Datasets:

bartoszmaj

/

arxiv_json

like 0

solv-int/9902004

Oleg Kiselev

O.M.Kiselev, B.I.Suleimanov (Institute of Mathematics, Ufa Science Centre, Russian Acad. of Sciences)

The solution of the Painleve equations as special functions of catastrophes, defined by a rejection in these equations of terms with derivative

Latex, 15 pages

solv-int nlin.SI

The relation between the Painleve equations and the algebraic equations with the catastrophe theory point of view are considered. The asymptotic solutions with respect to the small parameter of the Painleve equations different types are discussed. The qualitative analysis of the relation between algebraic and fast oscillating solutions is done for Painleve-2 as an example.

solv-int/9902005

Leon Jerome

J. Leon, M. Manna

Multiscale Analysis of Discrete Nonlinear Evolution Equations

published in J. Phys. A : Math. Gen. 32 (1999) 927-943

10.1088/0305-4470/32/15/012

solv-int nlin.SI

The method of multiscale analysis is constructed for dicrete systems of evolution equations for which the problem is that of the far behavior of an input boundary datum. Discrete slow space variables are introduced in a general setting and the related finite differences are constructed. The method is applied to a series of representative examples: the Toda lattice, the nonlinear Klein-Gordon chain, the Takeno system and a discrete version of the Benjamin-Bona-Mahoney equation. Among the resulting limit models we find a discrete nonlinear Schroedinger equation (with reversed space-time), a 3-wave resonant interaction system and a discrete modified Volterra model.

solv-int/9902006

Monica Ugaglia

Monica Ugaglia

On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEs

46 pages, Tex, to be published in "Differential Geometry and Applications"

solv-int nlin.SI

In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time-dependent symmetries. In particular we describe how the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time-dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painleve' equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds.

solv-int/9902007

Oleg Kiselev

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

Asymptotic approach for the rigid condition of appearance of the oscillations in the solution of the Painleve-2 equation

Latex, 18 pages

solv-int nlin.SI

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_*$ and fast oscillating behavior after the point $t_*$. In the transition layer the behavior of the asymptotic solution is more complicated. The leading term of the asymptotics satisfies the Painleve-1 equation and some elliptic equation with constant coefficients, where the solution of the Painleve-1 equation has poles. The uniform smooth asymptotics are constructed in the interval, containing the critical point $t_*$.

solv-int/9902008

Leonid Dickey

L.A.Dickey (Univ. of Oklahoma)

Modified KP and Discrete KP

LaTeX, 11 pages

solv-int nlin.SI

The discrete KP, or 1-Toda lattice hierarchy is the same as a properly defined modified KP hierarchy.

solv-int/9902009

Michael Baake

M. Baake, U. Grimm and R. J. Baxter

A critical Ising model on the Labyrinth

25 pages, 6 figures

Int. J. Mod. Phys. B 8 (1994) 3579-3600

10.1142/S0217979294001512

solv-int nlin.SI

A zero-field Ising model with ferromagnetic coupling constants on the so-called Labyrinth tiling is investigated. Alternatively, this can be regarded as an Ising model on a square lattice with a quasi-periodic distribution of up to eight different coupling constants. The duality transformation on this tiling is considered and the self-dual couplings are determined. Furthermore, we analyze the subclass of exactly solvable models in detail parametrizing the coupling constants in terms of four rapidity parameters. For those, the self-dual couplings correspond to the critical points which, as expected, belong to the Onsager universality class.

solv-int/9902010

Hendry Izaac Elim

Freddy P. Zen and Hendry I. Elim

Lax Pair Formulation and Multi-soliton Solution of the Integrable Vector Nonlinear Schrodinger Equation

11 pages, LaTeX2.09

solv-int adap-org hep-th math-ph math.DS math.MP nlin.AO nlin.PS nlin.SI patt-sol

The integrable vector nonlinear Schrodinger (vector NLS) equation is investigated by using Zakharov-Shabat (ZS) scheme. We get a Lax pair formulation of the vector NLS model. Multi-soliton solution of the equation is also derived by using inverse scattering method of ZS scheme. We also find that there is an elastic and inelastic collision of the bright and dark multi-solitons of the system.

solv-int/9902011

Chie Bing Wang

Chie Bing Wang

Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlev\'e II Equation

16 pages

10.1088/0305-4470/32/41/312

solv-int nlin.SI

We consider the polynomials $\phi_n(z)= \kappa_n (z^n+ b_{n-1} z^{n-1}+ >...)$ orthonormal with respect to the weight $\exp(\sqrt{\lambda} (z+ 1/z)) dz/2 \pi i z$ on the unit circle in the complex plane. The leading coefficient $\kappa_n$ is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlev\'e II equation. The leading coefficient and second leading coefficient of $\phi_n(z)$ can be expressed asymptotically in terms of the Painlev\'e II function.

solv-int/9902012

John Harnad

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

Modular Invariants and Generalized Halphen Systems

PlainTeX 15gs. Write - up of lecture by J. Harnad at International meeting: SIDE III, Sabaudia, May 16-22, 1998. To appear in CRM Proceedings and Lecture Notes series (1999/2000)

CRM Proceedings and Lecture Notes Series 25, Symmetries and Integrability of Difference Equations, pp. 181- 195 (eds. Decio Levy and Orlando Ragnisco, AMS, Providence R.I., 2000)

CRM 2597 (1999)

solv-int hep-th math-ph math.DS math.GR math.MP math.NT nlin.SI

Generalized Halphen systems are solved in terms of functions that uniformize genus zero Riemann surfaces, with automorphism groups that are commensurable with the modular group. Rational maps relating these functions imply subgroup relations between their automorphism groups and symmetrization relations between the associated differential systems.

solv-int/9902013

John Harnad

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

Picard-Fuchs Equations, Hauptmoduls and Integrable Systems

PlainTeX 15gs. Write - up of lecture given at international meeting: Integrability: Seiberg-Witten and Witham Equations, ICMS, Edinburgh, September 14-19, 1998

Integrability: The Seiberg-Witten and Witham Equations, Chapter 8, pp. 137-152 ( Ed. H.W. Braden and I.M. Krichever, Gordon and Breach, Amsterdam (2000))

CRM 2596 (1999)

solv-int hep-th math-ph math.AG math.GR math.MP math.NT nlin.SI

The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs equations for families of elliptic curves and associated surfaces. The inhomogeneous Picard-Fuchs equations associated to elliptic integrals with varying endpoints are derived and used to determine solutions of equations that are algebraically related to a class of Painlev\'e VI equations.

solv-int/9902014

Myrzakulov Ratbay

N.K.Bliev, G.N.Nugmanova, R.N.Syzdykova and R.Myrzakulov

Soliton equations in 2+1 dimensions: reductions, bilinearizations and simplest solutions

16 pages, Latex, no figures

CNLP-1997-05

solv-int nlin.SI

Soliton equations in 2+1 and their 1+1 = 2+0 reductions are considered.

solv-int/9902015

S. E. Derkachov

S.E. Derkachov

Baxter's Q-operator for the homogeneous XXX spin chain

23 pages, Latex

J.Phys.A32:5299-5316,1999

10.1088/0305-4470/32/28/309

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

Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator for the homogeneous XXX model as integral operator in standard representation of SL(2). The connection between Q-operator and local Hamiltonians is discussed. It is shown that operator of Lipatov's duality symmetry arises naturally as leading term of the asymptotic expansion of Q-operator for large values of spectral parameter.

solv-int/9902016

Vsevolod Adler

V.E. Adler, S.Ya. Startsev

Discrete analogues of the Liouville equation

LaTeX, 15 pages, submitted to Teor. i Mat. Fiz

Theoretical and Mathematical Physics 1999, Volume 121, Issue 2, pp 1484-1495

10.1007/BF02557219

solv-int nlin.SI

The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of twodimensional Toda lattice. The terminating of this sequence by zeroes is proved to be the necessary condition for existence of the integrals of the equation under consideration. The formulae are presented for the higher symmetries of the equations possessing integrals. The general theory is illustrated by examples of difference analogs of Liouville equation.

solv-int/9902017

Nicolas Regnault

Denis Bernard, Nicolas Regnault (Saclay-CNRS)

Vertex Operator Solutions of 2d Dimensionally Reduced Gravity

20 pages; added comparison with Belinskii-Zakharov method and remarks

Commun.Math.Phys.210:177-201,2000

10.1007/s002200050776

solv-int gr-qc hep-th nlin.SI

We apply algebraic and vertex operator techniques to solve two dimensional reduced vacuum Einstein's equations. This leads to explicit expressions for the coefficients of metrics solutions of the vacuum equations as ratios of determinants. No quadratures are left. These formulas rely on the identification of dual pairs of vertex operators corresponding to dual metrics related by the Kramer-Neugebauer symmetry.

solv-int/9903001

V. V. Mangazeev

H.E. Boos and V.V. Mangazeev

Functional relations and nested Bethe ansatz for sl(3) chiral Potts model at q^2=-1

20 pages, 6 figures, to appear in J. Phys. A: Math. and Gen

10.1088/0305-4470/32/16/012

Math. Res. Report No. MRR 051-98, ANU

solv-int nlin.SI

We obtain the functional relations for the eigenvalues of the transfer matrix of the sl(3) chiral Potts model for q^2=-1. For the homogeneous model in both directions a solution of these functional relations can be written in terms of roots of Bethe ansatz-like equations. In addition, a direct nested Bethe ansatz has also been developed for this case.

solv-int/9903002

Nugzar Makhaldiani

Dumitru Baleanu and Nugzar Makhaldiani

Nambu--Poisson reformulation of the finite dimensional dynamical systems

6 pages, latex, no figures

Communications of the JINR, Dubna, E2-98-348, 1998

solv-int nlin.SI

In this paper we introduce a system of nonlinear ordinary differential equations which in a particular case reduces to Volterra's system. We found in two simplest cases the complete sets of the integrals of motion using Nambu--Poisson reformulation of the Hamiltonian dynamics. In these cases we have solved the systems by quadratures.

solv-int/9903003

Ziad Maassarani

Z. Maassarani (University of Virginia)

New Integrable Models from Fusion

11 pages, Latex. v2: statement concerning symmetries qualified, 3 minor misprints corrected. J. Phys. A (1999) in press

J.Phys.A32:5123-5132,1999

10.1088/0305-4470/32/27/310

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

Integrable multistate or multiflavor/color models were recently introduced. They are generalizations of models corresponding to the defining representations of the U_q(sl(m)) quantum algebras. Here I show that a similar generalization is possible for all higher dimensional representations. The R-matrices and the Hamiltonians of these models are constructed by fusion. The sl(2) case is treated in some detail and the spin-0 and spin-1 matrices are obtained in explicit forms. This provides in particular a generalization of the Fateev-Zamolodchikov Hamiltonian.

solv-int/9903004

V. E. Vekslerchik

V.E. Vekslerchik

Finite genus solutions for the Ablowitz-Ladik hierarchy

14 pages, LaTeX 2e

Journal of Physics A, 32 (1999) 4983

10.1088/0305-4470/32/26/316

solv-int nlin.SI

The question of constructing the finite genus quasiperiodic solutions for the Ablowitz-Ladik hierarchy (ALH) is considered by establishing relations between the ALH and the Fay's identity for the theta-functions. It is shown that using a limiting procedure one can derive from the latter an infinite number of differential identities which can be arranged as an infinite set of differential-difference equations coinciding with the equations of the ALH, and that the original Fay's identity can be rewritten in a form similar to the functional equations representing the ALH which have been derived in the previous works of the author. This provides an algorithm for obtaining some class of quasiperiodic solutions for the ALH, which can be viewed as an alternative to the inverse scattering transform or the algebro-geometrical approach.

solv-int/9903005

Pierre van Moerbeke

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

The Pfaff lattice and skew-orthogonal polynomials

21 pages

Intern. Math. Research Notices, 1999

solv-int nlin.SI

Consider a semi-infinite skew-symmetric moment matrix, $m_{\iy}$ evolving according to the vector fields $\pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} ,$ where $\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\iy}:= Q^{-1} J Q^{\top -1} $ leads to the so-called Pfaff Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the affine symplectic algebra. The tau-functions for the system are shown to be pfaffians and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).

solv-int/9903006

Marcio J. Martins

M.J. Martins

Integrable mixed vertex models from braid-monoid algebra

To appear in the Festschriff in honor of Prof. Mc Guire published by World Scientific, plain latex, 10 pages

UFSCARF-TH-98-12

solv-int nlin.SI

We use the braid-monoid algebra to construct integrable mixed vertex models. The transfer matrix of a mixed SU(N) model is diagonalized by nested Bethe ansatz approach.

solv-int/9903007

Oliver B. Fringer

O. B. Fringer (1,2) and D. D. Holm (2) ((1) Stanford University, (2) Los Alamos National Laboratory)

Integrable vs Nonintegrable Geodesic Soliton Behavior

49 pages, 29 figures, animations at: http://rossby.stanford.edu/~fringer/pulsons

solv-int nlin.SI

We study confined solutions of certain evolutionary partial differential equations (pde) in 1+1 space-time. The pde we study are Lie-Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler-Poincare equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from the L2 norm of the velocity. These pde possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call ``pulsons.'' We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. In contrast, head-on antisymmetric collisons of pulsons tend to form singularities.

solv-int/9903008

Nugmanova G. N.

R.Myrzakulov

Singularity Structure Analysis, Integrability, Solitons and Dromions in (2+1)-Dimensional Zakharov Equations

15 pages, Latex, no figures

CNLP-1997-03

solv-int nlin.SI

The (2+1)-dimensional integrable Zakharov equations and their reductions are considered

solv-int/9903009

Pierre van Moerbeke

M. Adler & P. van Moerbeke

Symmetric random matrices and the Pfaff lattice

44 pages

solv-int nlin.SI

Consider a symmetric (finite) matrix ensemble, with a certain probability distribution. What is the probability that the spectrum belongs to a certain interval or union of intervals on the real line? In this paper, we show that, upon introducing an appropriate time parameter, this probability is intimately related to Pfaffians, which as a vector satisfy the so-called Pfaff lattice. The latter is a particular reduction of the 2d-Toda lattice. In particular, they satisfy a KP-like equation, but with a right hand side, depending on nearest neighbors. They also satisfy Virasoro constraints, which combined with the KP-like equation lead to inductive equations for the probabilities.

solv-int/9903010

V. V. Mangazeev

H.E. Boos and V.V. Mangazeev

Bethe ansatz for the three-layer Zamolodchikov model

22 pages, LaTeX, 5 figures

10.1088/0305-4470/32/28/308

solv-int nlin.SI

This paper is a continuation of our previous work (solv-int/9903001). We obtain two more functional relations for the eigenvalues of the transfer matrices for the $sl(3)$ chiral Potts model at $q^2=-1$. This model, up to a modification of boundary conditions, is equivalent to the three-layer three-dimensional Zamolodchikov model. From these relations we derive the Bethe ansatz equations.

solv-int/9903011

David H. Sattinger

R. Beals, D.H. Sattinger, and J. Szmigielski

Multipeakons and a theorem of Stieltjes

6 pages

Inverse Problems, volume 15 (1999), Letters, L1-L4

10.1088/0266-5611/15/1/001

solv-int nlin.SI

A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.

solv-int/9903012

Daisuke Takahashi

Daisuke Takahashi and Kenji Kajiwara

On integrability test for ultradiscrete equations

10 pages including 5 figures

solv-int nlin.SI

We consider an integrability test for ultradiscrete equations based on the singularity confinement analysis for discrete equations. We show how singularity pattern of the test is transformed into that of ultradiscrete equation. The ultradiscrete solution pattern can be interpreted as a perturbed solution. We can also check an integrability of a given equation by a perturbation growth of a solution, namely Lyapunov exponent. Therefore, singularity confinement test and Lyapunov exponent are related each other in ultradiscrete equations and we propose an integrability test from this point of view.

solv-int/9903013

Takayuki Tsuchida

T. Tsuchida, H. Ujino, M. Wadati (University of Tokyo)

Integrable semi-discretization of the coupled nonlinear Schr\"{o}dinger equations

27 pages, LaTeX2e (IOP style)

J. Phys. A: Math. Gen. 32 (1999) 2239-2262

10.1088/0305-4470/32/11/016

solv-int nlin.SI

A system of semi-discrete coupled nonlinear Schr\"{o}dinger equations is studied. To show the complete integrability of the model with multiple components, we extend the discrete version of the inverse scattering method for the single-component discrete nonlinear Schr\"{o}dinger equation proposed by Ablowitz and Ladik. By means of the extension, the initial-value problem of the model is solved. Further, the integrals of motion and the soliton solutions are constructed within the framework of the extension of the inverse scattering method.

solv-int/9903014

Masuda

K. Kajiwara, T. Masuda (U. of Doshisha)

A generalization of determinant formulas for the solutions of Painlev\'e II and XXXIV equations

20 pages, LaTeX 2.09(IOP style), submitted to J. Phys. A

10.1088/0305-4470/32/20/309

solv-int nlin.SI

A generalization of determinant formulas for the classical solutions of Painlev\'e XXXIV and Painlev\'e II equations are constructed using the technique of Darboux transformation and Hirota's bilinear formalism. It is shown that the solutions admit determinant formulas even for the transcendental case.

solv-int/9903015

Masuda

K. Kajiwara, T. Masuda (U. of Doshisha)

On the Umemura Polynomials for the Painlev\'e III equation

10 pages, LaTeX 2.09(Elsevier style), submitted to Phys. Lett. A

10.1016/S0375-9601(99)00577-0

solv-int nlin.SI

A determinant expression for the rational solutions of the Painlev\'e III (P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given. Degeneration of this determinant expression to that for the rational solutions of P$_{\rm II}$ is discussed by applying the coalescence procedure.

solv-int/9903016

Evgueni Sklyanin

E. K. Sklyanin

Canonicity of Baecklund transformation: r-matrix approach. I

7 pages, LATEX-2e, macros included

Translations of the American Mathematical Society-Series 2, 201 (2000) 277-282

solv-int nlin.SI

For the Hamiltonian integrable systems governed by SL(2)-invariant r-matrix (such as Heisenberg magnet, Toda lattice, nonlinear Schroedinger equation) a general procedure for constructing Baecklund transformation is proposed. The corresponding BT is shown to preserve the Poisson bracket. The proof is given by a direct calculation using the r-matrix expression for the Poisson bracket.

solv-int/9903017

Evgueni Sklyanin

E. K. Sklyanin

Canonicity of Baecklund transformation: r-matrix approach. II

7 pages, LATEX 2e, macros included

Proceedings of the Steklov Institute of Mathematics 226 (1999) 121-126

solv-int nlin.SI

This is the second part of the paper devoted to the general proof of canonicity of Baecklund transformation (BT) for a Hamiltonian integrable system governed by SL(2)-invariant r-matrix. Introducing an extended phase space from which the original one is obtained by imposing a 1st kind constraint, we are able to prove the canonicity of BT in a new way. The new proof allows to explain naturally the fact why the gauge transformation matrix M associated to the BT has the same structure as the Lax operator L. The technique is illustrated on the example of the DST chain.

solv-int/9904001

Bernard Deconinck

Bernard Deconinck and Harvey Segur

Pole Dynamics for Elliptic Solutions of the Korteweg-deVries Equation

22 pages, 12 figures. Submitted for publication

MSRI 1999-020

solv-int nlin.SI

The real, nonsingular elliptic solutions of the Korteweg-deVries equation are studied through the time dynamics of their poles in the complex plane. The dynamics of these poles is governed by a dynamical system with a constraint. This constraint is shown to be solvable for any finite number of poles located in the fundamental domain of the elliptic function, often in many different ways. Special consideration is given to those elliptic solutions that have a real nonsingular soliton limit.

solv-int/9904002

Pilar G. Estevez

P. G. Estevez (Universidad de Salamanca, Spain), P.A. Clarkson (University of Kent at Canterbury, UK)

Discrete equations and the singular manifold method

to appear in SIDE III proceedings

solv-int nlin.SI

The Painleve expansion for the second Painleve equation (PII) and fourth Painleve equation (PIV) have two branches. The singular manifold method therefore requires two singular manifolds. The double singular manifold method is used to derive Miura transformations from PII and PIV to modified Painleve type equations for which auto-Backlund transformations are obtained. These auto-Backlund transformations can be used to obtain discrete equations.

solv-int/9904003

Vadim Kuznetsov

A.N.W. Hone, V.B. Kuznetsov, O. Ragnisco

Backlund transformations for many-body systems related to KdV

LaTeX2e, 8 pages

J.Phys. A32 (1999) L299-L306

10.1088/0305-4470/32/27/102

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

We present Backlund transformations (BTs) with parameter for certain classical integrable n-body systems, namely the many-body generalised Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the fact that all these systems may be obtained as particular reductions (stationary or restricted flows) of the KdV hierarchy; alternatively they may be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide exact time-discretizations of the original (continuous) systems, preserving the Lax matrix and hence all integrals of motion, and satisfy the spectrality property with respect to the Backlund parameter.

solv-int/9904004

Andrei Maltsev

B.A.Dubrovin, A.Ya.Maltsev

Recurrent procedure for the determination of the Free Energy $\epsilon^{2}$-expansion in the Topological String Theory

18 pages, Latex

solv-int hep-th nlin.SI

We present here the iteration procedure for the determination of free energy $\epsilon^{2}$-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times $t_{1}, t_{2}, ...$ to find the $\epsilon^{2}$-expansion of $u(x,t_{1},t_{2},...)$ after the infinite number of shifts of $u(x,0,0,...) \equiv x$ along $t_{1}, t_{2}, ...$ in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to $u_{n}(x)$.

solv-int/9904005

Igor G. Korepanov

I.G. Korepanov

The tetrahedral analog of Veneziano amplitude

LaTeX, 7 pages. v2: some minor editing

solv-int nlin.SI

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

In solv-int/9812016 it was shown that the Veneziano amplitude in string theory comes naturally from one of the simplest solutions of the functional pentagon equation (FPE). More generally, FPE is intimately connected with the duality condition for scattering processes. Here I find the amplitude that comes the same way from a solution of the functional tetrahedron equation, with the duality replaced by the local Yang - Baxter equation.

solv-int/9904006

R. Radhakrishnan

R. Radhakrishnan and M. Lakshmanan

Suppression and Enhancement of Soliton Switching During Interaction in Periodically Twisted Birefringent Fiber

10 pages, 4 figures, Latex, submitted to Phys. Rev. E

10.1103/PhysRevE.60.2317

solv-int nlin.SI

Soliton interaction in periodically twisted birefringent optical fibers has been analysed analytically with refernce to soliton switching. For this purpose we construct the exact general two-soliton solution of the associated coupled system and investigate its asymptotic behaviour. Using the results of our analytical approach we point out that the interaction can be used as a switch to suppress or to enhance soliton switching dynamics, if one injects multi-soliton as an input pulse in the periodically twisted birefringent fiber.

solv-int/9904007

Q-Han Park and H.J. Shin(Kyunghee U.)

Complex sine-Gordon Equation in Coherent Optical Pulse Propagation

Journal of Korean Physical Society, Vol. 30; 336-340, 1997

solv-int nlin.SI

It is shown that the McCall-Hahn theory of self-induced transparency in coherent optical pulse propagation can be identified with the complex sine-Gordon theory in the sharp line limit. We reformulate the theory in terms of the deformed gauged Wess-Zumino-Witten sigma model and address various new aspects of self-induced transparency.

solv-int/9904008

Jongbae Kim (ETRI), Q-Han Park and H.J. Shin (Kyunghee U.)

Conservation Laws in Higher-Order Nonlinear Optical Effects

Phys. Rev. E {\bf 58} 6746, 1998

10.1103/PhysRevE.58.6746

solv-int nlin.SI

Conservation laws of the nonlinear Schr\"{o}dinger equation are studied in the presence of higher-order nonlinear optical effects including the third-order dispersion and the self-steepening. In a context of group theory, we derive a general expression for infinitely many conserved currents and charges of the coupled higher-order nonlinear Schr\"{o}dinger equation. The first few currents and charges are also presented explicitly. Due to the higher-order effects, conservation laws of the nonlinear Schr\"{o}dinger equation are violated in general. The differences between the types of the conserved currents for the Hirota and the Sasa-Satsuma equations imply that the higher-order terms determine the inherent types of conserved quantities for each integrable cases of the higher-order nonlinear Schr\"{o}dinger equation.

solv-int/9904009

Q-Han Park and H.J. Shin (Kyunghee U.)

Painlev\'{e} analysis of the coupled nonlinear Schr\"{o}dinger equation for polarized optical waves in an isotropic medium

Phys. Rev. E {\bf 59} 2373, 1999

10.1103/PhysRevE.59.2373

solv-int nlin.SI

Using the Painlev\'{e} analysis, we investigate the integrability properties of a system of two coupled nonlinear Schr\"{o}dinger equations that describe the propagation of orthogonally polarized optical waves in an isotropic medium. Besides the well-known integrable vector nonlinear Schr\"{o}dinger equation, we show that there exist a new set of equations passing the Painlev\'{e} test where the self and cross phase modulational terms are of different magnitude. We introduce the Hirota bilinearization and the B\"{a}cklund transformation to obtain soliton solutions and prove integrability by making a change of variables. The conditions on the third-order susceptibility tensor $\chi^{(3)} $ imposed by these new integrable equations are explained.

solv-int/9904010

Baryakhtar

I.V.Baryakhtar, V.G.Baryakhtar, E.N.Economou

Kinetic and Transport Equations for Localized Excitations in Sine-Gordon Model

23 pages, latex, no figures

10.1103/PhysRevE.60.6645

solv-int nlin.SI

We analyze the kinetic behavior of localized excitations - solitons, breathers and phonons - in Sine-Gordon model. Collision integrals for all type of localized excitation collision processes are constructed, and the kinetic equations are derived. We analyze the kinetic behavior of localized excitations - solitons, breathers and phonons - in Sine-Gordon model. Collision integrals for all type of localized excitation collision processes are constructed, and the kinetic equations are derived. We prove that the entropy production in the system of localized excitations takes place only in the case of inhomogeneous distribution of these excitations in real and phase spaces. We derive transport equations for soliton and breather densities, temperatures and mean velocities i.e. show that collisions of localized excitations lead to creation of diffusion, thermoconductivity and intrinsic friction processes. The diffusion coefficients for solitons and breathers, describing the diffusion processes in real and phase spaces, are calculated. It is shown that diffusion processes in real space are much faster than the diffusion processes in phase space.

solv-int/9904011

Nugmanova G. N.

R. Myrzakulov

On the M-XX equation

9 pages, Latex, no figures

solv-int nlin.SI

The (2+1)-dimensional integrable M-XX equation is considered.

solv-int/9904012

Paul Fendley

P. Fendley and H. Saleur

Differential equations and duality in massless integrable field theories at zero temperature

18 pages, harvmac

Nucl.Phys.B574:571-586,2000

10.1016/S0550-3213(99)00832-9

solv-int hep-th nlin.SI

Functional relations play a key role in the study of integrable models. We argue in this paper that for massless field theories at zero temperature, these relations can in fact be interpreted as monodromy relations. Combined with a recently discovered duality, this gives a way to bypass the Bethe ansatz, and compute directly physical quantities as solutions of a linear differential equation, or as integrals over a hyperelliptic curve. We illustrate these ideas in details in the case of the $c=1$ theory, and the associated boundary sine-Gordon model.

solv-int/9904013

Sergei M. Sergeev

S. Sergeev

Solitons in a 3d integrable model

LaTeX, 6 pages

10.1016/S0375-9601(99)00849-X

solv-int nlin.SI

Equations of motion for a classical 3d discrete model, whose auxialiary system is a linear system, are investigated. The Lagrangian form of the equations of motion is derived. The Lagrangian variables are a triplet of "tau functions". The equations of motion for the Triplet of Tau functions are Three Trilinear equations. Simple solitons for the trilinear equations are given. Both the dispersion relation and the phase shift reflect the triplet structure of equations.

solv-int/9904014

Robert Milson

Niky Kamran, Robert Milson, Peter Olver

Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems

28 pages. To appear in Advances in Mathematics

solv-int nlin.SI

We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasi-exactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization o f the annihilating differential operators for spaces of analytic functions are presented.

solv-int/9904015

Nikolay Asenov Kostov

N.A.Kostov, Z.T. Kostova

Nonlinear waves, differential resultant, computer algebra and completely integrable dynamical systems

33 pages, no figures

solv-int nlin.SI

The hierarchy of integrable equations are considered. The dynamical approach to the theory of nonlinear waves is proposed. The special solutions(nonlinear waves) of considered equations are derived. We use powerful methods of computer algebra such differential resultant and others.

solv-int/9904016

Nikolay Asenov Kostov

N.A. Kostov

Korteweg-de Vries hierarchy and related completely integrable systems: I. Algebro-geometrical approach

31 pages, no figures

Preprint TH-98/4, 1998, INRNE, BAS, Sofia

solv-int nlin.SI

We consider complementary dynamical systems related to stationary Korteweg-de Vries hierarchy of equations. A general approach for finding elliptic solutions is given. The solutions are expressed in terms of Novikov polynomials in general quais-periodic case. For periodic case these polynomials coincide with Hermite and Lam\'e polynomials. As byproduct we derive $2\times 2$ matrix Lax representation for Rosochatius-Wojciechiwski, Rosochatius, second flow of stationary nonlinear vectro Schr\"{o}dinger equations and complex Neumann system.

solv-int/9904017

Nikolay Asenov Kostov

P.L. Christiansen, J.C. Eilbeck, V.Z. Enolskii, and N.A. Kostov

Quasi-Periodic and Periodic Solutions for Systems of Coupled Nonlinear SCHR\"Odinger Equations

21 pages, no figures

10.1098/rspa.2000.0612

TH/2 INRNE, BAS, Sofia

solv-int nlin.SI

We consider travelling periodic and quasiperiodic wave solutions of a set of coupled nonlinear Schr\"odimger equations. In fibre optics these equations can be used to model single mode fibers with strong birefringence and two-mode optical fibres. Recently these equations appear as modes, which describe pulse-pulse interaction in wavelength-division-multiplexed channels of optical fiber transmission systems. Two phase quasi-periodic solutions for integrable Manakov system are given in tems of two-dimensional Kleinian functions. The reduction of quasi-periodic solutions to elliptic functions is dicussed. New solutions in terms of generalized Hermite polynomilas, which are associated with two-gap Treibich-Verdier potentials are found.

solv-int/9904018

Sudipta Nandy

Sasanka Ghosh, Anjan Kundu, Sudipta Nandy

Soliton solutions, Liouville integrability and gauge equivalence of Sasa Satsuma equation

14 pages, to be published in J. Math. Phys. April-May, 1999

10.1063/1.532845

solv-int nlin.SI

Exact integrability of the Sasa Satsuma eqation (SSE) in the Liouville sense is established by showing the existence of an infinite set of conservation laws. The explicit form of the conserved quantities in term of the fields are obtained by solving the Riccati equation for the associated 3x3 Lax operator. The soliton solutions in particular, one and two soliton solutions, are constructed by the Hirota's bilinear method. The one soliton solutions is also compared with that found through the inverse scattering method. The gauge equivalence of the SSE with a generalized Landau Lifshitz equation is established with the explicit construction o

solv-int/9904019

Sudipta Nandy

Sasanka Ghosh, Sudipta Nandy

Optical solitons in higher order nonlinear Schrodinger equation

10 pages

solv-int nlin.SI

We show the complete integrability and the existence of optical solitons of higher order nonlinear Schrodinger equation by inverse scattering method for a wide range of values of coefficients. This is achieved first by invoking a novel connection between the integrability of a nonlinear evolution equation and the dimensions of a family of matrix Lax pairs. It is shown that Lax pairs of different dimensions lead to the same evolution equation only with the coefficients of the terms in different integer ratios. Optical solitons, thus obtained by inverse scattering method, have been found by solving an n dimensional eigenvalue problem.

solv-int/9904020

Dr P. K. Panigrahi

C. Nagaraja Kumar and Prasanta K. Panigrahi (School of Physics, University of Hyderabad, Hyderabad, India)

Compacton-like Solutions for Modified KdV and other Nonlinear Equations

4 pages, RevTex

solv-int nlin.SI

We present compacton-like solution of the modified KdV equation and compare its properties with those of the compactons and solitons. We further show that, the nonlinear Schr{\"o}dinger equation with a source term and other higher order KdV-like equations also possess compact solutions of the similar form.

solv-int/9904021

Sudipta Nandy

Sasanka Ghosh and Sudipta Nandy

Inverse scattering method and vector higher order nonlinear Schrodinger equation

28 pages

10.1016/S0550-3213(99)00484-8

solv-int nlin.SI

A generalised inverse scattering method has been developed for arbitrary n dimensional Lax equations. Subsequently, the method has been used to obtain N soliton solutions of a vector higher order nonlinear Schrodinger equation, proposed by us. It has been shown that under suitable reduction, vector higher order nonlinear Schrodinger equation reduces to higher order nonlinear Schrodinger equation. The infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equation. A gauge equivalence is shown between the vector higher order nonlinear Schrodinger equation and the generalized Landau Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.

solv-int/9904022

Unal Goktas

Willy Hereman (Colorado School of Mines), Unal Goktas (Wolfram Research, Inc.)

Integrability Tests for Nonlinear Evolution Equations

uses 31x47jw.sty, chapter in: Computer Algebra Systems: A Practical Guide (Ed. Michael Wester), Wiley and Sons, New York, in press

solv-int nlin.SI

Discusses several integrability tests for nonlinear evolution equations.

solv-int/9904023

Andrew Pickering

Pilar R. Gordoa, Nalini Joshi and Andrew Pickering

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

To appear in Nonlinearity (22 pages)

10.1088/0951-7715/12/4/313

solv-int nlin.SI

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.

solv-int/9904024

Svetlana Pacheva-Nissimov

Henrik Aratyn, Emil Nissimov and Svetlana Pacheva

Multi-Component Matrix KP Hierarchies as Symmetry-Enhanced Scalar KP Hierarchies and Their Darboux-B"acklund Solutions

10 pages, LaTeX209

BGU-99/20/Apr-PH

solv-int nlin.SI

We show that any multi-component matrix KP hierarchy is equivalent to the standard one-component (scalar) KP hierarchy endowed with a special infinite set of abelian additional symmetries, generated by squared eigenfunction potentials. This allows to employ a special version of the familiar Darboux-B"acklund transformation techniques within the ordinary scalar KP hierarchy in the Sato formulation for a systematic derivation of explicit multiple-Wronskian tau-function solutions of all multi-component matrix KP hierarchies.

solv-int/9905001

Polterovich Iosif

Iosif Polterovich

From Agmon-Kannai expansion to Korteweg-de Vries hierarchy

7 pages, AMS-LaTeX

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

We present a new method for computation of the Korteweg-de Vries hierarchy via heat invariants of the 1-dimensional Schrodinger operator. As a result new explicit formulas for the KdV hierarchy are obtained. Our method is based on an asymptotic expansion of resolvent kernels of elliptic operators due to S.Agmon and Y.Kannai.

solv-int/9905002

Katrina Elfrieda Hibberd

Katrina Hibberd, Itzhak Roditi, Jon Links and Angela Foerster

Bethe ansatz solution of the closed anisotropic supersymmetric U model with quantum supersymmetry

7 pages (revtex), minor modifications. To appear in Mod. Phys. Lett. A

10.1142/S021773230000013X

solv-int nlin.SI

The nested algebraic Bethe ansatz is presented for the anisotropic supersymmetric $U$ model maintaining quantum supersymmetry. The Bethe ansatz equations of the model are obtained on a one-dimensional closed lattice and an expression for the energy is given.

solv-int/9905003

Alexander Turbiner

Alexander Turbiner, Pavel Winternitz

Solutions of Non-linear Differential and Difference Equations with Superposition Formulas

17 pages, REVTeX, submitted to Journ.Math.Phys

ICN-UNAM 99-03 (Mexico), CRM-2606 (Montreal)

solv-int nlin.SI

Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit solutions of certain classes of scalar and matrix Riccati equations are presented as an illustration of the general results.

solv-int/9905004

Takayuki Tsuchida

T. Tsuchida, M. Wadati

New integrable systems of derivative nonlinear Schr\"{o}dinger equations with multiple components

15 pages, LaTeX209, to appear in Phys. Lett. A

Phys. Lett. A 257 (1999) 53-64

10.1016/S0375-9601(99)00272-8

solv-int nlin.SI

The Lax pair for a derivative nonlinear Schr\"{o}dinger equation proposed by Chen-Lee-Liu is generalized into matrix form. This gives new types of integrable coupled derivative nonlinear Schr\"{o}dinger equations. By virtue of a gauge transformation, a new multi-component extension of a derivative nonlinear Schr\"{o}dinger equation proposed by Kaup-Newell is also obtained.

solv-int/9905005

Luis Martinez Alonso

Boris Konopelchenko and Luis Martinez Alonso

The KP Hierarchy in Miwa Coordinates

14 pages Latex2e

10.1016/S0375-9601(99)00373-4

solv-int nlin.SI

A systematic reformulation of the KP hierarchy by using continuous Miwa variables is presented. Basic quantities and relations are defined and determinantal expressions for Fay's identities are obtained. It is shown that in terms of these variables the KP hierarchy gives rise to a Darboux system describing an infinite-dimensional conjugate net.

solv-int/9905006

R. Radhakrishnan

R. Radhakrishnan, A. Kundu and M. Lakshmanan

Coupled nonlinear Schrodinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media

13 pages, 5 figures, LaTex, Submitted to Phys. Rev. E

10.1103/PhysRevE.60.3314

solv-int nlin.SI

We propose an integrable system of coupled nonlinear Schrodinger equations with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short optical soliton pulse propagation in non-Kerr media. Lax pair, conserved quantities and exact soliton solutions for the proposed integrable model are given. Explicit form of two-solitons are used to study soliton interaction showing many intriguing features including inelastic (shape changing) scattering. Another novel system of coupled equations with fifth-degree nonlinearity is derived, which represents vector generalization of the known chiral-soliton bearing system.

solv-int/9905007

Hubert Saleur

H. Saleur

The continuum limit of sl(N/K) integrable super spin chains

Nucl.Phys. B578 (2000) 552-576

10.1016/S0550-3213(00)00002-X

USC-99-002

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

I discuss in this paper the continuum limit of integrable spin chains based on the superalgebras sl(N/K). The general conclusion is that, with the full ``supersymmetry'', none of these models is relativistic. When the supersymmetry is broken by the generator of the sub u(1), Gross Neveu models of various types are obtained. For instance, in the case of sl(N/K) with a typical fermionic representation on every site, the continuum limit is the GN model with N colors and K flavors. In the case of sl(N/1) and atypical representations of spin j, a close cousin of the GN model with N colors, j flavors and flavor anisotropy is obtained. The Dynkin parameter associated with the fermionic root, while providing solutions to the Yang Baxter equation with a continuous parameter, thus does not give rise to any new physics in the field theory limit. These features are generalized to the case where an impurity is embedded in the system.

solv-int/9905008

Ruslan Sharipov

R. F. Bikbaev, R. A. Sharipov

Magnetization waves in Landau-Lifshitz Model

AmSTeX, Ver. 2.1h, 5 pages, amsppt style, 1 figure

Phys. Lett. 134A (1988), no. 2, 105-107.

10.1016/0375-9601(88)90943-7

solv-int nlin.SI

The solutions of the Landau-Lifshitz equation with finite-gap behavior at infinity are considered. By means of the inverse scattering method the large-time asymptotics is obtained.

solv-int/9905009

Takeo Kojima

H. Furutsu and T. Kojima (Nihon Univ.)

$U_q(\hat{sl}_n)$-analog of the XXZ chain with a boundary

24 pages, LaTEX2e

J.Math.Phys. 41 (2000) 4413-4436

10.1063/1.533351

solv-int hep-th nlin.SI

We study $U_q(\hat{sl}_n)$ analog of the XXZ spin chain with a boundary magnetic field h. We construct explicit bosonic formulas of the vacuum vector and the dual vacuum vector with a boundary magnetic field. We derive integral formulas of the correlation functions.

solv-int/9905010

Nalini Joshi

Nalini Joshi, Johannes A. Petersen, and Luke M. Schubert

Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili equations

17 pages in LaTeX2e, to appear in Stud. Appl. Math

solv-int nlin.SI

We study characteristic Cauchy problems for the Korteweg-deVries (KdV) equation $u_t=uu_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation $u_{yy}=\bigl(u_{xxx}+uu_x+u_t\bigr)_x$ with holomorphic initial data possessing nonnegative Taylor coefficients around the origin. For the KdV equation with initial value $u(0,x)=u_0(x)$, we show that there is no solution holomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\mathbb C}^2$ unless $u_0(x)=a_0+a_1x$. This also furnishes a nonexistence result for a class of $y$-independent solutions of the KP equation. We extend this to $y$-dependent cases by considering initial values given at $y=0$, $u(t,x,0)=u_0(x,t)$, $u_y(t,x,0)=u_1(x,t)$, where the Taylor coefficients of $u_0$ and $u_1$ around $t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic solution around the origin in ${\mathbb C}^3$ unless $u_0$ and $u_1$ are polynomials of degree 2 or lower.

solv-int/9905011

Askold Perelomov

L. Gavrilov (U. Paul Sabatier, Toulouse), A. Perelomov (MPIM Bonn)

On the explicit solutions of the elliptic Calogero system

18 pages, Latex

10.1063/1.533096

solv-int nlin.SI

Let $q_1,q_2,...,q_N$ be the coordinates of $N$ particles on the circle, interacting with the integrable potential $\sum_{j<k}^N\wp(q_j-q_k)$, where $\wp$ is the Weierstrass elliptic function. We show that every symmetric elliptic function in $q_1,q_2,...,q_N$ is a meromorphic function in time. We give explicit formulae for these functions in terms of genus $N-1$ theta functions.

solv-int/9905012

Doc. Dr. Ayse Humeyra Bilge

Ayse Humeyra Bilge

A System with a Recursion Operator but One Higher Local Symmetry of the Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$

7 pages, no figures

Lie Groups and Their Applications, Vol.1, No 2, pp.132-139, (1994)

solv-int nlin.SI

We construct a recursion operator for the system $(u_t,v_t)=(u_4+v^2,1/5 v_4)$, for which only one local symmetry is known and we show that the action of the recursion operator on $(u_t,v_t)$ is a local function.

solv-int/9905013

H. J. S. Dorren

H.J.S. Dorren and J.J.B. van den Heuvel

On pulse broadening for optical solitons

11 pages, the manuscript has undergone major revisions

solv-int nlin.SI

Pulse broadening for optical solitons due to birefringence is investigated. We present an analytical solution which describes the propagation of solitons in birefringent optical fibers. The special solutions consist of a combination of purely solitonic terms propagating along the principal birefringence axes and soliton-soliton interaction terms. The solitonic part of the solutions indicates that the decay of initially localized pulses could be due to different propagation velocities along the birefringence axes. We show that the disintegration of solitonic pulses in birefringent optical fibers can be caused by two effects. The first effect is similar as in linear birefringence and is related to the unequal propagation velocities of the modes along the birefringence axes. The second effect is related to the nonlinear soliton-soliton interaction between the modes, which makes the solitonic pulse-shape blurred.

solv-int/9906001

David H. Sattinger

Richard Beals, D.H. Sattinger, and J. Szmigielski

Multipeakons and the Classical Moment Problem

32 pages, 3 figures; to appear in Advances in Mathematics

solv-int nlin.SI

Classical results of Stieltjes are used to obtain explicit formulas for the peakon-antipeakon solutions of the Camassa-Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon-antipeakon pairs, and the details of the collisions are analyzed using results {}from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly

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.