT UWr 919/98

solv-int hep-th nlin.SI

We present the Lax operator for the N=3 KdV hierarchy and consider its extensions. We also construct a new infinite family of N=2 supersymmetric hierarchies by exhibiting the corresponding super Lax operators. The new realization of N=4 supersymmetry on the two general N=2 superfields, bosonic spin 1 and fermionic spin 1/2, is discussed.

solv-int/9802004

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

Computation of Higher-order Symmetries for Nonlinear Evolution and Lattice Equations

Submitted to: Advances in Computational Mathematics, 23 pages, Latex, uses the style file bal.sty

solv-int nlin.SI

A straightforward algorithm for the symbolic computation of higher-order symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the higher-order symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first-order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi-discrete lattice equations. With our Integrability Package, higher-order symmetries are obtained for several well-known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of higher-order symmetries exist. The existence of a sequence of such symmetries is a predictor for integrability.

solv-int/9802005

YU-Song Ju

Yu Song-Ju, Kouichi Toda and Takeshi Fukuyama

Hierarchy of Higher Dimensional Integrable System

10 pages, uses ioplppt.sty

solv-int nlin.SI

Integrable equations in ($1 + 1$) dimensions have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. In this paper we consider whether integrable equations in ($2 + 1$) dimensions have also the analogous hierarchies to those in ($1 + 1$) dimensions. Explicitly is discussed the Bogoyavlenskii-Schiff(BS) equation. For the BS hierarchy, there appears an ambiguity in the Painlev\'e test. Nevertheless, it may be concluded that the BS hierarchy is integrable.

solv-int/9802006

Myrzakulov Ratbay

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

Surfaces, curves and the Lakshmanan equivalent counterparts of the some Myrzakulov equations

8 pages, LaTex, no figures, [email protected]

solv-int nlin.SI

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

solv-int/9802007

Dionisio Bazeia

D. Bazeia

Chiral Solutions to Generalized Burgers and Burgers-Huxley Equations

17 pages, latex, no figures

MIT-CTP 2714

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

We investigate generalizations of the Burgers and Burgers-Huxley equations. The investigations we offer focus attention mainly on presenting explict analytical solutions by means of relating these generalized equations to relativistic 1+1 dimensional systems of scalar fields where topological solutions are known to play a role. Emphasis is given on chiral solutions, that is, on the possibility of finding solutions that travel with velocities determined in terms of the parameters that identify the generalized equation, with a definite sign.

solv-int/9802008

Fis. Teorica. Valladolid.

Angel Ballesteros and Orlando Ragnisco

A systematic construction of completely integrable Hamiltonians from coalgebras

26 pages, LaTeX

10.1088/0305-4470/31/16/009

UBU-Dfis-97-12

solv-int nlin.SI

A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of Hamiltonian systems with coalgebra symmetry. In order to illustrate this general method, the $so(2,1)$ algebra and the oscillator algebra $h_4$ are used to derive new classical integrable systems including a generalization of Gaudin-Calogero systems and oscillator chains. Quantum deformations are then used to obtain some explicit integrable deformations of the previous long-range interacting systems and a (non-coboundary) deformation of the $(1+1)$ Poincar\'e algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.

solv-int/9802009

Bireswar Basu-Mallick

B. Basu-Mallick

Multi-parameter deformed and nonstandard $Y(gl_M)$ Yangian symmetry in integrable variants of Haldane-Shastry spin chain

18 pages, latex, no figures

10.1143/JPSJ.67.2227

solv-int hep-th nlin.SI

By using `anyon like' representations of permutation algebra, which pick up nontrivial phase factors while interchanging the spins of two lattice sites, we construct some integrable variants of Haldane-Shastry (HS) spin chain. Lax equations for these spin chains allow us to find out the related conserved quantities. However, it turns out that such spin chains also possess a few additional conserved quantities which are apparently not derivable from the Lax equations. Identifying these additional conserved quantities, and the usual ones related to Lax equations, with different modes of a monodromy matrix, it is shown that the above mentioned HS like spin chains exhibit multi-parameter deformed and `nonstandard' variants of $Y(gl_M)$ Yangian symmetry.

solv-int/9802010

David H. Sattinger

R. Beals, D. H. Sattinger, and E. Williams

A Dirac Sea and thermodynamic equilibrium for the quantized three-wave interaction

10.1063/1.532306

solv-int nlin.SI

The classical version of the three wave interaction models the creation and destruction of waves; the quantized version models the creation and destruction of particles. The quantum three wave interaction is described and the Bethe Ansatz for the eigenfunctions is given in closed form. The Bethe equations are derived in a rigorous fashion and are shown to have a thermodynamic limit. The Dirac sea of negative energy states is obtained as the infinite density limit. Finite particle/hole excitations are determined and the asymptotic relation of energy and momentum is obtained. The Yang-Yang functional for the relative free energy of finite density excitations is constructed and is shown to be convex and bounded below. The equations of thermal equilibrium are obtained.

solv-int/9802011

Adam Doliwa

Quadratic reductions of quadrilateral lattices

24 pages

J. Geom. Phys. 30 (1999) 169-186

10.1016/S0393-0440(98)00053-9

solv-int nlin.SI

It is shown that quadratic constraints are compatible with the geometric integrability scheme of the multidimensional quadrilateral lattice equation. The corresponding Ribaucour reduction of the fundamental transformation of quadrilateral lattices is found as well, and superposition of the Ribaucour transformations is presented in the vectorial framework. Finally, the quadratic reduction approach is illustrated on the example of multidimensional circular lattices.

solv-int/9802012

Sergei M. Sergeev

I. G. Korepanov and S. M. Sergeev

Eigenvector and eigenvalue problem for 3D bosonic model

LaTeX, 18 pages

solv-int nlin.SI

In this paper we reformulate free field theory models defined on the rectangular $D+1$ dimensional lattices as $D+1$ evolution models. This evolution is in part a simple linear evolution on free (``creation'' and ``annihilation'') operators. Formal eigenvectors of this linear evolution can be directly constructed, and them play the role of the ``physical'' creation and annihilation operators. These operators being completed by a ``physical'' vacuum vector give the spectrum of the evolution operator, as well as the trace of the evolution operator give a correct expression for the partition function. As an example, Bazhanov -- Baxter's free bosonic model is considered.

solv-int/9802013

Francois Delduc

F. delduc, L. Gallot

Supersymmetric Drinfeld-Sokolov reduction

25 pages, LaTeX file

10.1063/1.532532

ENSLAPP-L-668/97

solv-int nlin.SI

The Drinfeld-Sokolov construction of integrable hierarchies, as well as its generalizations, may be extended to the case of loop superalgebras. A sufficient condition on the algebraic data for the resulting hierarchy to be invariant under supersymmetry transformation is given. The method used is a construction of the hierarchies in superspace, where supersymmetry is manifest. Several examples are discussed.

solv-int/9802014

Sergei M. Sergeev

S. M. Sergeev

3D symplectic map

LaTeX, 13 pages

10.1016/S0375-9601(99)00072-9

solv-int nlin.SI

Quantum 3D R-matrix in the classical (i.e. functional) limit gives a symplectic map of dynamical variables. The corresponding 3D evolution model is considered. An auxiliary problem for it is a system of linear equations playing the role of the monodromy matrix in 2D models. A generating function for the integrals of motion is constructed as a determinant of the auxiliary system.

solv-int/9802015

Wenli Yang

Bo-yu Hou and Wen-li Yang

The nondynamical r-matrix structure of the elliptic Ruijsenaars-Schneider model with N=2

7 pages, Latex file 17k

Commun.Theor.Phys.33:371-376,2000

IMPNWU-971219

solv-int nlin.SI

We demonstrate that in a certain gauge the elliptic Ruijsenaars-Shneider model with N=2 admits a nondynamical r-matrix structure and the corresponding classical r-matrix is the same as that of its non-relativistic counterpart (Calogero-Moser model) in the same gauge.The relation between our (classical)Lax operator and the Lax operator given by Ruijsenaars is also obtained.

solv-int/9802016

Juri Suris

Yuri B. Suris (Bremen/Berlin) and Orlando Ragnisco (Rome)

What is the relativistic Volterra lattice?

48 pp, LaTeX

Commun. Math. Phys., 1999, V. 200, p. 445--485.

10.1007/s002200050537

solv-int nlin.SI

We develop a systematic procedure of finding integrable ''relativistic'' (regular one-parameter) deformations for integrable lattice systems. Our procedure is based on the integrable time discretizations and consists of three steps. First, for a given system one finds a local discretization living in the same hierarchy. Second, one considers this discretization as a particular Cauchy problem for a certain 2-dimensional lattice equation, and then looks for another meaningful Cauchy problems, which can be, in turn, interpreted as new discrete time systems. Third, one has to identify integrable hierarchies to which these new discrete time systems belong. These novel hierarchies are called then ''relativistic'', the small time step $h$ playing the role of inverse speed of light. We apply this procedure to the Toda lattice (and recover the well-known relativistic Toda lattice), as well as to the Volterra lattice and a certain Bogoyavlensky lattice, for which the ''relativistic'' deformations were not known previously.

solv-int/9802017

Myrzakulov Ratbay

R. Myrzakulov

Solitons, Surfaces, Curves, and the Spin Description of Nonlinear Evolution Equations

25 pages, LaTex, no figures

solv-int nlin.SI

The briefly review on the common spin description of the nonlinear evolution equations.

solv-int/9802018

Ivanov Evgenyi

E. Ivanov

On gauge-equivalent formulations of N=4 SKdV hierarchy

7 pages, LaTeX

10.1142/S021773239800303X

solv-int hep-th nlin.SI

We point out that the N=4 supersymmetric KdV hierarchy, when written through the prepotentials of the bosonic chiral and antichiral N=2 supercurrents, exhibits a freedom related to the possibility to choose different gauges for the prepotentials. In particular, this implies that the Lax operator for the N=4 SKdV system and the associated realization of N=4 supersymmetry obtained in solv-int/9802003 are reduced to the previously known ones. We give the prepotential form of the `small' N=4 superconformal algebra, the second hamiltonian structure algebra of the N=4 SKdV hierarchy, for two choices of gauge.

solv-int/9803001

Metin Gurses

Metin Gurses (Bilkent University)

Motion of Curves on Two Dimensional Surfaces and Soliton Equations

Latex, 15 pp, to be published in Physics Letters A

10.1016/S0375-9601(98)00151-0

solv-int nlin.SI

A connection is established between the soliton equations and curves moving in a three dimensional space $V_{3}$. The sign of the self-interacting terms of the soliton equations are related to the signature of $V_{3}$. It is shown that there corresponds a moving curve to each soliton equations.

solv-int/9803002

Wen-Xiu Ma

Wen-Xiu MA

Extension of Hereditary Symmetry Operators

13 pages, LaTex

10.1088/0305-4470/31/35/009

solv-int nlin.SI

Two models of candidates for hereditary symmetry operators are proposed and thus many nonlinear systems of evolution equations possessing infinitely many commutative symmetries may be generated. Some concrete structures of hereditary symmetry operators are carefully analyzed on the base of the resulting general conditions and several corresponding nonlinear systems are explicitly given out as illustrative examples.

solv-int/9803003

Galina Gorbatina

Valery S. Dryuma, Makoto Matsumoto

Finsler-Geometrical Approach to the Studying of Nonlinear Dynamical Systems

22 pages, Latex; Reports of Math. Phys.(1998)

solv-int nlin.SI

A two dimensional Finsler space associated with the differential equation $y''=Y_3 y'^3+Y_2 y'^2+Y_1 y'+Y_0$ is characterized by a tensor equation and called the Douglas space. An application to the Lorenz nonlinear dynamical equation is discussed from the standpoint of Finsler geometry.

solv-int/9803004

Galina Gorbatina

Valery S. Dryuma

On the Law of Transformation of Affine Connection and its Integration. Part 1. Generalization of the Lame equations

18 pages, Latex

Buletinul Academiei de Stiinte a Republicii Moldova Matematica, v.1(26), 1998, p.55-68

solv-int nlin.SI

The law of transformation of affine connection for n-dimensional manifolds as the system of nonlinear equations on local coordinates of manifold is considered. The extension of the Darboux-Lame system of equations to the spaces of constant negative curvature is demonstrated. Geodesic deviation equation as well as the equations of geodesics are presented in the form of the matrix Darboux-Lame system of equations.

solv-int/9803005

Willy Hereman (1), Unal Goktas (1), Michael D. Colagrosso (1), Antonio J. Miller (2) ((1) Colorado School of Mines, (2) The Pennsylvania State University)

Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations

Submitted to: Computer Physics Communications, Latex, uses the style files elsart.sty and elsart12.sty

10.1016/S0010-4655(98)00121-0

solv-int nlin.SI

Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlev\'e test, which is applicable to polynomial systems of ordinary and partial differential equations. The second and third algorithms allow one to explicitly compute polynomial conserved densities and higher-order symmetries of nonlinear evolution and lattice equations. The first algorithm is implemented in the symbolic syntax of both Macsyma and Mathematica. The second and third algorithms are available in Mathematica. The codes can be used for computer-aided integrability testing of nonlinear differential and lattice equations as they occur in various branches of the sciences and engineering. Applied to systems with parameters, the codes can determine the conditions on the parameters so that the systems pass the Painlev\'e test, or admit a sequence of conserved densities or higher-order symmetries.

solv-int/9803006

Willy Hereman (Colorado School of Mines)

The Painlev\'e Integrability Test

For chapter in book `Computer Algebra in Germany', Eds.: J. Grabmeier et al. (Springer Verlag, 1998), Submitted to Werner Seiler, March 5, 1998, Latex

solv-int nlin.SI

The Painlev\'e test is a widely applied and quite successful technique to investigate the integrability of nonlinear ODEs and PDEs by analyzing the singularity structure of the solutions. The test is named after the French mathematician Paul Painlev\'e ....

solv-int/9803007

pilar Garcia Estevez

J.M. Cervero and P.G. Estevez

Miura Transformation between two Non-Linear Equations in 2+1 dimensions

14 pages, latex. Journal of Mathematical Physics (to appear)

10.1063/1.532421

AFTUS-97/15

solv-int nlin.SI

A Dispersive Wave Equation in 2+1 dimensions (2LDW) widely discussed by different authors is shown to be nothing but the modified version of the Generalized Dispersive Wave Equation (GLDW). Using Singularity Analysis and techniques based upon the Painleve Property leading to the Double Singular Manifold Expansion we shall find the Miura Transformation which converts the 2LDW Equation into the GLDW Equation. Through this Miura Transformation we shall also present the Lax pair of the 2LDW Equation as well as some interesting reductions to several already known integrable systems in 1+1 dimensions.

solv-int/9803008

A. N. Leznov

To the Gel'fand-Tsetlin realization of irreducible representations of classical semisimple algebras

13 pages, LaTeX

IIMAS-UNAM No. 77, 1998

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

It is shown that the Gel'fand-Tsetlin realization of irreducible representations of the $A_n$ algebra is directly connected with a linear exactly integrable system in the n-dimensional space. General solution for this system is explicitly given.

solv-int/9803009

Wen-Xiu Ma

Wen-Xiu Ma

A Class of Coupled KdV systems and Their Bi-Hamiltonian Formulations

8 pages, latex

10.1088/0305-4470/31/37/016

solv-int nlin.SI

A Hamiltonian pair with arbitrary constants is proposed and thus a sort of hereditary operators is resulted. All the corresponding systems of evolution equations possess local bi-Hamiltonian formulation and a special choice of the systems leads to the KdV hierarchy. Illustrative examples are given.

solv-int/9803010

Alexander Sorin

V.B. Derjagin, A.N. Leznov and A. Sorin

The solution of the N=(0|2) superconformal f-Toda lattice

12 pages, latex, no figures, some misprints corrected, one reference and report-no added

Nucl.Phys. B527 (1998) 643-656

10.1016/S0550-3213(98)00368-X

IIMAS-UNAM-80, JINR E2-98-49

solv-int hep-th nlin.SI

The general solution of the two-dimensional integrable generalization of the f-Toda chain with fixed ends is explicitly presented in terms of matrix elements of various fundamental representations of the SL(n|n-1) supergroup. The dominant role of the representation theory of graded Lie algebras in the problem of constructing integrable mappings and lattices is demonstrated.

solv-int/9803011

S. Vijayalakshmi

R. Myrzakulov (1), S. Vijayalakshmi (2), R.N. Syzdykova (1) and M. Lakshmanan(2) ((1) Centre for Nonlinear Dynamics,.Bharathidasan University, Tiruchirapalli, India (2) Center for Nonlinear Problems, Alma-Ata-35, Kazakstan)

On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schr\"odinger equations

32 pages, no figures, accepted for publication in J. Math. Phys

J. Math. Phys. vol.39 (1998) 2122-2140

10.1063/1.532279

solv-int nlin.SI

Using a moving space curve formalism, geometrical as well as gauge equivalence between a (2+1) dimensional spin equation (M-I equation) and the (2+1) dimensional nonlinear Schr\"odinger equation (NLSE) originally discovered by Calogero, discussed then by Zakharov and recently rederived by Strachan, have been estabilished. A compatible set of three linear equations are obtained and integrals of motion are discussed. Through stereographic projection, the M-I equation has been bilinearized and different types of solutions such as line and curved solitons, breaking solitons, induced dromions, and domain wall type solutions are presented. Breaking soliton solutions of (2+1) dimensional NLSE have also been reported. Generalizations of the above spin equation are discussed.

solv-int/9803012

Sonjiyu Yu

S.J. Yu, K. Toda and T. Fukuyama (Ritsumeikan Univ.)

N-Soliton Solutions to a New (2 + 1) Dimensional Integrable Equation

7 pages, uses ioplppt.sty

10.1088/0305-4470/31/50/013

solv-int nlin.SI

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, $\phi_{xt} + \phi_{xxxz}/4 + \phi_x \phi_{xz} + \phi_{xx} \phi_z/2 + \partial_x^{-1} \phi_{zzz}/4 = 0$. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations.

solv-int/9803013

Robert Conte

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

Towards second order Lax pairs to discrete Painlev\'e equations of first degree

16 pages, no figure, standard Latex, to appear in Chaos, solitons and fractals (1998). Proceedings of Integrability and chaos in discrete systems, Brussels 2--6 July 1997, eds. I. Antoniou and F. Lambert. Revision (one reference suppressed)

S98/018

solv-int nlin.SI

We investigate the question of finding discrete Lax pairs for the six discrete Painlev\'e equations (Pn). The choice we make is to discretize the pairs of Garnier, once converted to matricial form.

solv-int/9803014

Robert Conte

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

Rules of discretization for Painlev\'e equations

21 pages, no figure, standard Latex, to appear in Theory of nonlinear special functions : the Painlev\'e transcendents, eds. L. Vinet and P. Winternitz (Springer, Berlin, 1998). Proceedings of Montreal, 13--17 May 1996

S96/075

solv-int nlin.SI

The discrete Painlev\'e property is precisely defined, and basic discretization rules to preserve it are stated. The discrete Painlev\'e test is enriched with a new method which perturbs the continuum limit and generates infinitely many no-log conditions. A general, direct method is provided to search for discrete Lax pairs.

solv-int/9803015

Adam Doliwa

Adam Doliwa, Manuel Manas, Luis Martinez Alonso, Elena Medina and Paolo Maria Santini

Charged Free Fermions, Vertex Operators and Classical Theory of Conjugate Nets

28 pages, 3 Postscript figures

J.Phys.A32:1197-1216,1999

10.1088/0305-4470/32/7/010

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

We show that the quantum field theoretical formulation of the $\tau$-function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic vertex operators are Levy and adjoint Levy transformations and iii) the diagonal soliton vertex operators generate fundamental transformations. We also show that the bilinear identity for the multicomponent Kadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, a bilinear identity for the multidimensional quadrilateral lattice equations.

solv-int/9803016

Juri Suris

A.I.Bobenko, B.Lorbeer, Yu.B.Suris (TU Berlin)

Integrable discretizations of the Euler top

J. Math. Phys., 1998, V. 39, p. 6668-6683.

10.1063/1.532648

solv-int nlin.SI

Discretizations of the Euler top sharing the integrals of motion with the continuous time system are studied. Those of them which are also Poisson with respect to the invariant Poisson bracket of the Euler top are characterized. For all these Poisson discretizations a solution in terms of elliptic functions is found, allowing a direct comparison with the continuous time case. We demonstrate that the Veselov--Moser discretization also belongs to our family, and apply our methods to this particular example.

solv-int/9803017

Ladislav Hlavaty

L. Hlavaty

All generalized SU(2) chiral models have spectral dependent Lax formulation

5 pages, Latex2e, no figures

FJFI-98-3

solv-int nlin.SI

The equations that define the Lax pairs for generalized principal chiral models can be solved for any nondegenerate bilinear form on $su(2)$. The solution is dependent on one free variable that can serve as the spectral parameter.

solv-int/9804001

Oleg M. Kiselev

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

On lump instability of Davey--Stewartson II equation

Amstex, 9 pages

solv-int nlin.SI

We show that lumps (solitons) of the Davey--Stewartson II equation fail under small perturbations of initial data.

solv-int/9804002

Nugzar Makhaldiani

N. Makhaldiani (Dubna)

The system of three vortexes of two dimensional ideal hydrodinamics as a new example of the (integrable) Nambu- Poisson mechanics

LaTeX, 5 pages

JINR-E2-97-407

solv-int nlin.SI

A Nambu-Poisson formulation of the system of three ordinary differential equations describing dynamics of three vortexes of the ideal two-dimensional hydrodynamics is given. The system is integrated by quadratures.

solv-int/9804003

Micheline Musette

M. Musette (VUB, Brussels)

Painlev\'e analysis for nonlinear partial differential equations

61 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)

solv-int nlin.SI

The Painlev\'e analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlev\'e and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated B\"acklund transformation. A lot of remarkable properties are shared by these so-called ``integrable'' equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called ``partially integrable'' sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily % Conte agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and systems of Riccati equations which are linearisable, as well as the importance of the Weierstrass elliptic function, for building solitary waves or more elaborate solutions.

solv-int/9804004

Craig A. Tracy

Craig A. Tracy and Harold Widom

Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices

22 pages. LaTeX file. Minor correction

J. Statistical Physics 92 (1998), 809-835.

10.1023/A:1023084324803

solv-int math.SP nlin.SI

The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels for unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the representations of the correlation functions, cluster functions and spacing distributions in terms of them.

solv-int/9804005

Harold Widom

Harold Widom (University of California, Santa Cruz)

On the relation between orthogonal, symplectic and unitary matrix ensembles

13 pages. LaTeX file. Improved and simplified derivations of results

J.Statist.Phys. 94 (1999) 347-364

10.1023/A:1004536018336

solv-int hep-th math.SP nlin.SI

For the unitary ensembles of $N\times N$ Hermitian matrices associated with a weight function $w$ there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are $2\times2$ matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever $w'/w$ is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of $w'/w$. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations.

solv-int/9804006

John Harnad

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

Modular Solutions to Equations of Generalized Halphen Type

PlainTeX 36gs. (Formula for Hecke operator corrected.)

Proc.Roy.Soc.Lond. 456 (2000) 261-294

10.1098/rspa.2000.0517

CRM 2536 (1998)

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

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus zero and have $q$--series with integral coefficients. Rational maps relating these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps relating the associated differential systems.

solv-int/9804007

Edwin Langmann

Jonas Blom and Edwin Langmann

Novel integrable spin-particle models from gauge theories on a cylinder

12 pages, LaTex

Phys. Lett. B, 429 (1998) 336-342

10.1016/S0370-2693(98)00505-X

solv-int hep-th nlin.SI

We find and solve a large class of integrable dynamical systems which includes Calogero-Sutherland models and various novel generalizations thereof. In general they describe $N$ interacting particles moving on a circle and coupled to an arbitrary number, $m$, of $su(N)$ spin degrees of freedom with interactions which depend on arbitrary real parameters $x_j$, $j=1,2,...,m$. We derive these models from SU(N) Yang-Mills gauge theory coupled to non-dynamic matter and on spacetime which is a cylinder. This relation to gauge theories is used to prove integrability, to construct conservation laws, and solve these models.

solv-int/9804008

Robert Conte

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

On the exact solutions of the Bianchi IX cosmological model in the proper time

8 pages, no figure, standard Latex, to appear in Regular and chaotic dynamics (1998)

solv-int nlin.SI

It has recently been argued that there might exist a four-parameter analytic solution to the Bianchi IX cosmological model, which would extend the three-parameter solution of Belinskii et al. to one more arbitrary constant. We perform the perturbative Painlev\'e test in the proper time variable, and confirm the possible existence of such an extension.

solv-int/9804009

Masato Hisakado

The Davey Stewartson system and the B\"{a}cklund Transformations

13 pages, LaTeX

10.1143/JPSJ.67.3038

solv-int hep-th nlin.SI

We consider the (coupled) Davey-Stewartson (DS) system and its B\"{a}cklund transformations (BT). Relations among the DS system, the double Kadomtsev-Petviashvili (KP) system and the Ablowitz-Ladik hierarchy (ALH) are established. The DS hierarchy and the double KP system are equivalent. The ALH is the BT of the DS system in a certain reduction. {From} the BT of coupled DS system we can obtain new coupled derivative nonlinear Schr\"{o}dinger equations.

solv-int/9804010

Osamu Tsuchiya

O. Tsuchiya (University of Tokyo, Komaba)

Determinant formula for the six-vertex model with reflecting end

10 pages

10.1063/1.532606

UT-Komaba 98-5

solv-int nlin.SI

Using the Quantum Inverse Scattering Method for the XXZ model with open boundary conditions, we obtained the determinant formula for the six vertex model with reflecting end.

solv-int/9804011

Basil Grammaticos

B. Grammaticos, A. Ramani and S. Lafortune

The Gambier Mapping, Revisited

11 pages, no figures, to be published in Physica A

Physica A 253, 260-270 (1998)

10.1016/S0378-4371(97)00675-4

GMPIB-225

solv-int nlin.SI

We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete form of this equation, the Gambier mapping, and present conditions for its integrability. Finally, we obtain the reductions of the Gambier mapping, identify their integrable forms and compute their continuous limits.

solv-int/9804012

Basil Grammaticos

A.Ramani, B.Grammaticos and S.Lafortune

Again, Linearizable Mappings

14 pages, no figures, to be published in Physica A

Physica A 252, 138-150 (1998)

10.1016/S0378-4371(97)00614-6

GMPIB-222

solv-int nlin.SI

We examine a family of 3-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through the application of the singularity confinement criterion and are explicitly integrated.

solv-int/9804013

Arthur Vartanian

A. H. Vartanian

Higher Order Asymptotics of the Modified Non-Linear Schr\"{o}dinger Equation

54 pages, 7 figures, LaTeX, long appendix

solv-int nlin.SI

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the solution to the Cauchy problem for the modified non-linear Schr\"{o}dinger equation, $i \partial_{t} u + {1/2} \partial_{x}^{2} u + | u |^{2} u + i s \partial_{x} (| u |^{2} u) = 0$, $s \in \Bbb R_{> 0}$, which is a model for non-linear pulse propagation in optical fibres in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent non-linear evolution equations; in particular, the derivative non-linear Schr\"{o}dinger equation, $i \partial_{t} q + \partial_{x}^{2} q - i \partial_{x}(| q |^{2} q) = 0$.

solv-int/9804014

A. N. Leznov

The Gel'fand-Tsetlin Selection Rules and Representations of Quantum Algebras

16 pages, LaTeX

IIMAS-UNAM No. 79, 1998

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

The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general solution of this system is given explicitly and particular ones, which correspond to the irreducible representations are selected.

solv-int/9804015

Antonio L. Santos

H. Babujian, A. Lima-Santos and R. H. Poghossian

Knizhnik-Zamolodchikov-Bernard equations connected with the eight-vertex model

20 pages latex, macro: tcilatex

Int. Journ. Mod. Phys. A14 (1999) 615-630

10.1142/S0217751X99000300

UFSCAR-98-04

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

Using quasiclassical limit of Baxter's 8 - vertex R - matrix, an elliptic generalization of the Knizhnik-Zamolodchikov equation is constructed. Via Off-Shell Bethe ansatz an integrable representation for this equation is obtained. It is shown that there exists a gauge transformation connecting this equation with Knizhnik-Zamolodchikov-Bernard equation for SU(2)-WZNW model on torus.

solv-int/9804016

Igor Krichever

I.M. Krichever

Elliptic solutions to difference non-linear equations and nested Bethe ansatz equations

21 pages, Latex, no figures

solv-int hep-th nlin.SI

We outline an approach to a theory of various generalizations of the elliptic Calogero-Moser (CM) and Ruijsenaars-Shneider (RS) systems based on a special inverse problem for linear operators with elliptic coefficients. Hamiltonian theory of such systems is developed with the help of the universal symplectic structure proposed by D.H. Phong and the author. Canonically conjugated action-angle variables for spin generalizations of the elliptic CM and RS systems are found.

solv-int/9804017

Ernesto Raposo

D. Bazeia (Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge MA, USA, and Departamento de Fisica, Universidade Federal da Paraiba,Joao Pessoa PB, Brazil) and E.P. Raposo (Lyman Laboratory of Physics, Harvard University, Cambridge MA, USA)

Travelling Wave Solutions in Nonlinear Diffusive and Dispersive Media

10 pages, Latex

MIT-CTP-2734

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

We investigate the presence of soliton solutions in some classes of nonlinear partial differential equations, namely generalized Korteweg-de Vries-Burgers, Korteveg-de Vries-Huxley, and Korteveg-de Vries-Burgers-Huxley equations, which combine effects of diffusion, dispersion, and nonlinearity. We emphasize the chiral behavior of the travelling solutions, whose velocities are determined by the parameters that define the equation. For some appropriate choices, we show that these equations can be mapped onto equations of motion of relativistic 1+1 dimensional phi^{4} and phi^{6} field theories of real scalar fields. We also study systems of two coupled nonlinear equations of the types mentioned.

solv-int/9804018

A. Khare

Bishwajyoti Dey and Avinash Khare

On The Stability of the Compacton Solutions

9 pages, revtex style, no figures

Phys.Rev. E58 (1998) 2741-2744

10.1103/PhysRevE.58.R2741

IP-BBSR/98-15

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

The stability of the recently discovered compacton solutions is studied by means of both linear stability analysis as well as Lyapunov stability criteria. From the results obtained it follows that, unlike solitons, all the allowed compacton solutions are stable, as the stability condition is satisfied for arbitrary values of the nonlinearity parameter. The results are shown to be true even for the higher order nonlinear dispersion equations for compactons. Some new conservation laws for the higher order nonlinear dispersion equations are also presented.

solv-int/9804019

Nikita A. Slavnov

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

A nonlinear indentity for the scattering phase of integrable models

5 pages, Latex, no figures

10.1007/BF02557143

MI-98-27

solv-int nlin.SI

A nonlinear identity for the scattering phase of quantum integrable models is proved.

solv-int/9805001

Satoru Saito

Katsuhiko Yoshida and Satoru Saito

Analytical Study of the Julia Set of a Coupled Generalized Logistic Map

30pages, 22figures

10.1143/JPSJ.68.1513

TMUP-HEL-9806

solv-int nlin.SI

A coupled system of two generalized logistic maps is studied. In particular influence of the coupling to the behaviour of the Julia set in two dimensional complex space is analyzed both analytically and numerically. It is proved analytically that the Julia set disappears from the complex plane uniformly as a parameter interpolates from the chaotic phase to the integrable phase, if the coupling strength satisfies a certain condition.

solv-int/9805002

Ming-Hsien Tu

Jiin-Chang Shaw and Ming-Hsien Tu

On the Miura and Backlund transformations associated with the supersymmetric Gelfand-Dickey bracket

8 pages, Revtex, version to appear on Mod. Phys. Lett. A

Mod. Phys. Lett. A13 (1998) 979

10.1142/S0217732398001054

solv-int nlin.SI

The supersymmetric version of the Miura and B\"acklund transformations associated with the supersymmetric Gelfand-Dickey bracket are investigated from the point of view of the Kupershmidt-Wilson theorem.

solv-int/9805003

Alexander Turbiner

Alexander Turbiner

Hidden Algebra of Three-Body Integrable Systems

11 pages, AMS-LaTeX, no figures, minor typos corrected, to appear in Mod.Phys.Lett.A

Modern Physics Letters A, 13(1998)1473-1483

10.1142/S0217732398001558

Minneapolis TPI-MINN-98/04 and M\'exico ICN-UNAM 98-02

solv-int cond-mat.stat-mech hep-th math-ph math.MP math.RT math.SP nlin.SI

It is shown that all 3-body quantal integrable systems that emerge in the Hamiltonian reduction method possess the same hidden algebraic structure. All of them are given by a second degree polynomial in generators of an infinite-dimensional Lie algebra of differential operators. It leads to new families of the orthogonal polynomials in two variables.

solv-int/9805004

Fis. Teorica. Valladolid.

Angel Ballesteros and Francisco J. Herranz

Long range integrable oscillator chains from quantum algebras

17 pages, LaTeX

UBU-Dfis-98-01

solv-int math.QA nlin.SI

Completely integrable Hamiltonians defining classical mechanical systems of $N$ coupled oscillators are obtained from Poisson realizations of Heisenberg--Weyl, harmonic oscillator and $sl(2,\R)$ coalgebras. Various completely integrable deformations of such systems are constructed by considering quantum deformations of these algebras. Explicit expressions for all the deformed Hamiltonians and constants of motion are given, and the long-range nature of the interactions is shown to be linked to the underlying coalgebra structure. The relationship between oscillator systems induced from the $sl(2,\R)$ coalgebra and angular momentum chains is presented, and a non-standard integrable deformation of the hyperbolic Gaudin system is obtained.

solv-int/9805005

Manuel Manas

Q. P. Liu and M. Manas

Reduced Vectorial Ribaucour Transformation for the Darboux-Egoroff Equations

15 pages LaTeX2e with AMSLaTeX and Babel packages

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

The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain those vectorial Ribaucour transformations which preserve the Egoroff reduction. We also show that a permutability property holds for all these transformations. Finally, as an example, we apply these transformations to the Cartesian background.

solv-int/9805006

Henrik Aratyn

H. Aratyn

On Grassmannian Description of the Constrained KP Hierarchy

LaTeX, 17 pgs

10.1016/S0393-0440(98)00062-X

solv-int nlin.SI

This note develops an explicit construction of the constrained KP hierarchy within the Sato Grassmannian framework. Useful relations are established between the kernel elements of the underlying ordinary differential operator and the eigenfunctions of the associated KP hierarchy as well as between the related bilinear concomitant and the squared eigenfunction potential.

solv-int/9805007

Ziemowit Popowicz

Z.Popowicz

Integrable Extensions of N=2 Supersymmetric KdV Hierarchy Associated with the Nonuniqueness of the Roots of the Lax operator

9 pages Latex,e-mail [email protected]

10.1016/S0375-9601(98)00731-2

solv-int nlin.SI

We preesent a new supersymmetric integrable extensions of the a=4,N=2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generate new hierarchy of integrable equations, for which we investigate the hamiltonian structure. In special case our system describes the interaction of the KdV equation with the two MKdV equations.

solv-int/9805008

Ernesto Raposo

E.P. Raposo (Lyman Laboratory of Physics, Harvard University, Cambridge MA, USA) and D. Bazeia (Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge MA, USA, and Departamento de Fisica, Universidade Federal da Paraiba,Joao Pessoa PB, Brazil)

Exact Kink Solitons in the Presence of Diffusion, Dispersion, and Polynomial Nonlinearity

11 pages, Latex

10.1016/S0375-9601(99)00067-5

MIT-CTP-2742

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

We describe exact kink soliton solutions to nonlinear partial differential equations in the generic form u_{t} + P(u) u_{x} + \nu u_{xx} + \delta u_{xxx} = A(u), with polynomial functions P(u) and A(u) of u=u(x,t), whose generality allows the identification with a number of relevant equations in physics. We emphasize the study of chirality of the solutions, and its relation with diffusion, dispersion, and nonlinear effects, as well as its dependence on the parity of the polynomials $P(u)$ and $A(u)$ with respect to the discrete symmetry $u\to-u$. We analyze two types of kink soliton solutions, which are also solutions to 1+1 dimensional phi^{4} and phi^{6} field theories.

solv-int/9805009

Ziad Maassarani

Z. Maassarani (Laval University)

Multiplicity A_m Models

11 pages, Latex, one figure. Some clarifications added

Eur. Phys. J. B vol. 7 (1999) 627-633 - Erratum: vol. 9 (1999) 371

LAVAL-PHY-20/98

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

Models generalizing the su(2) XX spin-chain were recently introduced. These XXC models also have an underlying su(2) structure. Their construction method is shown to generalize to the chains based on the fundamental representations of the A_m Lie algebras. Integrability of the new models is shown in the context of the quantum inverse scattering method. Their R-matrix is found and shown to yield a representation of the Hecke algebra. The diagonalization of the transfer matrices is carried out using the algebraic Bethe Ansatz. I comment on eventual generalizations and possible links to reaction-diffusion processes.

solv-int/9805010

Manuel Manas

Manuel Manas and Luis Martinez Alonso

From Ramond Fermions to Lame Equations for Orthogonal Curvilinear Coordinates

14 pages, LaTeX2e with AMSLaTeX and Babel packages

10.1016/S0370-2693(98)00851-X

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

We show how Ramond free neutral Fermi fields lead to a $\tau$-function theory of BKP type which describes iso-orthogonal deformations of systems of ortogonal curvilinear coordinates. We also provide a vertex operator representation for the classical Ribaucour transformation.

solv-int/9805011

Andrei Kapaev

Andrei A. Kapaev (St.Petersburg Department of Steklov Mathematical Institute)

Connection formulae for degenerated asymptotic solutions of the fourth Painleve equation

39 pages, LaTeX

solv-int nlin.SI

All possible 1-parametric classical and transcendent degenerated solutions of the fourth Painleve equation with the corresponding connection formulae of the asymptotic parameters are described.

solv-int/9805012

Sergei Yu. Sakovich

Sergei Yu. Sakovich

On integrability of a (2+1)-dimensional perturbed Kdv equation

J. Nonlinear Math. Phys. 5 (1998) 230-233

10.2991/jnmp.1998.5.3.1

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

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlev\'e test for integrability well, and its 4$\times $4 Lax pair with two spectral parameters is found. The results show that the Painlev\'e classification of coupled KdV equations by A. Karasu should be revised.

solv-int/9805013

Robert Milson

R. Milson, D. Richter

Quantization of cohomology in semi-simple Lie algebras

Length: 16 pages. To appear in the Journal of Lie Theory, Volume 8, #2, 1998

solv-int math.RT nlin.SI

The space of realizations of a finite-dimensional Lie algebra by first order differential operators is naturally isomorphic to H^1 with coefficients in the module of functions. The condition that a realization admits a finite-dimensional invariant subspace of functions seems to act as a kind of quantization condition on this H^1. It was known that this quantization of cohomology holds for all realizations on 2-dimensional homogeneous spaces, but the extent to which quantization of cohomology is true in general was an open question. The present article presents the first known counter-examples to quantization of cohomology; it is shown that quantization can fail even if the Lie algebra is semi-simple, and even if the homogeneous space in question is compact. A explanation for the quantization phenomenon is given in the case of semi-simple Lie algebras. It is shown that the set of classes in H^1 that admit finite-dimensional invariant subspaces is a semigroup that lies inside a finitely-generated abelian group. In order for this abelian group be a discrete subset of H^1, i.e. in order for quantization to take place, some extra conditions on the isotropy subalgebra are required. Two different instances of such necessary conditions are presented.

solv-int/9806001

Manuel Manas

Boris G. Konopelchenko, Luis Martinez Alonso and Elena Medina

Singular sector of the KP hierarchy, $\bar{\partial}$-operators of non-zero index and associated integrable systems

45 pages, LaTeX 2.09 with epsf,amstex and amssymb styles

solv-int nlin.SI

Integrable hierarchies associated with the singular sector of the KP hierarchy, or equivalently, with $\dbar$-operators of non-zero index are studied. They arise as the restriction of the standard KP hierarchy to submanifols of finite codimension in the space of independent variables. For higher $\dbar$-index these hierarchies represent themselves families of multidimensional equations with multidimensional constraints. The $\dbar$-dressing method is used to construct these hierarchies. Hidden KdV, Boussinesq and hidden Gelfand-Dikii hierarchies are considered too.

solv-int/9806002

Gregorio Falqui

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

Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP Hierarchy

Latex, 27 pages. To appear in Commun. Math. Phys

10.1007/s002200050452

SISSA 82/97/FM

solv-int nlin.SI

We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, to study the KP equations. In this approach they have the form of local conservation laws, and can be traded for a system of ordinary differential equations of Riccati type, which we call the Central System. We show that the latter can be linearized by means of a Darboux covering, and we use this procedure as an alternative technique to construct rational solutions of the KP equations.

solv-int/9806003

Robert Milson

Robert Milson

Imprimitively generated Lie-algebraic Hamiltonians and separation of variables

32 pages. To appear in the Canadian Journal of Mathematics

solv-int math.DG nlin.SI

Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the A_2 root system.

solv-int/9806004

Manuel Manas

Q. P. Liu and Manuel Manas

Pfaffian form of the Grammian determinant solutions of the BKP hierarchy

8 pages, LaTeX2e with AMSLaTeX and Babel packages

solv-int nlin.SI

The Grammian determinant type solutions of the KP hierarchy, obtained through the vectorial binary Darboux transformation, are reduced, imposing suitable differential constraint on the transformation data, to Pfaffian solutions of the BKP hierarchy.

solv-int/9806005

Manuel Manas

Q. P. Liu and Manuel Manas

Pfaffian Solutions for the Manin-Radul-Mathieu SUSY KdV and SUSY sine-Gordon Equations

10 pages, LaTeX2e with AMSLaTeX and Babel packages

Phys. Lett. B436 (1998) 306-310

10.1016/S0370-2693(98)00852-1

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

We reduce the vectorial binary Darboux transformation for the Manin-Radul supersymmetric KdV system in such a way that it preserves the Manin-Radul-Mathieu supersymmetric KdV equation reduction. Expressions in terms of bosonic Pfaffians are provided for transformed solutions and wave functions. We also consider the implications of these results for the supersymmetric sine-Gordon equation.

solv-int/9806006

Ismagil T. Habibullin

I.T.Habibullin

Initial boundary value problem on a half-line for the MKdV equation

16 pages, special macros

solv-int nlin.SI

Initial boundary value problem on a half-line for the Modified KdV equation is considered with the boundary conditions equal to zero at the origin and initial condition chosen arbitrary decreasing rapidly enough and this problem is plunged into the scheme of the inverse scattering method. Here the inverse scattering problem is reduced to the Riemann problem on a system of rays on the complex plane.

solv-int/9806007

Juhi-Lian Julian Ting

S. Lee, Julian J.-L. Ting and S. Kim

Phonon Scattering by Breathers in the Discrete Nonlinear Schroedinger Equation

13 pages 11 figures in EPS, RevTeX, Phys. Rev. E accepted

solv-int cond-mat.dis-nn nlin.SI quant-ph

Linear theory for phonon scattering by discrete breathers in the discrete nonlinear Schroedinger equation using the transfer matrix approach is presented. Transmission and reflection coefficients are obtained as a function of the wave vector of the input phonon. The occurrence of a nonzero transmission, which in fact becomes perfect for a symmetric breather, is shown to be connected with localized eigenmodes thresholds. In the weak-coupling limit, perfect reflection are shown to exist, which requires two scattering channels. A necessary condition for a system to have a perfect reflection is also considered in a general context.

solv-int/9806008

Leonid Bogdanov

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

M\"obius invariant integrable lattice equations associated with KP and 2DTL hierarchies

13 pages, LaTeX; talk at SIDE III conference, Sabaudia, Italy, May 1998

10.1016/S0375-9601(99)00199-1

solv-int nlin.SI

Integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. These equation generate the corresponding continuous hierarchy of singular manifold equations, its B\"acklund transformations and different forms of superposition principles. They possess rather special form of compatibility representation. The distinctive feature of these equations is invariance under the action of M\"obius transformation. Geometric interpretation of these discrete equations is given.

solv-int/9806009

G.F. Helminck, J.W. van de Leur

Geometric B\"acklund--Darboux transformations for the KP hierarchy

44 pages Latex2e

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

We shown that, if you have two planes in the Segal-Wilson Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by B\"acklund-Darboux transformations (BDT). The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from elementary BDT's and is given here in a closed form. The geometric description of elementary DBT's requires that one has a geometric interpretation of the dual wavefunctions involved. This is done here with the help of a suitable algebraic characterization of the wavefunction. The BDT's also induce transformations of the tau-function associated to a plane in the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric characterization of the BDT'ss that preserves these subsystems of the KP hierarchy. This generalizes the classical Darboux-transformations. we also determine an explicit expression for the squared eigenfunction potentials. Next a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy. It is shown that infinite flags in the Grassmannian yield solutions of the latter hierarchy. these flags can be constructed by means of BDT's, starting from some plane. Other applications of these BDT's are a geometric way to characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy and the construction of a vast collection of orthogonal polynomials, playing a role in matrix models.

solv-int/9807001

Juhi-Lian Julian Ting

Julian Juhi-Lian Ting

DNA Transcription Mechanism with a Moving Enzyme

paper published long time ago. 11 pages RevTeX 4 EPS files

Int. J. Bifurcat. Chaos.7:5, 1125-1132 (1997)

solv-int nlin.SI q-bio

Previous numerical investigations of an one-dimensional DNA model with an extended modified coupling constant by transcripting enzyme are integrated to longer time and demonstrated explicitly the trapping of breathers by DNA chains with realistic parameters obtained from experiments. Furthermore, collective coordinate method is used to explain a previously observed numerical evidence that breathers placed far from defects are difficult to trap, and the motional effect of RNA-polymerase is investigated.

solv-int/9807002

Wen-Xiu Ma

Wen-Xiu Ma and Maxim Pavlov

Extending Hamiltonian Operators to Get Bi-Hamiltonian Coupled KdV Systems

13 pages, latex

10.1016/S0375-9601(98)00555-6

solv-int nlin.SI

An analysis of extension of Hamiltonian operators from lower order to higher order of matrix paves a way for constructing Hamiltonian pairs which may result in hereditary operators. Based on a specific choice of Hamiltonian operators of lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a consequence of bi-Hamiltonian structure, they all possess infinitely many symmetries and infinitely many conserved densities.

solv-int/9807003

Lafortune

A. Ramani, B. Grammaticos and S. Lafortune

Schlesinger Transformations for Linearisable Equations

14 pages, no figures, Tex file

Lett.Math.Phys. 46, 131-145 (1998).

Preprint 98

solv-int nlin.SI

We introduce the Schlesinger transformations of the Gambier equation. The latter can be written, in both the continuous and discrete cases, as a system of two coupled Riccati equations in cascade involving an integer parameter n. In the continuous case the parameter appears explicitly in the equation while in the discrete case it corresponds to the number of steps for singularity confinement. Two Schlesinger transformations are obtained relating the solutions for some value $n$ to that corresponding to either n+1 or n+2.

solv-int/9807004

I. A. B. Strachan

I.A.B.Strachan

Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations

28 pages, LaTeX

J. Math. Phys. 40, 5058 (1999);

10.1063/1.533015

solv-int nlin.SI

The bi-Hamiltonian structure of certain multi-component integrable systems, generalizations of the dispersionless Toda hierarchy, is studies for systems derived from a rational Lax function. One consequence of having a rational rather than a polynomial Lax function is that the corresponding bi-Hamiltonian structures are degenerate, i.e. the metric which defines the Hamiltonian structure has vanishing determinant. Frobenius manifolds provide a natural setting in which to study the bi-Hamiltonian structure of certain classes of hydrodynamic systems. Some ideas on how this structure may be extanded to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, are given.

solv-int/9807005

V. E. Vekslerchik

V.E. Vekslerchik

'Universality' of the Ablowitz-Ladik hierarchy

21 pages, LaTeX

IC/98/52

solv-int nlin.SI

The aim of this paper is to summarize some recently obtained relations between the Ablowitz-Ladik hierarchy (ALH) and other integrable equations. It has been shown that solutions of finite subsystems of the ALH can be used to derive a wide range of solutions for, e.g., the 2D Toda lattice, nonlinear Schr\"odinger, Davey-Stewartson, Kadomtsev-Petviashvili (KP) and some other equations. Similar approach has been used to construct new integrable models: O(3,1) and multi-field sigma models. Such 'universality' of the ALH becomes more transparent in the framework of the Hirota's bilinear method. The ALH, which is usually considered as an infinite set of differential-difference equations, has been presented as a finite system of functional-difference equations, which can be viewed as a generalization of the famous bilinear identities for the KP tau-functions.

solv-int/9807006

Helge Holden

F. Gesztesy, H. Holden

Dubrovin equations and integrable systems on hyperelliptic curves

LaTeX2e

solv-int nlin.SI

We introduce the most general version of Dubrovin-type equations for divisors on a hyperelliptic curve of arbitrary genus, and provide a new argument for linearizing the corresponding completely integrable flows. Detailed applications to completely integrable systems, including the KdV, AKNS, Toda, and the combined sine-Gordon and mKdV hierarchies, are made. These investigations uncover a new principle for 1+1-dimensional integrable soliton equations in the sense that the Dubrovin equations, combined with appropriate trace formulas, encode all hierarchies of soliton equations associated with hyperelliptic curves. In other words, completely integable hierarchies of soliton equations determine Dubrovin equations and associated trace formulas and, vice versa, Dubrovin-type equations combined with trace formulas permit the construction of hierarchies of soliton equations.

solv-int/9807007

H. J. S. Dorren

H.J.S. Dorren

On the integrability of nonlinear partial differential equations

12 pages Latex

10.1063/1.532843

solv-int nlin.SI

We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV equation to other NPDEs. The method is based upon a linearization principle which can be applied on nonlinearities which have a polynomial form. We illustrate the potential of the method by finding solutions of the (coupled) nonlinear Schr\"{o}dinger equation and the Manakov equation which play an important role in optical fiber communication. Finally, it is shown that the method can also be generalized to higher-dimensions.

solv-int/9807008

Takashi Takebe

Evgueni K. Sklyanin, Takashi Takebe

Separation of Variables in the Elliptic Gaudin Model

24 pages, Latex; minor corrections

Communications in Mathematical Physics 204:1 (1999) 17-38

10.1007/s002200050635

UTMS 98-28, PDMI 15/98

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

For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of separating integral operator is constructed. The simplest one-degree-of-freedom case is studied in detail.

solv-int/9807009

Pilar G. Estevez

Pilar G. Estevez (Universidad de Salamanca) and Pilar R. Gordoa (Universidad de Salamanca)

Non-classical symmetries and the singular manifold method: A further two examples

9 pages (latex), to appear in Journal of Physics A

10.1088/0305-4470/31/37/011

AFTUS-98-15

solv-int nlin.SI

This paper discusses two equations with the conditional Painleve property. The usefulness of the singular manifold method as a tool for determining the non-classical symmetries that reduce the equations to ordinary differential equations with the Painleve property is confirmed once more

solv-int/9807010

Wen-Xiu Ma

Benno Fuchssteiner and Wen-Xiu Ma

An Approach to Master Symmetries of Lattice Equations

14 pages, latex, to appear in Proceedings of SIDEII, UK

solv-int nlin.SI

An approach to master symmetries of lattice equations is proposed by the use of discrete zero curvature equation. Its key is to generate non-isospectral flows from the discrete spectral problem associated with a given lattice equation. A Volterra-type lattice hierarchy and the Toda lattice hierarchy are analyzed as two illustrative examples.

solv-int/9808001

Roman Paunov

H. Belich and R. Paunov

$A_n^{(1)}$ Toda Solitons: a Relation between Dressing transformations and Vertex Operators

17 pages, Latex, Talk given at the IV International Conference on Non Associative Algebra and its Applications, University of Sao Paulo, July 19-24, 1988

CBPF/NF/050/98

solv-int hep-th nlin.SI

Affine Toda equations based on simple Lie algebras arise by imposing zero curvature condition on a Lax connection which belongs to the corresponding loop Lie algebra in the principal gradation. In the particular case of $A_n^{(1)}$ Toda models, we exploit the symmetry of the underlying linear problem to calculate the dressing group element which generates arbitrary $N$-soliton solution from the vacuum. Starting from this result we recover the vertex operator representation of the soliton tau functions.

solv-int/9808002

Antonio Lima Santos

A. Lima-Santos

Polynomial rings of the chiral $SU(N)_{2}$ models

10 pages, LaTex (ioplppt.sty)

J. Phys. A: Math. Gen 30 (1997) 8653-8660

10.1088/0305-4470/30/24/024

solv-int nlin.SI

Via explicit diagonalization of the chiral $SU(N)_{2}$ fusion matrices, we discuss the possibility of representing the fusion ring of the chiral SU(N) models, at level K=2, by a polynomial ring in a single variable when $N$ is odd and by a polynomial ring in two variables when $N$ is even.

solv-int/9808003

Henrik Aratyn

H. Aratyn, E. Nissimov and S. Pacheva

From One-Component KP Hierarchy to Two-Component KP Hierarchy and Back

LaTeX, 9 pgs., contribution to Festschrift for A.H. Zimerman

solv-int hep-th nlin.SI

We show that the system of the standard one-component KP hierarchy endowed with a special infinite set of abelian additional symmetries, generated by squared eigenfunction potentials, is equivalent to the two-component KP hierarchy.

solv-int/9808004

Henrik Aratyn

H. Aratyn, E. Nissimov and S. Pacheva

Berezinian Construction of Super-Solitons in Supersymmetric Constrained KP Hierarchies

LaTeX, 9 pgs., contribution to Festschrift for A.H. Zimerman

solv-int hep-th nlin.SI

We consider a broad class of consistently reduced Manin-Radul supersymmetric KP hierarchies (MR-SKP) which are supersymmetric analogs of the ordinary bosonic constrained KP models. Compatibility of these reductions with the MR fermionic isospectral flows is achieved via appropriate modification of the latter preserving their (anti-)commutation algebra. Unlike the general unconstrained MR-SKP case, Darboux-Backlund transformations do preserve the fermionic isospectral flows of the reduced MR-SKP hierarchies. This allows for a systematic derivation of explicit Berezinian solutions for the super-tau-functions (super-solitons) for these models.

solv-int/9808005

Pijush Kanti Ghosh

Pijush K. Ghosh and Avinash Khare

Relationship Between the Energy Eigenstates of Calogero-Sutherland Models With Oscillator and Coulomb-like Potentials

23 pages, RevTeX, no figure, some clarifications added, version to appear in Journal of Physics A

Journal of Physics A : Math. & Gen. 32 (1999) 2129-2140

10.1088/0305-4470/32/11/008

IMSC/98/07/47, IP/BBSR/98-25

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

We establish a simple algebraic relationship between the energy eigenstates of the rational Calogero-Sutherland model with harmonic oscillator and Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in both of these models which plays a crucial role in such an identification. Further, we show that our analysis is in fact valid for any many-particle system in arbitrary dimensions whose potential term (apart from the oscillator or the Coulomb-like potential) is a homogeneous function of coordinates of degree -2. The explicit coordinate transformation which maps the Coulomb-like problem to the oscillator one has also been determined in some specific cases.

solv-int/9808006

Zixiang Zhou

Zixiang Zhou (Institute of Mathematics, Fudan University, Shanghai, China)

Darboux transformations for twisted so(p,q) system and local isometric immersion of space forms

LaTeX, 21 pages, 5 Postscript figures, to appear in Inverse Problems (1998)

Inverse Problems 14 (1998) 1353-1370

10.1088/0266-5611/14/5/018

solv-int nlin.SI

For the n-dimensional integrable system with a twisted so(p,q) reduction, Darboux transformations given by Darboux matrices of degree 2 are constructed explicitly. These Darboux transformations are applied to the local isometric immersion of space forms with flat normal bundle and linearly independent curvature normals to give the explicit expression of the position vector. Some examples are given from the trivial solutions and standard imbedding T^n\to R^{2n}.

solv-int/9808007

Vadim V. Varlamov

Vadim V. Varlamov

Equations of Geodesic Deviation and the Inverse Scattering Transform

32 pages, LaTeX2e, to appear in "Relativity, Gravitation, Cosmology" (Nova Science Publishers, New York)

solv-int gr-qc nlin.SI

Solutions of equations of geodesic deviation in three- and four- dimensional spaces obtained by the inverse scattering transform are considered. It is shown that in the case of three-dimensional space solutions of geodesic deviation equations are reduced to solutions of the well-known Zakharov-Shabat problem. In four- dimensional space system of geodesic deviation equations is associated with $3\times 3$ matrix Schr\"{o}dinger equation, and dependence on parameters defined by the nonlinear equations of three-wave interaction.

solv-int/9808008

J. vandeLeur

J.W. van de Leur and R.Martini

The construction of Frobenius manifolds from KP tau-functions

29 pages, latex2e, no figures

10.1007/s002200050691

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

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux-Egoroff equations. This system of partial differential equations appears as a specific subset of the $n$-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux-Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function.

solv-int/9808009

Boris Lorbeer

Boris Lorbeer

Finite gap integration of a discrete Euler top

21 pages, 6 figures

solv-int nlin.SI

In [1] new discretizations of the Euler top have been found. They can be discribed with a Lax pair with a spectral parameter on an elliptic curve. This is used in this paper to perform a finite gap integration.

solv-int/9808010

Wen-Xiu Ma

Yishen Li and Wen-Xiu Ma

Binary Nonlinearization of AKNS Spectral Problem under Higher-Order Symmetry Constraints

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

solv-int nlin.SI

Binary nonlinearization of AKNS spectral problem is extended to the cases of higher-order symmetry constraints. The Hamiltonian structures, Lax representations, $r$-matrices and integrals of motion in involution are explicitly proposed for the resulting constrained systems in the cases of the first four orders. The obtained integrals of motion are proved to be functionally independent and thus the constrained systems are completely integrable in the Liouville sense.

solv-int/9808011

Zixiang Zhou

Zixiang Zhou

Localized solitons of hyperbolic su(N) AKNS system

15 pages, 5 figures, to appear in Inverse Problems

Inverse Problems 14 (1998) 1371-1383

10.1088/0266-5611/14/5/019

solv-int nlin.SI

Using the nonlinear constraint and Darboux transformation methods, the (m_1,...,m_N) localized solitons of the hyperbolic su(N) AKNS system are constructed. Here "hyperbolic su(N)" means that the first part of the Lax pair is F_y=JF_x+U(x,y,t)F where J is constant real diagonal and U^*=-U. When different solitons move in different velocities, each component U_{ij} of the solution U has at most m_i m_j peaks as t tends to infinity. This corresponds to the (M,N) solitons for the DSI equation. When all the solitons move in the same velocity, U_{ij} still has at most m_i m_j peaks if the phase differences are large enough.

solv-int/9808012

Antonio Lima Santos

A. Lima-Santos and Wagner Utiel

On The KMS Condition for the critical Ising model

8 pages, TcilaTex

Physics Letter A 226 (1997) 65-68

10.1016/S0375-9601(96)00919-X

solv-int hep-th nlin.SI

Using the KMS condition and exchange algebras we discuss the monodromy and modular properties of two-point KMS states of the critical Ising model.

solv-int/9808013

Antonio Lima Santos

A. Lima-Santos

On fusion algebra of chiral $SU(N)_{k}$ models

11 pages, ioplppt

J.Phys.A: Math. Gen. 30 (1997) 5123-5131

10.1088/0305-4470/30/14/021

solv-int nlin.SI

We discuss some algebraic setting of chiral $SU(N)_{k}$ models in terms of the statistical dimensions of their fields. In particular, the conformal dimensions and the central charge of the chiral $SU(N)_{k}$ models are calculated from their braid matrices. Futhermore, at level K=2, we present the characteristic polynomials of their fusion matrices in a factored form.

solv-int/9808014

David H. Sattinger

M. Haragus-Courcelle and D.H. Sattinger

Inversion of the linearized Korteweg-deVries equation at the multi-soliton solutions

39 pages, 1 figure

Zeit fur Angew. Math. und Physik (ZAMP), vol 49, (1998), pp. 436-469

10.1007/s000000050101

solv-int nlin.SI

Uniform estimates for the decay structure of the $n$-soliton solution of the Korteweg-deVries equation are obtained. The KdV equation, linearized at the $n$-soliton solution is investigated in a class $\WW$ consisting of sums of travelling waves plus an exponentially decaying residual term. An analog of the kernel of the time-independent equation is proposed, leading to solvability conditions on the inhomogeneous term. Estimates on the inversion of the linearized KdV equation at the $n$-soliton are obtained.