111 research outputs found
Nonlinear Dynamics of Accelerator via Wavelet Approach
In this paper we present the applications of methods from wavelet analysis to
polynomial approximations for a number of accelerator physics problems. In the
general case we have the solution as a multiresolution expansion in the base of
compactly supported wavelet basis. The solution is parametrized by the
solutions of two reduced algebraical problems, one is nonlinear and the second
is some linear problem, which is obtained from one of the next wavelet
constructions: Fast Wavelet Transform, Stationary Subdivision Schemes, the
method of Connection Coefficients. According to the orbit method and by using
construction from the geometric quantization theory we construct the symplectic
and Poisson structures associated with generalized wavelets by using
metaplectic structure. We consider wavelet approach to the calculations of
Melnikov functions in the theory of homoclinic chaos in perturbed Hamiltonian
systems and for parametrization of Arnold-Weinstein curves in Floer variational
approach.Comment: 16 pages, no figures, LaTeX2e, aipproc.sty, aipproc.cl
Grassmannian flows and applications to nonlinear partial differential equations
We show how solutions to a large class of partial differential equations with
nonlocal Riccati-type nonlinearities can be generated from the corresponding
linearized equations, from arbitrary initial data. It is well known that
evolutionary matrix Riccati equations can be generated by projecting linear
evolutionary flows on a Stiefel manifold onto a coordinate chart of the
underlying Grassmann manifold. Our method relies on extending this idea to the
infinite dimensional case. The key is an integral equation analogous to the
Marchenko equation in integrable systems, that represents the coodinate chart
map. We show explicitly how to generate such solutions to scalar partial
differential equations of arbitrary order with nonlocal quadratic
nonlinearities using our approach. We provide numerical simulations that
demonstrate the generation of solutions to
Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal
nonlinearities. We also indicate how the method might extend to more general
classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure
On families of constrictions in model of overdamped Josephson junction and Painlev\'e 3 equation
The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns
the Josephson junction: two superconductors separated by a narrow dielectric.
It states existence of a supercurrent through it and equations governing it.
The overdamped Josephson junction is modeled by a family of differential
equations on 2-torus depending on 3 parameters: (abscissa), (ordinate),
(frequency). We study its rotation number as a
function of with fixed . The phase-lock areas are the level
sets with non-empty interiors; they exist for
(Buchstaber, Karpov, Tertychnyi). Each is an infinite chain of domains
going vertically to infinity and separated by points called constrictions
(expect for those with ). We show that: 1) all the constrictions in
lie in its axis (confirming a conjecture of Tertychnyi,
Kleptsyn, Filimonov, Schurov); 2) each constriction is positive: some its
punctured neighborhood in the vertical line lies in
(confirming another conjecture). We first prove deformability of each
constriction to another one, with arbitrarily small , of the same
, and type (positive or not), using equivalent
description of model by linear systems of differential equations on
(Buchstaber, Karpov, Tertychnyi) and studying their
isomonodromic deformations described by Painlev\'e 3 equations. Then
non-existence of ghost constrictions (i.e., constrictions either with
, or of non-positive type) with a given for small
is proved by slow-fast methods. In Section 6 we present applications of results
and elaborated methods and open problems.Comment: 72 pages, 10 figure
The turnpike property in finite-dimensional nonlinear optimal control
Turnpike properties have been established long time ago in finite-dimensional
optimal control problems arising in econometry. They refer to the fact that,
under quite general assumptions, the optimal solutions of a given optimal
control problem settled in large time consist approximately of three pieces,
the first and the last of which being transient short-time arcs, and the middle
piece being a long-time arc staying exponentially close to the optimal
steady-state solution of an associated static optimal control problem. We
provide in this paper a general version of a turnpike theorem, valuable for
nonlinear dynamics without any specific assumption, and for very general
terminal conditions. Not only the optimal trajectory is shown to remain
exponentially close to a steady-state, but also the corresponding adjoint
vector of the Pontryagin maximum principle. The exponential closedness is
quantified with the use of appropriate normal forms of Riccati equations. We
show then how the property on the adjoint vector can be adequately used in
order to initialize successfully a numerical direct method, or a shooting
method. In particular, we provide an appropriate variant of the usual shooting
method in which we initialize the adjoint vector, not at the initial time, but
at the middle of the trajectory
Mathematical methods of factorization and a feedback approach for biological systems
The first part of the thesis is devoted to factorizations of linear and
nonlinear differential equations leading to solutions of the kink type. The
second part contains a study of the synchronization of the chaotic dynamics of
two Hodgkin-Huxley neurons by means of the mathematical tools belonging to the
geometrical control theory.Comment: Ph. D. Thesis at IPICyT, San Luis Potosi, Mexico, 102 pp, 40 figs.
Supervisors: Dr. H.C. Rosu and Dr. R. Fema
Computing stability of multi-dimensional travelling waves
We present a numerical method for computing the pure-point spectrum
associated with the linear stability of multi-dimensional travelling fronts to
parabolic nonlinear systems. Our method is based on the Evans function shooting
approach. Transverse to the direction of propagation we project the spectral
equations onto a finite Fourier basis. This generates a large, linear,
one-dimensional system of equations for the longitudinal Fourier coefficients.
We construct the stable and unstable solution subspaces associated with the
longitudinal far-field zero boundary conditions, retaining only the information
required for matching, by integrating the Riccati equations associated with the
underlying Grassmannian manifolds. The Evans function is then the matching
condition measuring the linear dependence of the stable and unstable subspaces
and thus determines eigenvalues. As a model application, we study the stability
of two-dimensional wrinkled front solutions to a cubic autocatalysis model
system. We compare our shooting approach with the continuous orthogonalization
method of Humpherys and Zumbrun. We then also compare these with standard
projection methods that directly project the spectral problem onto a finite
multi-dimensional basis satisfying the boundary conditions.Comment: 23 pages, 9 figures (some in colour). v2: added details and other
changes to presentation after referees' comments, now 26 page
Partial differential systems with nonlocal nonlinearities: Generation and solutions
We develop a method for generating solutions to large classes of evolutionary
partial differential systems with nonlocal nonlinearities. For arbitrary
initial data, the solutions are generated from the corresponding linearized
equations. The key is a Fredholm integral equation relating the linearized flow
to an auxiliary linear flow. It is analogous to the Marchenko integral equation
in integrable systems. We show explicitly how this can be achieved through
several examples including reaction-diffusion systems with nonlocal quadratic
nonlinearities and the nonlinear Schrodinger equation with a nonlocal cubic
nonlinearity. In each case we demonstrate our approach with numerical
simulations. We discuss the effectiveness of our approach and how it might be
extended.Comment: 4 figure
Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
Nonlinear SUSY approach to preparation of quantum systems with pre-planned
spectral properties is reviewed. Possible multidimensional extensions of
Nonlinear SUSY are described. The full classification of ladder-reducible and
irreducible chains of SUSY algebras in one-dimensional QM is given. Emergence
of hidden symmetries and spectrum generating algebras is elucidated in the
context of Nonlinear SUSY in one- and two-dimensional QM.Comment: 75 pages, Minor corrections, Version published in Journal of Physics
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