Nonlinear Evolution Equations and Dynamical Systems 2007
Papers resulting from the NEEDS-07 workshop held in L'Ametlla Mar, Spain
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...
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.
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
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....
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...
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...
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.
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...
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...
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...
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...
Pages: 112 - 123
We construct the free energy associated with the waterbag model of dToda. Also, relations for conserved densities are investigated.
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),...
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...
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...
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
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...
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.
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...
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...
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...
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.
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.
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...
Pages: 251 - 263
We define the deformation quantization in the Fedosov sense for a limit model of Taubes in
white noise analysis.
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...
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
Pages: 288 - 299
This paper pursues the study carried out in , 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...
Pages: 300 - 309
We investigate the relationship between integrating factors and -symmetries for ordinary
differential equations of arbitrary order. Some results on the existence of -symmetries are
used to prove an independent existence theorem for integrating factors. A new method to
calculate integrating factors...
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
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...
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...
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  where T1 and T2 are differential
operators of second order and a space of functions of one variable defined...
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...
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.
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...
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...
Pages: 396 - 406
We present a geometric analysis of the model of Stirling et al. . 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...
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...
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...
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. ; a physiological justification for this
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...