# Journal of Nonlinear Mathematical Physics

Volume 8, Issue Supplement, February 2001

### Proceedings of the 13th Workshop NEEDS'99: Nonlinear Evolution Equations and Dynamical Systems

**Research Article**

## 1. Complex Angle Variables for Constrained Integrable Hamiltonian Systems

S. Abenda, Yu Fedorov

Pages: 1 - 4

We propose Dirac formalism for constraint Hamiltonian systems as an useful tool for the algebro-geometrical and dynamical characterizations of a class of integrable systems, the so called hyperelliptically separable systems. As a model example, we apply it to the classical geodesic flow on an ellipsoid.

**Research Article**

## 2. Taming Spatiotemporal Chaos by Impurities in the Parametrically Driven Damped Nonlinear Schrödinger Equation

N.V. Alexeeva, I.V. Barashenkov, G.P. Tsironis

Pages: 5 - 12

Solitons of the parametrically driven, damped nonlinear Schrödinger equation become unstable and seed spatiotemporal chaos for sufficiently large driving amplitudes. We show that the chaos can be suppressed by introducing localized inhomogeneities in the parameters of the equation. The pinning of the...

**Research Article**

## 3. Nonlinear Models in Quantum Optics through Quantum Algebras

Angel Ballesteros, Sergey Chumakov

Pages: 13 - 17

The suq(2) algebra is shown to provide a natural dynamical algebra for some nonlnear models in Quantum Optics. Applications to the computation of eigenvalues and eigenvectors for the Hamiltonian describing second harmonics generation are proposed.

**Research Article**

## 4. Two-Photon Algebra and Integrable Hamiltonian Systems

Angel Ballesteros, Francisco J. Herranz

Pages: 18 - 22

The two-photon algebra h6 is used to define an infinite class of N-particle Hamiltonian systems having (N -2) additional constants of the motion in involution. By constrution, all these systems are h6-coalgebra invariant. As a straightforward application, a new family of (quasi)integrable N-dimensional...

**Research Article**

## 5. Peakon-Antipeakon Interaction

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

Pages: 23 - 27

Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon.

**Research Article**

## 6. Response Functions of Spiral Wave Solutions of the Complex GinzburgLandau Equation

I.V. Biktasheva, V.N. Biktashev

Pages: 28 - 34

Dynamics of spiral waves in perturbed two-dimensional autowave media can be dscribed asymptotically in terms of Aristotelean dynamics. We apply this general thory to the spiral waves in the Complex GinzburgLandau equation (CGLE). The RFs are found numerically. In this work, we study the dependence of...

**Research Article**

## 7. Threshold Behavior for Nonlinear Wave Equations

Piotr Bizoń

Pages: 35 - 41

In this brief contribution, which is based on my talk at the conference, I discuss the dynamics of solutions of nonlinear wave equations near the threshold of singularity formation. The heuristic picture of threshold behavior is first presented in a general setting and then illustrated with three examples.

**Research Article**

## 8. Sp(2) Quantization of Solitonic Theories

V. Calian

Pages: 42 - 47

The gauge-field theoretical formulation of solitonic theories is quantized by using an extended version of the BRST Sp(2) symmetric formalism. The proposed method is based on a modified triplectic geometry which allows us to incorporate the linear and/or nonlinear global symmetries of the model and to...

**Research Article**

## 9. Bilinear Approach to Supersymmetric KdV Equation

A.S. Carstea

Pages: 48 - 52

Extending the gauge-invariance principle for functions of the standard bilinear fomalism to the supersymmetric case, we define N = 1 supersymmetric Hirota operators. Using them, we bilinearize SUSY KdV equation. The solution for multiple collisions of super-solitons is given.

**Research Article**

## 10. A Hidden Hierarchy for GD4

Paolo Casati

Pages: 53 - 57

A reduction process to construct hidden hierarchies corresponding to the Gelfand Dickey ones is outlined in a specific example, not yet treated in the literature.

**Research Article**

## 11. Algebraic Linearization of Hyperbolic RuijsenaarsSchneider Systems

R. Caseiro, J.P. Françoise

Pages: 58 - 61

In this article, we present an explicit linearization of dynamical systems of RuijsenaarSchneider (RS) type and of the perturbations introduced by F Calogero [2] of these systems with all orbits periodic of the same period. The existence of this linearization and its algebraic nature relies on the dynamical...

**Research Article**

## 12. On Huygens' Principle for Dirac Operators and Nonlinear Evolution Equations

Fabio A.C.C. Chalub, Jorge P. Zubelli

Pages: 62 - 68

We exhibit a class of Dirac operators that possess Huygens' property, i.e., the support of their fundamental solutions is precisely the light cone. This class is obtained by considering the rational solutions of the modified Korteweg-de Vries hierarchy.

**Research Article**

## 13. CalogeroMoser Systems and Super YangMills with Adjoint Matter

Eric d'Hoker, D.H. Phong

Pages: 69 - 78

We review the construction of Lax pairs with spectral parameter for twisted and utwisted elliptic Calogero-Moser systems defined by a general simple Lie algebra G, and the corresponding solution of N = 2 SUSY G Yang-Mills theories with a hypermultplet in the adjoint representation of G.

**Research Article**

## 14. Amplitude-Dependent Oscillations in Gases

O.L. de Lange, J. Pierrus

Pages: 79 - 81

We consider the following question: Suppose part of the boundary of a cavity contaiing a gas is set into oscillation, the damping in the boundary being small. What is the nature of the oscillations in the gas? We treat the low-frequency limit (wavelength much greater than dimensions of the cavity). Experiment...

**Research Article**

## 15. Inhomogeneous Burgers Lattices

S. De Lillo, V.V. Konotop

Pages: 82 - 87

We study statistical properties of inhomogeneous Burgers lattices which are solved by the discrete ColeHopf transformation. Using exact solutions we investigate effect of various kinds of noise on the dynamics of solutions.

**Research Article**

## 16. Asymptotic Lattices and W-Congruences in Integrable Discrete Geometry

Adam Doliwa

Pages: 88 - 92

The asymptotic lattices and their transformations are included into the theory of quadrilateral lattices.

**Research Article**

## 17. Quantization of Soliton Cellular Automata

Demosthenes Ellinas, Elena P. Papadopoulou, Yiannis G. Saridakis

Pages: 93 - 99

A method of quantization of classical soliton cellular automata (QSCA) is put forward that provides a description of their time evolution operator by means of quantum cicuits that involve quantum gates from which the associated Hamiltonian describing a quantum chain model is constructed. The intrinsic...

**Research Article**

## 18. Brownian Motion on a Smash Line

Demosthenes Ellinas, Ioannis Tsohantjis

Pages: 100 - 105

Brownian motion on a smash line algebra (a smash or braided version of the algebra resulting by tensoring the real line and the generalized paragrassmann line algebras), is constructed by means of its Hopf algebraic structure. Further, statistical moments, non stationary generalizations and its diffusion...

**Research Article**

## 19. Painlevé Analysis and Singular Manifold Method for a (2 + 1) Dimensional Non-Linear Schrödinger Equation

P.G. Estévez, G.A. Hernáez

Pages: 106 - 111

The real version of a (2 + 1) dimensional integrable generalization of the nonlinear Schrödinger equation is studied from the point of view of Painlevé analysis. In this way we find the Lax pair, Darboux transformations and Hirota's functions as well as solitonic and dromionic solutions from an iterative...

**Research Article**

## 20. Analysis of Timing Jitter for Ultrashort Soliton Communication Systems Using Perturbation Methods

Margarida Facão, Mário Ferreira

Pages: 112 - 117

We analyse timing jitter of ultrashort soliton systems taking into account the major higher order effects, namely, intrapulse Raman scattering and third order dispersion and using adiabatic perturbation theory. We obtain an expression for the soliton arrival time variance that depends on the quintic...

**Research Article**

## 21. Bihamiltonian Geometry and Separation of Variables for Toda Lattices

Gregorio Falqui, Franco Magri, Marco Pedroni

Pages: 118 - 127

We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical HamiltonJacobi method in the so-called DarbouxNijenhuis coordinates.

**Research Article**

## 22. Asymptotic Solutions of the Whitham Equations

T. Grava

Pages: 128 - 132

We extend a previous result, namely we show that the solution of the Whitham equtions is asymptotically self-similar for generic monotone polynomial initial data with smooth perturbation.

**Research Article**

## 23. KAM Theorem for the Nonlinear Schrödinger Equation

Benoît Grébert, Thomas Kappeler

Pages: 133 - 138

We prove the persistence of finite dimensional invariant tori associated with the dfocusing nonlinear Schrödinger equation under small Hamiltonian perturbations. The invariant tori are not necessarily small.

**Research Article**

## 24. Beyond Nonlinear Schrödinger Equation Approximation for an Anharmonic Chain with Harmonic Long Range Interactions

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

Pages: 139 - 144

Multi-scales method is used to analyze a nonlinear differential-difference equation. In the order 3 the NLS eq. is found to determine the space-time evolution of the leading amplitude. In the next order this has to satisfy a complex mKdV eq. (the next in the NLS hierarchy) in order to eliminate secular...

**Research Article**

## 25. Symplectic Structure of the Painlevé Test

Jishan Hu, Min Yan

Pages: 145 - 148

In this note, we present a result to show that the symplectic structures have been naturally encoded into the Painlevé test. In fact, for every principal balance, there is a symplectic change of dependent variables near movable poles.

**Research Article**

## 26. Some Recent Results on Integrable Bilinear Equations

Xing-Biao Hu, Hon-Wah Tam

Pages: 149 - 155

This paper shows that several integrable lattices can be transformed into coupled biliear differential-difference equations by introducing auxiliary variables. By testing the Bäcklund transformations for this type of coupled bilinear equations, a new integrable lattice is found. By using the Bäcklund...

**Research Article**

## 27. Singular Solution of the Liouville Equation under Perturbation

L.A. Kalyakin

Pages: 156 - 160

The Cauchy problem for the Liouville equation with a small perturbation is considered. We are interested in the asymptotics of the perturbed solution under the assumption that one has singularity. The main goal is to study both the asymptotic approximation of the singular lines and the asymptotic approximation...

**Research Article**

## 28. Nonlinear Gauge Transformation for a Quantum System Obeying an Exclusion-Inclusion Principle

G. Kaniadakis, A. Lavagno, P. Quarati, A.M. Scarfone

Pages: 161 - 165

We introduce a nonlinear and noncanonical gauge transformation which allows the rduction of a complex nonlinearity, contained in a Schrödinger equation, into a real one. This Schrödinger equation describes a canonical system, whose kinetics is governed by a generalized Exclusion-Inclusion Principle....

**Research Article**

## 29. On Integrable Discretization of the Inhomogeneous Volterra Lattice

V.V. Konotop

Pages: 166 - 171

An integrable discretization of an inhomogeneous Volterra lattice is introduced. Some aspects of dynamics of a soliton affected by a random force are discussed.

**Research Article**

## 30. Some Fourth-Order Ordinary Differential Equations which Pass the Painlevé Test

Nicolai A. Kudryashov

Pages: 172 - 177

An approach to the Painlevé analysis of fourth-order ordinary differential equations is presented. Some fourth-order ordinary differential equations which pass the Painlevé test are found.

**Research Article**

## 31. Construction of Variable Mass Sine-Gordon and Other Novel Inhomogeneous Quantum Integrable Models

Anjan Kundu

Pages: 178 - 182

The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model,...

**Research Article**

## 32. Nonlinear Renormalization Group Flow and Optimization

Sen-Ben Liao

Pages: 183 - 187

Renormalization group flow equations for scalar 4 are generated using smooth smearing functions. Numerical results for the critical exponent in d = 3 are caculated by polynomial truncation of the blocked potential. It is shown that the covergence of with the order of truncation can be improved by fine...

**Research Article**

## 33. Torsional Travelling Waves in DNA

Tomasz Lipniacki

Pages: 188 - 194

The simple model of the non-linear DNA dynamics [4] is pursued in order to study the local untwisting of DNA double helix. It is shown how the advancing RNA polymerase may force the motion of the torsional solitary wave along DNA.

**Research Article**

## 34. A New Class of Integrable Newton Systems

Hans Lundmark

Pages: 195 - 199

A new class of integrable Newton systems in Rn is presented. They are characterized by the existence of two quadratic integrals of motion of so-called cofactor type, and are therefore called cofactor pair systems. This class includes as special cases conservative systems separable in elliptic or parabolic...

**Research Article**

## 35. Integrability and Non-Integrability of Planar Hamiltonian Systems of Cosmological Origin

Andrzej J. Maciejewski, Marek Szydlowski

Pages: 200 - 206

We study the problem of non-integrability (integrability) of cosmological dynamcal systems which are given in the Hamiltonian form with indefinite kinetic energy form T = 1 2 g(v, v), where g is a two-dimensional pseudo-Riemannian metric with a Lorentzian signature (+, -), and v TxM is a tangent vector...

**Research Article**

## 36. A Simple Family of Non-Local Poisson Brackets

Oscar McCarthy

Pages: 207 - 211

A dispersionless integrable system with repeated eigenvalues is presented. For N 3 components the system has no local Hamiltonian structure. Infinitely many simple compatible non-local Hamiltonian structures are given, using a result of Ferapontov.

**Research Article**

## 37. WhithamToda Hierarchy in the Laplacian Growth Problem

M. Mineev-Weinstein, A. Zabrodin

Pages: 212 - 218

The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version...

**Research Article**

## 38. Topological Solitons in Discrete Space-Time as the Model of Fermions

A.I. Musienko

Pages: 219 - 224

In the present work we discuss arguments in favour of the view that massive fermions represent dislocations (i.e. topological solitons) in discrete space-time, with Burgers vectors parallel to the axis of time. If we assume that the symmetrical parts of tensors of distortions (i.e. derivatives of atomic...

**Research Article**

## 39. Self-Dual ChernSimons Solitons and Quantum Potential

Oktay K. Pashaev, Jyh-Hao Lee

Pages: 225 - 229

An influence of the quantum potential on the ChernSimons solitons leads to quatization of the statistical parameter = me2 /g, and the quantum potential strenght s = 1 - m2 . A new type of exponentially localized ChernSimons solitons for the Bloch electrons near the hyperbolic energy band boundary are...

**Research Article**

## 40. Soliton Resonances, Black Holes and Madelung Fluid

Oktay K. Pashaev, Jyh-Hao Lee

Pages: 230 - 234

The reaction-diffusion system realizing a particular gauge fixing condition of the JackiwTeitelboim gravity is represented as a coupled pair of Burgers equations with positive and negative viscosity. For acoustic metric in the Madelung fluid representtion the space-time points where dispersion change...

**Research Article**

## 41. The Solution of Chain of Quantum Kinetic Equations of Bogoliubov for Bose Systems, Interacting by Delta Potential

M. Yu. Rasulova

Pages: 235 - 239

The BBGKY's chain of quantum kinetic equations that describes the system of Bose particles interacting by delta potential is solved by the operator method with the help of nonlinear Schrödinger's equations. The solution of the chain is defined in terms of the Bethe ansatz.

**Research Article**

## 42. Reflectionless Analytic Difference Operators (AOs): Examples, Open Questions and Conjectures

S.N.M. Ruijsenaars

Pages: 240 - 248

We present a scenario concerning the existence of a large class of reflectionless seladjoint analytic difference operators. In order to exemplify this scenario, we summarize our results on reflectionless self-adjoint difference operators of relativistic CalogerMoser type.

**Research Article**

## 43. Elbow Scattering and Boundary Value Problems of NLPDE

Pierre C. Sabatier

Pages: 249 - 253

We use Lax equations to define a scattering problem on an infinite elbow shaped line of the (x, t) plane. The evolution of scattering coefficients when the elbow is translated in the plane shows how convenient scannings may reconstruct the solution V (x, t) of the nonlinear equation associated to the...

**Research Article**

## 44. Universal Lax Pair for Generalised CalogeroMoser Models

R. Sasaki

Pages: 254 - 260

In this talk we introduce generalised CalogeroMoser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m),...

**Research Article**

## 45. A Finite Dimensional Completely Integrable System Associated with the WKI- and Heisenberg Hierarchies

Rudolf Schmid, Taixi Xu, Zhongding Li

Pages: 261 - 265

We consider the following spectral problem y1 y2 x = -w u v w y1 y2 M y1 y2 , (1) where u, v, w are smooth functions. It produces a hierarchy of evolution equations with an arbitrary function Am-1. This hierarchy includes the WKI [8] and Heiseberg [7] hierarchies by properly selecting the special function...

**Research Article**

## 46. The KaldorKalecki Model of Business Cycle as a Two-Dimensional Dynamical System

Marek Szydłowski, Adam Krawiec

Pages: 266 - 271

In the paper we analyze the KaldorKalecki model of business cycle. The time dlay is introduced to the capital accumulation equation according to Kalecki's idea of delay in investment processes. The dynamics of this model is represented in terms of time delay differential equation system. In the special...

**Research Article**

## 47. The Investigation into New Equations in (2 + 1) Dimensions

Kouichi Toda, Song-Ju Yu

Pages: 272 - 277

First of all, we show the existence of the Lax pair for the Calogero Korteweg-de Vries(CKdV) equation. Next we modify T operator that is one of the Lax pair for the CKdV equation for the search of the (2 + 1)-dimensional case and propose a new equation in (2 + 1) dimensions. We call it the (2 + 1)-dimensional...

**Research Article**

## 48. Change of the Time for the Toda Lattice

A.V. Tsiganov

Pages: 278 - 282

For the Toda lattice we consider properties of the canonical transformations of the extended phase space, which preserve integrability. At the special values of integrals of motion the integral trajectories, separated variables and the action variables are invariant under change of the time. On the other...

**Research Article**

## 49. On Representation of the PQ Pair Solution at the Singular Point Neighborhood

N.V. Ustinov

Pages: 283 - 288

The compatible expansion in series of solutions of both the equations of PQ pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition...

**Research Article**

## 50. A Dynamical System: Mars and its Satellite

Piotr Wąż

Pages: 289 - 293

A description of the most accurate analytical theory of the motion of Phobos, so far constructed, is presented. Several elements of the gravitational field of Mars, gravitational interactions between Phobos and Mars, Deimos and Jupiter, as well as tidal effects due to the interaction between the Sun...

**Research Article**

## 51. A Pivotal Model for the (1 + 1)-Dimensional Heisenberg and Sigma Models

A.E. Winn

Pages: 294 - 299

An integrable interpolative (Pivotal) model for the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models is proposed and some solutions classifiable by an integer winding number examined.

**Research Article**

## 52. Quasi-Bi-Hamiltonian Systems Obtained from Constrained Flows

Yunbo Zeng

Pages: 300 - 304

An infinite number of families of quasi-bi-Hamiltonian (QBH) systems can be costructed from the constrained flows of soliton equations. The Nijenhuis coordinates for the QBH systems are proved to be exactly the same as the separation variables introduced by the Lax matrices for the constrained flows.

**Research Article**

## 53. Dynamics of Soliton-Like Excitations in a Chain of a Polymer Crystal: Influence of Neighbouring Chains Mobility

Elena A. Zubova, N.K. Balabaev

Pages: 305 - 311

We investigate influence of mobility of neighbouring chains on dynamics of soliton-like excitations in a chain of the simplest polymer crystal (polyethylene in the "united atoms" approximation) using molecular dynamics simulation. We present results for point-like structural defects: static and moving...