# Journal of Nonlinear Mathematical Physics

Volume 15, Issue supplement 3, October 2008

### Nonlinear Evolution Equations and Dynamical Systems 2007

Papers resulting from the NEEDS-07 workshop held in L'Ametlla Mar, Spain

**Research Article**

## 2. Quantum Integrability of the Dynamics on a Group Manifold

V. Aldaya, M. Calixto, J. Guerrero, F.F. Lopez-Ruiz

Pages: 1 - 12

We study the dynamics of a particle moving on the SU(2) group manifold. An exact quantization of this system is accomplished by finding the unitary and irreducible representations of a finite-dimensional Lie subalgebra of the whole Poisson algebra in phase space. In fact, the basic position and momentum...

**Research Article**

## 3. Existence of Periodic Solutions of a Type of Nonlinear Impulsive Delay Differential Equations with a Small Parameter

Jehad O. Alzabut

Pages: 13 - 21

The Banach fixed point theorem is used to prove the existence of a unique( w) periodic solution of a new type of nonlinear impulsive delay differential equation with a small parameter.

**Research Article**

## 4. Boundary Algebra and Exact Solvability of the Asymmetric Exclusion Process

Boyka Aneva

Pages: 22 - 33

We consider a lattice driven diffusive system withUq(su(2)) invariance in the bulk. Within the matrix product states approach the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra. Boundary processes amount to the appearance of parameter dependent...

**Research Article**

## 5. Solutions of Adler's Lattice Equation Associated with 2-Cycles of the Bäcklund Transformation

James Atkinson, Frank Nijhoff

Pages: 34 - 42

The Bäcklund transformation (BT) of Adler's lattice equation is inherent in the equation itself by virtue of its multidimensional consistency. We refer to a solution of the equation that is related to itself by the composition of two BTs (with different Bäcklund parameters) as a 2-cycle of the BT. In...

**Research Article**

## 6. Superintegrable Anharmonic Oscillators on N-dimensional Curved Spaces

Angel Ballesteros, Alberto Enciso, Francisco José Herranz, Orlando Ragnisco

Pages: 43 - 52

The maximal superintegrability of the intrinsic harmonic oscillator potential on N-dimensional spaces with constant curvature is revisited from the point of view of sl(2)-Poisson coalgebra symmetry. It is shown how this algebraic approach leads to a straightforward definition of a new large family of...

**Research Article**

## 7. The Motion of a Gyrostat in a Central Gravitational Field: Phase Portraits of an Integrable Case

M.C. Balsas, E.S. Jiménez, J.A. Vera

Pages: 53 - 64

In this paper we describe the Hamiltonian dynamics, in some invariant manifolds of the mo- tion of a gyrostat in Newtonian interaction with a spherical rigid body. Considering a first integrable approximation of this roto-translatory problem, by means of Liouville-Arnold the- orem and some specifics...

**Research Article**

## 8. Existence of Dark Soliton Solutions of the Cubic Nonlinear Schrödinger Equation with Periodic Inhomogeneous Nonlinearity

Juan Belmonte-Beitia, Pedro J Torres

Pages: 65 - 72

In this paper, we give a proof of the existence of stationary dark soliton solutions of the cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, together with an analytical example of a dark soliton.

**Research Article**

## 9. Nonlinear-Integral-Equation Construction of Orthogonal Polynomials

Carl M. Bender, E. Ben-Naim

Pages: 73 - 80

The nonlinear integral equation P(x) = dyw(y)P(y)P(x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions Pn(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations...

**Research Article**

## 10. Symmetry Analysis and Solutions for a Generalization of a Family of BBM Equations

M.S. Bruzon, M. L. Gandarias, J. C. Camacho

Pages: 81 - 90

In this paper, the family of BBM equation with strong nonlinear dispersive B(m,n) is considered. We apply the classical Lie method of infinitesimals. The symmetry reductions are derived from the optimal system of subalgebras and lead to systems of ordinary differential equations. We obtain for special...

**Research Article**

## 11. Coupling Nonlinear Sigma-Matter to Yang-Mills Fields: Symmetry Breaking Patterns

M. Calixto, V. Aldaya, F. F. Lopez-Ruiz, E. Sanchez-Sastre

Pages: 91 - 101

We extend the traditional formulation of Gauge Field Theory by incorporating the (non- Abelian) gauge group parameters (traditionally simple spectators) as new dynamical (nonlinearsigma- model-type) fields. These new fields interact with the usual Yang-Mills fields through a generalized minimal coupling...

**Research Article**

## 12. Symbolic Dynamics and Chaotic Synchronization in Coupled Duffing Oscillators

Acilina Caneco, Clara Gracio, J. Leonel Rocha

Pages: 102 - 111

In this work we discuss the complete synchronization of two identical double-well Duffing oscillators unidirectionally coupled, from the point of view of symbolic dynamics. Working with Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the...

**Research Article**

## 13. Remarks on the Waterbag Model of Dispersionless Toda Hierarchy

Jen-Hsu Chang

Pages: 112 - 123

We construct the free energy associated with the waterbag model of dToda. Also, relations for conserved densities are investigated.

**Research Article**

## 14. Approximation of Solitons in the Discrete NLS Equation

Jesus Cuevas, Guillaume James, Panayotis G. Kevrekidis, Boris A. Malomed, Bernardo Sanchez-Rey

Pages: 124 - 136

We study four different approximations for finding the profile of discrete solitons in the one- dimensional Discrete Nonlinear Schrödinger (DNLS) Equation. Three of them are discrete approximations (namely, a variational approach, an approximation to homoclinic orbits and a Green-function approach),...

**Research Article**

## 15. Interchannel Soliton Collisions in Periodic Dispersion Maps in the Presence of Third Order Dispersion

Francisco J. Diaz-Otero, Pedro Chamorro-Posada

Pages: 137 - 143

We study the effects of third order dispersion (TOD) on the collision of wavelength division multiplexed solitons in periodic dispersion maps. The analysis is based on a proposed ODE model obtained using the variational method which takes into account third order dispersion. The impact of TOD on the...

**Research Article**

## 16. Evolution of Kink Network in Inhomogenous Systems

T. Dobrowolski, P. Tatrocki

Pages: 144 - 154

The purpose of this report is to show the influence of imperfections on creation and evolution of a kink network. Our main finding is a mechanism for reduction of the kinetic energy of kinks which works in both the overdamped and underdamped regimes. This mechanism reduces mobility of kinks and therefore...

**Research Article**

## 17. Partially Solvable Spin Chains and QES Spin Models

A. Enisco, F. Finkel, A. Gonzalez-Lopez, M.A. Rodriguez

Pages: 155 - 165

In this paper we prove an extension of the usual freezing trick argument which can be applied to a number of quasi-exactly solvable spin models of CalogeroÂSutherland type. In order to illustrate the application of this method we analyze a partially solvable spin chain presenting near-neighbors interactions...

**Research Article**

## 18. Lump Solutions for PDE's: Algorithmic Construction and Classification

P.G. Estévez, J. Prada

Pages: 166 - 175

In this paper we apply truncated Painleve expansions to the Lax pair of a PDE to derive gauge Backlund transformations of this equation. It allows us to construct an algorithmic method to derive solutions by starting from the simplest one. Actually, we use this method to obtain an infinite set of lump...

**Research Article**

## 19. Harmonic Maps Between Noncompact Manifolds

Anestis Fotiadis

Pages: 176 - 184

We describe the problem of finding a harmonic map between noncompact manifold. Given some sufficient conditions on the domain, the target and the initial map, we prove the existence of a harmonic map that deforms the given map.

**Research Article**

## 20. Nonclassical Potential System Approach for a Nonlinear Diffusion Equation

M.L. Gandarias, M.S. Bruzon

Pages: 185 - 196

In this paper we consider a class of equations that model diffusion. For some of these equations nonclassical potential symmetries are derived by using a modified system approach. These symmetries allow us to increase the number of exact known solutions. These solutions are unobtainable from classical...

**Research Article**

## 21. On the Caudrey-Beals-Coifman System and the Gauge Group Action

Georgi G. Grahovski, Marissa Condon

Pages: 197 - 208

The generalized ZakharovÂShabat systems with complex-valued Cartan elements and the systems studied A.V. Mikhailov, and later on by Caudrey, Beals and Coifman (CBC systems), and their gauge equivalent are studied. This includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent...

**Research Article**

## 22. Solitary Waves in a Madelung Fluid Description of Derivative NLS Equations

Dan Grecu, Alexandru Tudor Grecu, Anca Visinescu, Renato Fedele, Sergio De Nicola

Pages: 209 - 219

Recently using a Madelung fluid description a connection between envelope-like solutions of NLS-type equations and soliton-like solutions of KdV-type equations was found and investigated by R. Fedele et al. (2002). A similar discussion is given for the class of derivative NLS-type equations. For a motion...

**Research Article**

## 23. Justification of an Asymptotic Expansion at Infinity

Leonid Kalyakin

Pages: 220 - 226

A family of asymptotic solutions at infinity for a system of ordinary differential equations is considered. Existence of exact solutions which have these asymptotics is proved.

**Research Article**

## 24. Corrugated Surfaces with Slow Modulation and Quasiclassical Weierstrass Representation

B.G. Konopelchenko

Pages: 227 - 236

Quasiclassical generalized Weierstrass representation for highly corrugated surfaces in R3 with slow modulation is proposed. Integrable deformations of such surfaces are described by the dispersionless modified Veselov-Novikov hierarchy.

**Research Article**

## 25. Changing Solitons in Classical & Quantum Integrable Defect and Variable Mass Sine-Gordon Model

Anjan Kundu

Pages: 237 - 250

Sine-Gordon (SG) models with position dependent mass or with isolated defects appear in many physical situations, ranging from fluxon or semi-fluxon in nonuniform Josephson junction to spin-waves in quantum spin chain with variable coupling or DNA solitons in the active promoter region. However such...

**Research Article**

## 26. Fedosov Quantization in White Noise Analysis

Rémi Léandre

Pages: 251 - 263

We define the deformation quantization in the Fedosov sense for a limit model of Taubes in white noise analysis.

**Research Article**

## 27. Continuous and Discontinuous Piecewise Linear Solutions of the Linearly Forced Inviscid Burgers Equation

Hans Lundmark, Jacek Szmigielski

Pages: 264 - 276

We study a class of piecewise linear solutions to the inviscid Burgers equation driven by a linear forcing term. Inspired by the analogy with peakons, we think of these solutions as being made up of solitons situated at the breakpoints. We derive and solve ODEs governing the soliton dynamics, first for...

**Research Article**

## 28. A Common Integrable Structure in the Hermitian Matrix Model and Hele-Shaw Flows

L. Martinez Alonso, E. Medina

Pages: 277 - 287

It is proved that the system of string equations of the dispersionless 2-Toda hierarchy which arises in the planar limit of the hermitian matrix model also underlies certain processes in Hele-Shaw flows.

**Research Article**

## 29. Hopf Bifurcations in a Watt Governor with a Spring

Jorge Sotomayor, Luis Fernando Mello, Denis de Carvalho Braga

Pages: 288 - 299

This paper pursues the study carried out in [10], focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor system. Here are studied Hopf bifurcations of codimensions two, three and four and the pertinent Lyapunov stability coefficients and bifurcation diagrams. This allows to...

**Research Article**

## 30. Integrating Factors and λ—Symmetries

C. Muriel, J. L. Romero

Pages: 300 - 309

We investigate the relationship between integrating factors and λ-symmetries for ordinary differential equations of arbitrary order. Some results on the existence of λ-symmetries are used to prove an independent existence theorem for integrating factors. A new method to calculate integrating...

**Research Article**

## 31. The Sato Grassmannian and the CH Hierarchy

Gregorio Falqui, Giovanni Ortenzi

Pages: 310 - 322

We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmannian. We discuss the geometry of an extension of the negative flows of the CH hierarchy, recover the well-known CH equations, and frame the problem within the theory of pseudo-differential operators.

**Research Article**

## 32. Multiscale Expansion and Integrability Properties of the Lattice Potential KdV Equation

Rafael Hernandez Heredero, Decio Levi, Matteo Petrera, Christian Scimiterna

Pages: 323 - 333

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear SchrÂ¨odinger equation, the Lax pair gives at the same order...

**Research Article**

## 33. The Klein-Gordon Equation on the Half Line: a Riemann-Hilbert Approach

Beatrice Pelloni, Dimitrios A. Pinotsis

Pages: 334 - 344

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems...

**Research Article**

## 34. On Differential Operators on Sequence Spaces

M. Maldonado, J. Prada, M.J. Senosiain

Pages: 345 - 352

Two differential operators T1 and T2 on a space are said to be equivalent if there is an isomorphism S from onto such that ST1 = T2 S. The notion was first introduced by Delsarte in 1938 [2] where T1 and T2 are differential operators of second order and a space of functions of one variable defined for...

**Research Article**

## 35. Where Do Braid Statistics and Discrete Motion Meet Each Other?

Luigi Martina, Alexander Protogenov, Valery Verbus

Pages: 353 - 361

We consider universal statistical properties of systems that are characterized by phase states with macroscopic degeneracy of the ground state. A possible topological order in such systems is described by non-linear discrete equations. We focus on the discrete equations which take place in the case of...

**Research Article**

## 36. Symmetry Preserving Discretization of SL(2,R) Invariant Equations

Anne Bourlioux, Raphaël Rebelo, Pavel Winternitz

Pages: 362 - 372

Nonlinear ODEs invariant under the group SL(2,R) are solved numerically. We show that solution methods incorporating the Lie point symmetries provide better results than standard methods.

**Research Article**

## 37. New Quasi-Exactly Solvable Difference Equation

Ryu Sasaki

Pages: 373 - 384

Exact solvability of two typical examples of the discrete quantum mechanics, i.e. the dynamics of the Meixner-Pollaczek and the continuous Hahn polynomials with full parameters, is newly demonstrated both at the SchrÂ¨odinger and Heisenberg picture levels. A new quasiexactly solvable difference equation...

**Research Article**

## 38. Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

M.B. Sheftel, A.A. Malykh

Pages: 385 - 395

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of three- and twodimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions...

**Research Article**

## 39. The Point of Maximum Curvature as a Marker for Physiological Time Series

James Robert Stirling, Maria Zakynthinaki

Pages: 396 - 406

We present a geometric analysis of the model of Stirling et al. [14]. In particular we analyze the curvature of a heart rate time series in response to a step like increment in the exercise intensity. We present solutions for the point of maximum curvature which can be used as a marker of physiological...

**Research Article**

## 40. On the Origin of Fractional Shapiro Steps in Systems of Josephson Junctions with Few Degrees of Freedom

A. Valizadeh, M.R. Kolahchi, J.P. Straley

Pages: 407 - 416

We investigate the origin of fractional Shapiro steps in arrays consisting of a few overdamped Josephson junctions. We show that when the symmetry reduces the equations to that of a single junction equation, only integer steps appear. Otherwise, fractional steps will appear when the evolution equations...

**Research Article**

## 41. Symmetry Reductions of Second Heavenly Equation and 2+1-Dimensional Hamiltonian Integrable Systems

D. Yazici, M. B. Sheftel

Pages: 417 - 425

Second heavenly equation of Pleba~nski, presented in a two-component form, is known to be a 3 +1 dimensional multi-Hamiltonian integrable system. We show that one symmetry reduction of this equation yields a two component 2+1Âdimensional multi-Hamiltonian integrable system. For this system, we present...

**Research Article**

## 42. A Model of Heart Rate Kinetics in Response to Exercise

James Robert Stirling, Maria Zakynthinaki, Ignacio Refoyo, Javier Sampedro

Pages: 426 - 436

We present a mathematical model, in the form of two coupled ordinary differential equations, for the heart rate kinetics in response to exercise. Our heart rate model is an adaptation of the model of oxygen uptake kinetics of Stirling et al. [21]; a physiological justification for this adaptation, as...

**Research Article**

## 43. Combination of Inverse Spectral Transform Method and Method of Characteristics: Deformed Pohlmeyer Equation

A.I. Zenchuk

Pages: 437 - 448

We apply a version of the dressing method to a system of four-dimensional nonlinear Partial Differential Equations (PDEs), which contains both Pohlmeyer equation (i.e. nonlinear PDE integrable by the Inverse Spectral Transform Method) and nonlinear matrix PDE integrable by the method of characteristics...