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solv-int/9701018

Kirill Vaninsky

K.L. Vaninsky

Symplectic Structures and Volume Elements in the Function Space for the Cubic Schrodinger Equation

20 pages, AMS-TEX

Duke Math. J, vol 92, no. 1, pp. 381-402 (1998)

solv-int nlin.SI

We consider various trace formulas for the cubic Schrodinger equation in the space of infinitely smooth functions subject to periodic boundary conditions. The formulas relate conventional integrals of motion to the periods of some Abelian differentials (holomorphic one-forms) on the spectral curve. We show that the periods of Abelian differentials are global coordinates on the moduli space of spectral curves. The exterior derivatives of the holomorphic one-forms are the basic and higher symplectic structures on the phase space. We write explicitly these symplectic structures in $QP$ coordinates. We compute the ratio of two symplectic volume elements in the infinite genus limit.

solv-int/9701019

Alexander Sorin

A.N. Leznov and A. Sorin

The Solution of the N=2 Supersymmetric f-Toda Chain with Fixed Ends

15 pages, latex, no figures

Phys.Lett. B402 (1997) 87-100

10.1016/S0370-2693(97)00449-8

JINR E2-97-21

solv-int hep-th nlin.SI

The integrability of the recently introduced N=2 supersymmetric f-Toda chain, under appropriate boundary conditions, is proven. The recurrent formulae for its general solutions are derived. As an example, the solution for the simplest case of boundary conditions is presented in explicit form.

solv-int/9701020

Alexander Sorin

A. Sorin

The Discrete Symmetries of the N=2 Supersymmetric GNLS Hierarchies

8 pages, latex, no figures, report-no added

JINR E2-97-37

solv-int hep-th nlin.SI

The discrete symmetry transformations of the N=2 supersymmetric (n,m)-GNLS hierarchy are constructed. Their bosonic limit is analyzed and new discrete symmetries of the modified GNLS hierarchy are derived. The explicit relations connecting the integrable hierarchy, produced by the junction of the Lax operators for the N=2 supersymmetric a=4 KdV and (n-1,m)-GNLS hierarchies, to the N=2 supersymmetric (n,m)-GNLS hierarchy are established.

solv-int/9701021

Marco Ameduri

Marco Ameduri, Costas J. Efthimiou

Is the classical Bukhvostov-Lipatov model integrable? A Painlev\'e analysis

J. Nonlinear Math. Phys. 5 (1998), no. 2, 132-139

10.2991/jnmp.1998.5.2.4

JNMP 4/2002 (Letter)

solv-int hep-th nlin.SI

In this work we apply the Weiss, Tabor and Carnevale integrability criterion (Painlev\'e analysis) to the classical version of the two dimensional Bukhvostov-Lipatov model. We are led to the conclusion that the model is not integrable classically, except at a trivial point where the theory can be described in terms of two uncoupled sine-Gordon models.

solv-int/9701022

Masato Hisakado

Coupled Nonlinear Schr\"{o}dinger equation and Toda equation (the Root of Integrability)

11 pages, LateX, to apper in J. Phys. Soc. Jpn. Vol. 66, No 7

10.1143/JPSJ.66.1939

solv-int hep-th nlin.SI

We consider the relation between the discrete coupled nonlinear Schr\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\"{o}dinger equation. In the same way we can show the integrability in coupled case of dark and bright equations. Using this method we obtain several integrable equations.

solv-int/9702001

Kenji Kajiwara

Kenji Kajiwara, Kazushi Yamamoto and Yasuhiro Ohta

Rational Solutions for the Discrete Painlev\'e II Equation

12 pages, latex

10.1016/S0375-9601(97)00397-6

solv-int nlin.SI

The rational solutions for the discrete Painlev\'e II equation are constructed based on the bilinear formalism. It is shown that they are expressed by the determinant whose entries are given by the Laguerre polynomials. Continuous limit to the Devisme polynomial representation of the rational solutions for the Painlev\'e II equation is also discussed.

solv-int/9702002

Q. P. Liu

Fully Supersymmetric Hierarchies From A Energy Dependent Super Hill Operator

15 pages, AMS-LaTex

J. Phys. A: Math. Gen., 30 (1997) 8661

10.1088/0305-4470/30/24/025

solv-int hep-th nlin.SI

A super Hill operator with energy dependent potentials is proposed and the associated integrable hierarchy is constructed explicitly. It is shown that in the general case, the resulted hierarchy is multi-Hamiltonian system. The Miura type transformations and modified hierarchies are also presented.

solv-int/9702003

Juri Suris

Yuri B. Suris (University of Bremen, Germany)

On an integrable discretization of the modified Korteweg-de Vries equation

23 pages, LaTeX

Phys. Lett. A, 1997, V.234, p. 91-102.

10.1016/S0375-9601(97)00592-6

solv-int nlin.SI

We find time discretizations for the two ''second flows'' of the Ablowitz-Ladik hierachy. These discretizations are described by local equations of motion, as opposed to the previously known ones, due to Taha and Ablowitz. Certain superpositions of our maps allow a one-field reduction and serve therefore as valid space-time discretizations of the modified Korteweg-de Vries equation. We expect the performance of these discretizations to be much better then that of the Taha-Ablowitz scheme. The way of finding interpolating Hamiltonians for our maps is also indicated, as well as the solution of an initial value problem in terms of matrix factorizations.

solv-int/9702004

Galina A. Korepanova

I.G. Korepanov

Some eigenstates for a model associated with solutions of tetrahedron equation. II. A bit of algebraization

LaTeX, 8 pages

solv-int nlin.SI

This paper adds two observations to the work solv-int/9701016 where some eigenstates for a model based on tetrahedron equation have been constructed. The first observation is that there exists a more "algebraic" construction of one-particle states, resembling the 1+1-dimensional algebraic Bethe ansatz. The second observation is that the strings introduced in solv-int/9701016 are symmetries of a transfer matrix, rather than just eigenstates.

solv-int/9702005

Vadim Vereschagin

V.L.Vereschagin

Asymptotics for Solution to the Cauchy Problem for Volterra Lattice with Step-Like Initial Values

solv-int nlin.SI

The connection between modulated Riemann surface of genus one and solution to Volterra lattice that tends to constants at infinity is studied. The main term of asymptotics for large time of solution to the mentioned Cauchy problem is written out.

solv-int/9702006

G. Tondo

C. Morosi and G. Tondo

Quasi-BiHamiltonian Systems and Separability

10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May 1997)

10.1088/0305-4470/30/8/023

solv-int nlin.SI

Two quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.

solv-int/9702007

Harold Widom

Harold Widom (University of California, Santa Cruz)

An Integral Operator Solution to the Matrix Toda Equations

8 pages, LaTeX file. An argument improved

J. Int. Eqs. Appl. 10 (1998) 363

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

In previous work the author found solutions to the Toda equations that were expressed in terms of determinants of integral operators. Here it is observed that a simple variant yields solutions to the matrix Toda equations. As an application another derivation is given of a differential equation of Sato, Miwa and Jimbo for a particular Fredholm determinant.

solv-int/9702008

Unal Goktas and Willy Hereman (Colorado School of Mines)

Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations

31 pages, Latex, uses jsc.sty, submitted to J. Symbolic Computation

MCS-96-06

solv-int nlin.SI

A new algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented. The algorithm is implemented in Mathematica. The program condens.m automatically carries out the lengthy symbolic computations for the construction of conserved densities. The code is tested on several well-known partial differential equations from soliton theory. For systems with parameters, condens.m can be used to determine the conditions on these parameters so that a sequence of conserved densities might exist. The existence of a large number of conservation laws is a predictor for integrability of the system.

solv-int/9703001

Leonid Dickey

L. A. Dickey

Poisson brackets with divergence terms in field theories: two examples

7 pages, LaTeX

solv-int nlin.SI

In field theories one often works with the functionals which are integrals of some densities. These densities are defined up to divergence terms (boundary terms). A Poisson bracket of two functionals is also a functional, i.e., an integral of a density. Suppose the divergence term in the density of the Poisson bracket be fixed so that it becomes a bilinear form of densities of two functionals. Then the left-hand side of the Jacobi identity written in terms of densities is not necessarily zero but a divergence of a trilinear form. The question is: what can be said about this trilinear form, what kind of a higher Jacobi identity (involving four fields) it enjoys? Two examples whose origin is the theory of integrable systems are given.

solv-int/9703002

Sergei Ya. Startsev

S. Ya. Startsev

An analog of the variational derivative and constructive necessary integrability condition for hyperbolic equation

6 pages, Latex

solv-int nlin.SI

An algorithm is constructed which allows to express conserved flows of hyperbolic equations in terms of corresponding conserved densities and to eliminate these flows from conservation laws of hyperbolic equations. The application of this algorithm to canonical conservation laws gives constructive necessary integrability conditions of hyperbolic equations in terms of the generalized Laplace invariants of these equations.

solv-int/9703003

Mts

V.V. Dmitrieva and R.A. Sharipov

On the point transformations for the second order differential equations. I

AmSTeX, Version 2.1, 15 pages

solv-int nlin.SI

Point transformations for the ordinary differential equations of the form $y''=P(x,y)+3 Q(x,y) y'+3 R(x,y) (y')^2+S(x,y) (y')^3$ are considered. Some classical results are resumed. Solution for the equivalence problem for the equations of general position is described.

solv-int/9703004

Juri Suris

Yuri B. Suris (University of Bremen)

A collection of integrable systems of the Toda type in continuous and discrete time, with 2x2 Lax representations

33 pp, LaTeX

solv-int nlin.SI

A fairly complete list of Toda-like integrable lattice systems, both in the continuous and discrete time, is given. For each system the Newtonian, Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax representation and r-matrix structure. The material is given in the "no comment" style, in particular, all proofs are omitted.

solv-int/9703005

Andrew Hone

Andrew N.W. Hone

Non-autonomous H\'{e}non-Heiles Systems

25 pages, Latex. Some minor corrections

10.1016/S0167-2789(98)00010-4

solv-int nlin.SI

Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable H\'{e}non-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivatives of the tau-functions (inherited from the original PDEs). The ODEs for the similarity solutions also have inherited B\"{a}cklund transformations, which may be used to generate sequences of rational solutions as well as other special solutions related to the first Painlev\'{e} transcendent.

solv-int/9703006

Manna Miguel

M. A. Manna and V. Merle

Modified Korteweg-de Vries Hierachies in Multiple-Times Variables and the Solutions of Modified Boussinesq Equations

RevTex file, submitted to Proc. Roy. Soc. London A

10.1098/rspa.1998.0215

solv-int nlin.SI

We study solitary-wave and kink-wave solutions of a modified Boussinesq equation through a multiple-time reductive perturbation method. We use appropriated modified Korteweg-de Vries hierarchies to eliminate secular producing terms in each order of the perturbative scheme. We show that the multiple-time variables needed to obtain a regular perturbative series are completely determined by the associated linear theory in the case of a solitary-wave solution, but requires the knowledge of each order of the perturbative series in the case of a kink-wave solution. These appropriate multiple-time variables allow us to show that the solitary-wave as well as the kink-wave solutions of the modified Botussinesq equation are actually respectively a solitary-wave and a kink-wave satisfying all the equations of suitable modified Korteweg-de Vries hierarchies.

solv-int/9703007

Leon Jerome

J. Leon and A.V. Mikhailov

Raman Solitons and Raman spikes

RevTex file, 4 pages

solv-int nlin.SI

Stimulated Raman scattering of a laser pump pulse seeded by a Stokes pulse generically leaves a two-level medium initially at rest in an excited state constituted of static solitons and radiation. The soliton birth manifests as sudden very large variations of the phase of the output pump pulse. This is proved by building the IST solution of SRS on the semi-line, which shows moreover that initial Stokes phase flips induce Raman spikes in the pump output also for short pulse experiments.

solv-int/9703008

Jarmo Hietarinta

R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta

Inelastic Collision and Switching of Coupled Bright Solitons in Optical Fibers

9 pages in LaTeX, 1 PostScript figure. To appear in Phys. Rev. E

10.1103/PhysRevE.56.2213

solv-int nlin.SI

By constructing the general six-parameter bright two-soliton solution of the integrable coupled nonlinear Schrodinger equation (Manakov model) using the Hirota method, we find that the solitons exhibit certain novel inelastic collision properties, which have not been observed in any other (1+1) dimensional soliton system so far. In particular, we identify the exciting possibility of switching solitons between modes by changing the phase. However, the standard elastic collision property of solitons is regained with specific choices of parameters.

solv-int/9703009

Jose Geraldo Pereira

R. A. Kraenkel, J. G. Pereira and E. C. de Rey Neto (IFT-UNESP, Sao Paulo, Brazil)

Linearizability of the Perturbed Burgers Equation

10 pages, RevTeX, no figures

10.1103/PhysRevE.58.2526

IFT-P.020/97

solv-int nlin.SI

We show in this letter that the perturbed Burgers equation $u_t = 2uu_x + u_{xx} + \epsilon ( 3 \alpha_1 u^2 u_x + 3\alpha_2 uu_{xx} + 3\alpha_3 u_x^2 + \alpha_4 u_{xxx} )$ is equivalent, through a near-identity transformation and up to order \epsilon, to a linearizable equation if the condition $3\alpha_1 - 3\alpha_3 - 3/2 \alpha_2 + 3/2 \alpha_4 = 0$ is satisfied. In the case this condition is not fulfilled, a normal form for the equation under consideration is given. Then, to illustrate our results, we make a linearizability analysis of the equations governing the dynamics of a one-dimensional gas.

solv-int/9703010

Galina A. Korepanova

I.G. Korepanov

Some eigenstates for a model associated with solutions of tetrahedron equation. III. Tetrahedral Zamolodchikov algebras and perturbed strings

LaTeX, 7 pages

solv-int nlin.SI

This paper continues the series begun with works solv-int/9701016 and solv-int/9702004. Here we show how to construct eigenstates for a model based on tetrahedron equation using the tetrahedral Zamolodchikov algebras. This yields, in particular, new eigenstates for the model on infinite lattice -- `perturbed', or `broken', strings.

solv-int/9703011

Andres Gomberoff

Andres Gomberoff and Sergio A. Hojman

Non-standard Construction of Hamiltonian Structures

13 pages, Revtex

J.Phys.A30:5077-5084,1997

10.1088/0305-4470/30/14/018

solv-int hep-th nlin.SI

Examples of the construction of Hamiltonian structures for dynamical systems in field theory (including one reputedly non-Hamiltonian problem) without using Lagrangians, are presented. The recently developed method used requires the knowledge of one constant of the motion of the system under consideration and one solution of the symmetry equation.

solv-int/9703012

Hasan Gumral

H. Gumral

Lagrangian Description, Symplectic Structure, and Invariants of 3D Fluid Flow

Plain Latex, 15 pages

10.1016/S0375-9601(97)00404-0

RIBS-PH-5/97

solv-int nlin.SI

Three dimensional unsteady flow of fluids in the Lagrangian description is considered as an autonomous dynamical system in four dimensions. The condition for the existence of a symplectic structure on the extended space is the frozen field equations of the Eulerian description of motion. Integral invariants of symplectic flow are related to conservation laws of the dynamical equation. A scheme generating infinite families of symmetries and invariants is presented. For the Euler equations these invariants are shown to have a geometric origin in the description of flow as geodesic motion; they are also interpreted in connection with the particle relabelling symmetry.

solv-int/9703013

Robert Carroll

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

WDVV and DZM

Latex, 14 pages

10.1016/S0375-9601(97)00588-4

solv-int nlin.SI

We show how the WDVV equations and the DZM system can be characterized via a background family of functions.

solv-int/9704001

Andrei Mironov

A.Zabrodin

A survey of Hirota's difference equations

LaTeX, 43 pages, LaTeX figures (with emlines2.sty)

10.1007/BF02634165

ITEP/TH-10/97

solv-int hep-th nlin.SI

A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.

solv-int/9704002

Laszlo Feher

Laszlo Feher, Ian Marshall

Extended matrix Gelfand-Dickey hierarchies: reduction to classical Lie algebras

plain TeX, 12 pages

10.1088/0305-4470/30/16/022

solv-int hep-th nlin.SI

The Drinfeld-Sokolov reduction method has been used to associate with $gl_n$ extensions of the matrix r-KdV system. Reductions of these systems to the fixed point sets of involutive Poisson maps, implementing reduction of $gl_n$ to classical Lie algebras of type $B, C, D$, are here presented. Modifications corresponding, in the first place to factorisation of the Lax operator, and then to Wakimoto realisations of the current algebra components of the factorisation, are also described.

solv-int/9704003

G. Cicogna

Convergent Normal Forms of Symmetric Dynamical Systems

11 pag., Plain TeX

10.1088/0305-4470/30/17/013

solv-int nlin.SI

It is shown that the presence of Lie-point-symmetries of (non-Hamiltonian) dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical sytem (or vector field) into Poincar\'e-Dulac normal form.

solv-int/9704004

Kanehisa Takasaki

Kanehisa Takasaki (Kyoto University)

Spectral Curves and Whitham Equations in Isomonodromic Problems of Schlesinger Type

41 pages, latex, no figures; typos in references are corrected

Asian J.Math. 4 (2) (1998), 1049-1078

KUCP-0105

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

It has been known since the beginning of this century that isomonodromic problems --- typically the Painlev\'e transcendents --- in a suitable asymptotic region look like a kind of ``modulation'' of isospectral problem. This connection between isomonodromic and isospectral problems is reconsidered here in the light of recent studies related to the Seiberg-Witten solutions of $N = 2$ supersymmetric gauge theories. A general machinary is illustrated in a typical isomonodromic problem, namely the Schlesinger equation, which is reformulated to include a small parameter $\epsilon$. In the small-$\epsilon$ limit, solutions of this isomonodromic problem are expected to behave as a slowly modulated finite-gap solution of an isospectral problem. The modulation is caused by slow deformations of the spectral curve of the finite-gap solution. A modulation equation of this slow dynamics is derived by a heuristic method. An inverse period map of Seiberg-Witten type turns out to give general solutions of this modulation equation. This construction of general solution also reveals the existence of deformations of Seiberg-Witten type on the same moduli space of spectral curves. A prepotential is also constructed in the same way as the prepotential of the Seiberg-Witten theory.

solv-int/9704005

Guest

R. Myrzakulov (High Energy Physics Institute, National Academy of Sciences, Alma-Ata, Kazakstan), S. Vijayalakshmi, G. N. Nugmanova, and M. Lakshmanan (Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli, India)

A (2+1) dimensional integrable spin model: Geometrical and gauge equivalent counterpart, solitons and localized coherent structures

14 pages, LaTex, no figures; email of first author: [email protected] and [email protected]

Physics Letters A, v.233, N4-6, 391-396 (1997)

10.1016/S0375-9601(97)00457-X

solv-int nlin.SI

A non-isospectral (2+1) dimensional integrable spin equation is investigated. It is shown that its geometrical and gauge equivalent counterparts is the (2+1) dimensional nonlinear Schr\"odinger equation introduced by Zakharov and studied recently by Strachan. Using a Hirota bilinearised form, line and curved soliton solutions are obtained. Using certain freedom (arbitrariness) in the solutions of the bilinearised equation, exponentially localized dromion-like solutions for the potential is found. Also, breaking soliton solutions (for the spin variables) of the shock wave type and algebraically localized nature are constructed.

solv-int/9704006

Hisao Konuma

Satoru Saito, Noriko Saitoh, Hisao Konuma and Katsuhiko Yoshida

Complex Analysis of a Piece of Toda Lattice

17 pages, LaTeX

10.1088/0305-4470/30/19/029

solv-int hep-th nlin.SI

We study a small piece of two dimensional Toda lattice as a complex dynamical system. In particular the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.

solv-int/9704007

Hisao Konuma

Satoru Saito

The Correspondence between Discrete Surface and Difference Geometry of the KP-hierarchy

solv-int hep-th nlin.SI

The correspondence between two geometrical descriptions of the KP-hierarchy, one by discrete surface and another by difference analogue of differential geometry, is given.

solv-int/9704008

Hisao Konuma

Satoru Saito

Dual Resonance Model Solves the Yang-Baxter Equation

10 pages, LaTeX

10.1088/0305-4470/30/23/025

solv-int hep-th nlin.SI

The duality of dual resonance models is shown to imply that the four point string correlation function solves the Yang-Baxter equation. A reduction of transfer matrices to $A_l$ symmetry is described by a restriction of the KP $\tau$ function to Toda molecules.

solv-int/9704009

Ken Umeno

Ken Umeno

Singularity analysis towards nonintegrability of nonhomogeneous nonlinear lattices

Latex 6pages, use crckapb.sty

Hamiltonian Systems with Three or More Degrees of Freedom, Edited by C. Simo, pp.614-617 (Kluwer,1999).

solv-int nlin.SI

We show non-integrability of the nonlinear lattice of Fermi-Pasta-Ulam type via the singularity analysis(Picard-Vessiot theory) of normal variational equations of Lam\'e type.

solv-int/9704010

Gregorio Falqui

Gregorio Falqui, Cesare Reina, and Alessandro Zampa

Krichever Maps, Faa' di Bruno Polynomials, and Cohomology in KP Theory

16 pages, LaTex using amssymb.sty. To be published in Lett. Math. Phys

SISSA/ISAS/37/97/FM

solv-int nlin.SI

We study the geometrical meaning of the Faa' di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faa' di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.

solv-int/9704011

Michael Shapiro

M.Gekhtman and M. Shapiro

Non-commutative and commutative integrability of generic Toda flows in simple Lie algebras

AMSTeX, 24 pages, no figures, available via http://www.math.kth.se/~mshapiro/

solv-int nlin.SI

In this paper we prove the complete integrability of Toda flows on generic coadjoint orbits in simple Lie algebras.

solv-int/9704012

Igor Loutsenko

Y. Berest, I. Loutsenko

Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation

23 pages, LaTeX, to be published in Comm.Math.Phys (1997)

10.1007/s002200050235

solv-int nlin.SI

A new class of linear second order hyperbolic partial differential operators satisfying Huygens' principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens' principle and the theory of solitary wave solutions of the Korteweg-de Vries equation.

solv-int/9704013

Galina A. Korepanova

I.G. Korepanov

Some eigenstates for a model associated with solutions of tetrahedron equation. IV. String-particle marriage

LaTeX, 6 pages

solv-int nlin.SI

This paper continues the series begun with works solv-int/9701016, solv-int/9702004 and solv-int/9703010. Here we construct more sophisticated strings, combining ideas from those papers and some considerations involving solutions of tetrahedron equation due to Sergeev, Mangazeev and Stroganov.

solv-int/9704014

Kaptsov

O. V. Kaptsov, Yu. V. Shan'ko (Computing Center, Academy of Sciences, Krasnoyarsk, Russia)

Trilinear representation and the Moutard transformation for the Tzitzeica equation

16 pages (30 Kbytes), standard LaTeX 2.09, run twice to get the right cross-references

solv-int nlin.SI

In the paper we present a trilinear form and a Darboux-type transformation to an equation considered by Tzitzeica in 1910. This equation equivalent to the Bullough-Dodd-Jiber-Shabat equation. Soliton solutions are constructed by dressing the trivial solution.

solv-int/9704015

Ovidiu Lipan

O. Lipan, P.B. Wiegmann and A. Zabrodin

Fusion rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds

LaTex (MPLA macros included) 10 pages, 1 figure, included in the text

Mod.Phys.Lett. A12 (1997) 1369-1378

10.1142/S0217732397001394

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

We show that the set of transfer matrices of an arbitrary fusion type for an integrable quantum model obey these bilinear functional relations, which are identified with an integrable dynamical system on a Grassmann manifold (higher Hirota equation). The bilinear relations were previously known for a particular class of transfer matrices corresponding to rectangular Young diagrams. We extend this result for general Young diagrams. A general solution of the bilinear equations is presented.

solv-int/9704016

Unal Goktas, Willy Hereman, Grant Erdmann (Colorado School of Mines)

Computation of conserved densities for systems of nonlinear differential-difference equations

submitted to Phys. Lett A, 10 pages, latex

10.1016/S0375-9601(97)00750-0

MCS-97-02

solv-int nlin.SI

A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability.

solv-int/9705001

Loriano Bonora

L.Bonora, S.Krivonos

Hamiltonian structure and coset construction of the supersymmetric extensions of N=2 KdV hierarchy

11 pages, Latex, a few modifications in the text

10.1142/S0217732397003162

SISSA 59/97/EP

solv-int hep-th nlin.SI

A manifestly N=2 supersymmetric coset formalism is applied to analyse the "fermionic" extensions of N=2 $a=4$ and $a=-2$ KdV hierarchies. Both these hierarchies can be obtained from a manifest N=2 coset construction. This coset is defined as the quotient of some local but non-linear superalgebra by a $\hat{U(1)}$ subalgebra. Three superextensions of N=2 KdV hierarchy are proposed, among which one seems to be entirely new.

solv-int/9705002

Jarmo Hietarinta

Jarmo Hietarinta and Kenji Kajiwara

Rational solutions to d-PIV

11 pages, LaTeX2e with epic. To appear in the proceedings of SIDE II, Canterbury 1996

solv-int nlin.SI

We study the rational solutions of the discrete version of Painleve's fourth equation d-PIV. The solutions are generated by applying Schlesinger transformations on the seed solutions -2z and -1/z. After studying the structure of these solutions we are able to write them in a determinantal form that includes an interesting parameter shift that vanishes in the continuous limit.

solv-int/9705003

V.S. Dryuma, B.G. Konopelchenko

On equation of geodesic deviation and its solutions

17 pages, Latex

Bulletin of Moldavian Academy of Sciences, ser. math. N3, (1996) 31-48

solv-int nlin.SI

Equations of geodesic deviation for the 3-dimensional and 4-dimensional Riemann spaces are discussed. Availability of wide classes of exact solutions of such equations, due to recent results for the matrix Schr\"odinger equation, is demonstrated. Particular classes of exact solutions for the geodesic deviation equation as well as for the Raychaudhuri and generalized Raychaudhuri equation are presented. Solutions of geodesic deviation equation for the Schwarzshild and Kasner metrics are found.

solv-int/9705004

Andrey V. Tsiganov

A.V. Tsiganov

On superintegrable systems closed to geodesic motion

22 pages, LaTeX

solv-int nlin.SI

In this work we consider superintegrable systems in the classical $r$-matrix method. By using other authomorphisms of the loop algebras we construct new superintegrable systems with rational potentials from geodesic motion on $R^{2n}$.

solv-int/9705005

Galina A. Korepanova

I.G. Korepanov

Some eigenstates for a model associated with solutions of tetrahedron equation. V. Two cases of string superposition

LaTeX, 7 pages

solv-int nlin.SI

In paper IV (solv-int/9704013) we have considered a string living in the infinite lattice that was, in a sense, generated by a "particle". Here we show how to construct multi-string eigenstates generated by several particles. It turns out that, at least in some cases, this allows us to bypass the difficulties of constructing multi-particle states. We also present and discuss the "dispersion relations" for our particles-strings.

solv-int/9705006

Dr. L. Bordag

L.A. Bordag (Leipzig) and V.S. Dryuma (Kishinev)

Investigation of dynamical systems using tools of the theory of invariants and projective geometry

18 pages, Latex, to appear in J. of Applied Mathematics (ZAMP)

10.1007/s000330050061

NTZ-Preprint 24/95, Leipzig, 1995

solv-int chao-dyn nlin.CD nlin.SI

The investigation of nonlinear dynamical systems of the type $\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to some ordinary differential equations of the second order in the form $y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main backbone of this investigation was provided by the theory of invariants developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century and the projective geometry of E. Cartan. In our work two, in some sense supplementary, systems are considered: the Lorenz system $\dot{x}=\sigma (y-x), \dot{y}=rx-y-zx,\dot{z}=xy-bz $ and the R\"o\ss ler system $\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+xz-cz.$. The invarinats for the ordinary differential equations, which correspond to the systems mentioned abouve, are evaluated. The connection of values of the invariants with characteristics of dynamical systems is established.

solv-int/9705007

Basile Grammaticos

Stephane Lafortune, Basil Grammaticos, Alfred Ramani

Constructing Integrable Third Order Systems:The Gambier Approach

14 pages, TEX FILE

Inverse Problems 14, 287-298 (1998)

10.1088/0266-5611/14/2/005

solv-int nlin.SI

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlev\'e method for ODE's and the singularity confinement method for mappings).

solv-int/9705008

Dita Petre

Petre Dita and Nicolae Grama

On Adomian's Decomposition Method for Solving Differential Equations

11 pages, Latex, no figure

solv-int nlin.SI

We show that with a few modifications the Adomian's method for solving second order differential equations can be used to obtain the known results of the special functions of mathematical physics. The modifications are necessary in order to take correctly into account the behaviour of the solutions in the neighborhood of the singular points.

solv-int/9705009

Leonid Bogdanov

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

Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies

43 pages, Latex

10.1063/1.532531

solv-int nlin.SI

Analytic-bilinear approach for construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. Resolution of functional equations also leads to the $\tau $-function and addition formulae to it.

solv-int/9705010

Renat Zhdanov

Renat Zhdanov (Institute of Mathematics, Kyiv.)

Integrability of Riccati equations and the stationary KdV equations

6 pages, LaTeX

solv-int nlin.SI

Using the S.Lie's infinitesimal approach we establish the connection between integrability of a one-parameter family of the Riccati equations and the stationary KdV hierarchy.

solv-int/9705011

Renat Zhdanov

Renat Zhdanov, Ihor Revenko and Wilhelm Fushchych (Institute of Mathematics, Kyiv)

Stationary mKdV hierarchy and integrability of the Dirac equations by quadratures

6 pages, LaTeX

10.1016/S0375-9601(98)00114-5

solv-int nlin.SI

Using the Lie's infinitesimal method we establish that the Dirac equation in one variable is integrable by quadratures if the potential V(x) is a solution of one of the equations of the stationary mKdV hierarchy.

solv-int/9705012

Q. P. Liu and M. Manas

Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy

26 pages, some formulae corrected. To appear in J. Nonlin. Sci

solv-int hep-th nlin.SI

We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the $n$-th Gel'fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy we get the line soliton, the lump solution and the Johnson-Thompson lump, and the corresponding determinant formulae for the non-linear superposition of several of them. For Kadomtsev-Petviashvili II apart from the line solitons we get singular rational solutions with its singularity set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel'fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases.

solv-int/9705013

Kanehisa Takasaki

Partha Guha and Kanehisa Takasaki

Dispersionless Hierarchies, Hamilton-Jacobi Theory and Twistor Correspondences

20 pages, latex, no figures

J. Geom. Phys. 25 (3-4) (1998), 326-340

10.1016/S0393-0440(97)00034-X

RIMS-1124

solv-int hep-th nlin.SI

The dispersionless KP and Toda hierarchies possess an underlying twistorial structure. A twistorial approach is partly implemented by the method of Riemann-Hilbert problem. This is however still short of clarifying geometric ingredients of twistor theory, such as twistor lines and twistor surfaces. A more geometric approach can be developed in a Hamilton-Jacobi formalism of Gibbons and Kodama. AMS Subject Classifiation (1991): 35Q20, 58F07,70H99

solv-int/9705014

Wen-Xiu Ma

W. X. Ma, R. K. Bullough, P. J. Caudrey and W. I. Fushchych

Time-Dependent Symmetries of Variable-Coefficient Evolution Equations and Graded Lie Algebras

11 pages, latex, to appear in J. Phys. A: Math. Gen

10.1088/0305-4470/30/14/023

solv-int nlin.SI

Polynomial-in-time dependent symmetries are analysed for polynomial-in-time dependent evolution equations. Graded Lie algebras, especially Virasoro algebras, are used to construct nonlinear variable-coefficient evolution equations, both in 1+1 dimensions and in 2+1 dimensions, which possess higher-degree polynomial-in-time dependent symmetries. The theory also provides a kind of new realisation of graded Lie algebras. Some illustrative examples are given.

solv-int/9705015

Wen-Xiu Ma

W. X. Ma, R. K. Bullough and P. J. Caudrey

Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP equations

19 pages, latex, to appear in J. Nonlinear Math. Phys

10.2991/jnmp.1997.4.3-4.6

solv-int nlin.SI

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with $m$ arbitrary time-dependent coefficients are obtained possessing symmetries involving $m$ arbitrary functions of time. A particular graded symmetry algebra for the modified KP equations is derived in this connection homomorphic to the Virasoro algebras.

solv-int/9705016

Kanehisa Takasaki

Kanehisa Takasaki

Dual Isomonodromic Problems and Whitham Equations

15 pages, latex, no figures. Several sentences are added in order to clarify the contents of Sections 5 and 6

Lett.Math.Phys. 43 (1998) 123-135

KUCP-0106

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

The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter $\epsilon$. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the Seiberg- Witten solutions of low energy supersymmetric gauge theories.

solv-int/9705017

Eugene Ferapontov

E.V. Ferapontov (Institute for Mathematical Modelling, Moscow)

Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences

22 pages, Latex

10.1088/0305-4470/30/19/023

solv-int nlin.SI

The conserved densities of hydrodynamic type system in Riemann invariants satisfy a system of linear second order partial differential equations. For linear systems of this type Darboux introduced Laplace transformations, generalising the classical transformations in the scalar case. It is demonstrated that Laplace transformations can be pulled back to the transformations of the corresponding hydrodynamic type systems. We discuss periodic Laplace sequences of with the emphasize on the simplest nontrivial case of period 2. For 3-component systems in Riemann invariants a complete discription of closed quadruples is proposed. They turn to be related to a special quadratic reduction of the (2+1)-dimensional 3-wave system which can be reduced to a triple of pairwize commuting Monge-Ampere equations. In terms of the Lame and rotation coefficients Laplace transformations have a natural interpretation as the symmetries of the Dirac operator, associated with the (2+1)-dimensional n-wave system. The 2-component Laplace transformations can be interpreted also as the symmetries of the (2+1)-dimensional integrable equations of Davey-Stewartson type. Laplace transformations of hydrodynamic type systems originate from a canonical geometric correspondence between systems of conservation laws and line congruences in projective space.

solv-int/9705018

Fritz Gesztesy

Fritz Gesztesy and Rudi Weikard

A Characterization of All Elliptic Solutions of the AKNS Hierarchy

LaTeX

solv-int nlin.SI

An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are elliptic of the second kind for n-th order ordinary differential equations with elliptic coefficients associated with a common period lattice. As by-products we offer a detailed Floquet analysis of Dirac-type differential expressions with periodic coefficients, specifically emphasizing algebro-geometric coefficients, and a constructive reduction of singular hyperelliptic curves and their Baker-Akhiezer functions to the nonsingular case.

solv-int/9705019

Fritz Gesztesy

W. Bulla, F. Gesztesy, H. Holden, and G. Teschl

Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies

LaTeX, to appear in Memoirs of the Amer. Math. Soc

Memoirs of the Amer. Math. Soc. 135/641, 1998

10.1090/memo/0641

solv-int math.SP nlin.SI

Combining algebro-geometric methods and factorization techniques for finite difference expressions we provide a complete and self-contained treatment of all real-valued quasi-periodic finite-gap solutions of both the Toda and Kac-van Moerbeke hierarchies. In order to obtain our principal new result, the algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we employ particular commutation methods in connection with Miura-type transformations which enable us to transfer whole classes of solutions (such as finite-gap solutions) from the Toda hierarchy to its modified counterpart, the Kac-van Moerbeke hierarchy, and vice versa.

solv-int/9706001

Harry Braden

H. W. Braden

R-Matrices and Generalized Inverses

11 pages, Latex

MS-97-006

solv-int nlin.SI

Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of $ad L$. For generic $L$ a generalized inverse (and indeed the Moore-Penrose inverse) is explicitly constructed. The R-matrices are in general momentum dependent and dynamical. The construction applies equally to Lax matrices with spectral parameter.

solv-int/9706002

John Harnad

J. Harnad and Alexander R. Its

Integrable Fredholm Operators and Dual Isomonodromic Deformations

PlainTeX 32gs

Commun.Math.Phys.226:497-530,2002

10.1007/s002200200614

CRM 2477 (1997)

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

The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed as nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M, whose Poisson quotient, under a parametric family of Hamiltonian group actions, is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem method is used to identify the auxiliary space M with the data defining the integral kernel of the resolvent operator at the endpoints of the curve segments. A second associated isomonodromic family of covariant derivative operators D_z is derived, having rank n=2m, and r finite regular singular points at the values of the exponents defining the kernel of K. This family is similarly embedded into the algebra Lgl_R(n) through a dual parametric family of Poisson quotients of M. The operators D_z are shown to be analogously associated to the integral operator obtained from K through a Fourier-Laplace transform.

solv-int/9706003

Mts

R. A. Sharipov (Bashkir State University, Ufa, Russia)

On the point transformations for the equation $y''= P + 3Qy' + 3R{y'}^2 + S{y'}^3$

AmS-TeX, Version 2.1, amsppt style, 36 pages

solv-int nlin.SI

For the equations $y''=P(x,y) + 3Q(x,y)y' + 3R(x,y){y'}^2 + S(x,y){y'}^3$ the problem of equivalence is considered. Some classical results are resumed in order to prepare the background for the study of special subclass of such equations, which arises in the theory of dynamical systems admitting the normal shift.

solv-int/9706004

J. vandeLeur

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

An analytic description of the vector constrained KP hierarchy

15 pages, Latex2e

10.1007/s002200050341

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

In this paper we give a geometric description in terms of the Grassmann manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the vector $k$-constrained KP hierarchy. We also show in a geometric way that these hierarchies are equivalent to Krichever's general rational reductions of the KP hierarchy.

solv-int/9706005

Jose Carlos Brunelli

J. C. Brunelli and A. Das

A Lax Description for Polytropic Gas Dynamics

9 pages, TeX

Phys.Lett. A235 (1997) 597-602

10.1016/S0375-9601(97)00708-1

solv-int hep-th nlin.SI

We give a Lax description for the system of polytropic gas equations. The special structure of the Lax function naturally leads to the two infinite sets of conserved charges associated with this system. We obtain closed form expressions for the conserved charges as well as the generating functions for them. We show how the study of these generating functions can naturally lead to the recursion relation between the conserved quantities as well as the higher order Hamiltonian structures.

solv-int/9706006

J. vandeLeur

Victor Kac, Johan van de Leur

The geometry of spinors and the multicomponent BKP and DKP hierarchies

46 pages, Latex2e

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

We develop a formalism of multicomponent BKP hierarchies using elementary geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence all their reductions like KdV, NLS, AKNS or DS) are reductions of the multicomponent BKP.

solv-int/9706007

Robert Milson

Robert Milson

On the Liouville transformation and exactly-solvable Schrodinger equations

16 pages, 6 figures

solv-int nlin.SI

The present article discusses the connection between exactly-solvable Schrodinger equations and the Liouville transformation. This transformation yields a large class of exactly-solvable potentials, including the exactly-solvable potentials introduced by Natanzon. As well, this class is shown to contain two new families of exactly solvable potentials.

solv-int/9706008

Yuly Billig

Yuly Billig

An Extension of the KdV Hierarchy Arising from a Representation of a Toroidal Lie Algebra

22 pages, plain tex, no figures

solv-int nlin.SI

In this article we show how to construct hierarchies of partial differential equations from the vertex operator representations of toroidal Lie algebras. In the smallest example - rank 2 toroidal cover of $sl_2$ - we obtain an extension of the KdV hierarchy. We use the action of the corresponding infinite-dimensional group to construct solutions for these non-linear PDEs.

solv-int/9706009

Pierre van Moerbeke

Pierre van Moerbeke

The spectrum of random matrices and integrable systems

17 pages, Latex, group21.sty

Group21, Physical applications and Mathematical aspects of Geometry, Groups and Algebras, Vol II, 835--852, Eds.: Doebner, Scherer, Schulte, World Scientific, Singapore, 1997

solv-int nlin.SI

What is the connection of random matrices with integrable systems? Is this connection really useful? Introducing apprpriate times in the distribution of the ensemble of matrices, one shows that the corresponding distribution of the eigenvalues satisfies the KP-equation, the 1-Toda lattice or the 2-Toda lattice, depending on the original distribution. The probability distribution also satisfies Virasoro type constraints, which contain a time-part and a boundary-part. These equations taken together lead to a system of PDE's for the distribution of the spectrum in terms of the boundary of the set, under consideration.

solv-int/9706010

Pierre van Moerbeke

Mark Adler and Pierre van Moerbeke

Matrix Integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials

50 pages, Latex

Duke Math Journal, 80, 863--911, 1995

solv-int nlin.SI

The relationship is made between matrix integrals, Toda master-symmetries, Virasoro constraints and orthogonal polynomials.

solv-int/9707001

Craig A. Tracy

Craig A. Tracy, Harold Widom

The Distribution of the Largest Eigenvalue in the Gaussian Ensembles

13 pages

in Calogero-Moser-Sutherland Models, eds. J.F. van Diejen and L. Vinet, CRM Series in Mathematical Physics 4, Springer-Verlag, New York, 2000, pp. 461-472

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

The focus of this survey paper is on the distribution function for the largest eigenvalue in the finite N Gaussian ensembles (GOE,GUE,GSE) in the edge scaling limit of N->infinity. These limiting distribution functions are expressible in terms of a particular Painleve II function. Comparisons are made with finite N simulations as well as a discussion of the universality of these distribution functions.

solv-int/9707002

Kirill Vaninsky

K.L. Vaninsky

Trace Formula for a System of Particles with Elliptic Potential

corrected version

Pacific J. Math, vol. 189, no. 1, 159--178 (1999)

solv-int nlin.SI

We consider classical particles on the line with the Weierstrass $\wp$ function as potential. This system parameterizes special solutions of the KP equation. We derive the trace formula which relates the Hamiltonian of the particle system to the residues of some Abelian differential (meromorphic one-form) on the spectral curve. Such formula is important for the construction action-angle variables and study invariant Gibbs' states.

solv-int/9707003

Shigeki Matsutani

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

AMS-Tex Use

10.1088/0305-4470/31/11/017

solv-int nlin.SI

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

solv-int/9707004

Vadim Vereschagin

V.L.Vereschagin

Nonlinear Quasiclassics and the Painlev\'e Equations

5 pp., Latex

solv-int nlin.SI

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

solv-int/9707005

Alexander Turbiner

Marcos Rosenbaum, Alexander Turbiner and Antonio Capella

Solvability of the G_2 Integrable System

18 pages, LaTeX, some minor typos corrected

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

10.1142/S0217751X98001815

Mexico ICN-UNAM 97-05

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

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

solv-int/9707006

Shigeki Matsutani

Quantum Coupled Nonlinear Schr\"odinger System with Different Masses

AMS-Tex Use

solv-int nlin.SI

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

solv-int/9707007

Shigeki Matsutani

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

AMS-Tex Use

10.1088/0305-4470/31/15/021

solv-int nlin.SI

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

solv-int/9707008

V. E. Vekslerchik

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

Functional representation of the Ablowitz-Ladik hierarchy

15 pages, LaTeX

10.1088/0305-4470/31/3/018

solv-int nlin.SI

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

solv-int/9707009

Fritz Gesztesy

Fritz Gesztesy and Ratnam Ratnaseelan

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

LaTeX, submitted to Reviews in Mathematical Physics

10.1142/S0129055X98000112

solv-int nlin.SI

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

solv-int/9707010

Fritz Gesztesy

Fritz Gesztesy and Helge Holden

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

LaTeX; emphasis put on the mKdV hierarchy

solv-int hep-th nlin.SI

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

solv-int/9707011

Tamara Grava

T. Grava

Bifurcation diagram of a one-parameter family of dispersive waves

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

solv-int nlin.SI

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

solv-int/9707012

Q. P. Liu

The Constrained MKP Hierarchy and the Generalized Kupershmidt-Wilson Theorem

9 pages, LaTex

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

solv-int hep-th nlin.SI

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

solv-int/9707013

Luis Eduardo Saltini

L.E. Saltini, A. Zadra

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

10 pages, LaTeX, no figures

solv-int hep-th nlin.SI

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

solv-int/9707014

Shen-Jane Chang

Jiin-Chang Shaw and Ming-Hsien Tu

The constrained modified KP hierarchy and the generalized Miura transformations

8 pages, latex, no figures

J. Phys. A30 (1997) L725

10.1088/0305-4470/30/21/004

solv-int nlin.SI

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

solv-int/9707015

Vsevolod Adler

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

B\"acklund transformation for the Krichever-Novikov equation

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

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

10.1155/S1073792898000014

solv-int nlin.SI

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

solv-int/9707016

Adrian-Stefan Carstea

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

Continuum limit of nonlinear discrete systems with long range interaction potentials

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

IFIN-HH, F.T. 430-1997

solv-int nlin.SI

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

solv-int/9707017

Gregorio Falqui

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

Bihamiltonian Reductions and W_n Algebras

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

10.1016/S0393-0440(97)00060-0

SISSA 72/97/FM

solv-int nlin.SI

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

solv-int/9708001

Atsushi Nagai

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

Two-dimensional soliton cellular automaton of deautonomized Toda-type

11 pages, LaTeX file

10.1016/S0375-9601(97)00591-4

solv-int nlin.SI

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

solv-int/9708002

Francois Delduc

F. Delduc, L. Feher, L. Gallot

Nonstandard Drinfeld-Sokolov reduction

19 pages, LaTeX file

10.1088/0305-4470/31/25/006

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

solv-int hep-th nlin.SI

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

solv-int/9708003

Sedra Moulay Brahim

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

Three Graded Modified Classical Yang-Baxter Equations and Integrable Systems

22 pages, Revtex

solv-int hep-th nlin.SI

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

solv-int/9708004

Vladimir Gerdjikov

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

Criterion and Regions of Stability for Quasi-Equidistant Soliton Trains

14 pages, LaTeX (revtex style), 5 figures

INRNE-TH-97-4

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

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

solv-int/9708005

Q. P. Liu

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

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

Phys. Lett. A, 235 (1997) 335

10.1016/S0375-9601(97)00638-5

solv-int hep-th nlin.SI

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

solv-int/9708006

Jarmo Hietarinta

J. Hietarinta

Introduction to the Hirota bilinear method

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

10.1007/BFb0113694

solv-int nlin.SI

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

solv-int/9708007

E. Sklyanin

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

Generating function of correlators in the sl_2 Gaudin model

16 pages, LaTex 209, macros included

Letters in Mathematical Physics 47 (1999) 275-292

10.1023/A:1007585716273

solv-int nlin.SI

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

solv-int/9708008

Ming-Hsien Tu

Jiin-Chang Shaw and Ming-Hsien Tu

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

14 pages, RevTex, no figures, typos corrected

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

10.1142/S0217751X98001384

solv-int nlin.SI

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

solv-int/9708009

Stephane Gourmelen

F. Gieres, S. Gourmelen

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

29 pages, LaTeX

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

10.1063/1.532446

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

solv-int hep-th nlin.SI

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

solv-int/9708010

Jarmo Hietarinta

Jarmo Hietarinta

Pure quantum integrability

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

10.1016/S0375-9601(98)00535-0

solv-int nlin.SI

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

solv-int/9709001

Alexander V. Shapovalov

Ya. V. Lisitsyn and A. V. Shapovalov

Separation of variables via integral transformations

14 LaTex pages

solv-int nlin.SI

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

solv-int/9709002

Park Q.-Han

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

Field Theory for Coherent Optical Pulse Propagation

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

10.1103/PhysRevA.57.4621

SNUTP 97-110

solv-int nlin.SI

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