71,470 research outputs found
Space-modulated Stability and Averaged Dynamics
In this brief note we give a brief overview of the comprehensive theory,
recently obtained by the author jointly with Johnson, Noble and Zumbrun, that
describes the nonlinear dynamics about spectrally stable periodic waves of
parabolic systems and announce parallel results for the linearized dynamics
near cnoidal waves of the Korteweg-de Vries equation. The latter are expected
to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees
partielles", Roscoff 201
Two novel classes of solvable many-body problems of goldfish type with constraints
Two novel classes of many-body models with nonlinear interactions "of
goldfish type" are introduced. They are solvable provided the initial data
satisfy a single constraint (in one case; in the other, two constraints): i.
e., for such initial data the solution of their initial-value problem can be
achieved via algebraic operations, such as finding the eigenvalues of given
matrices or equivalently the zeros of known polynomials. Entirely isochronous
versions of some of these models are also exhibited: i.e., versions of these
models whose nonsingular solutions are all completely periodic with the same
period.Comment: 30 pages, 2 figure
Traveling waves of nonlinear Schr\"{o}dinger equation including higher order dispersions
The solitary wave solution and periodic solutions expressed in terms of
elliptic Jacobi's functions are obtained for the nonlinear Schr\"{o}dinger
equation governing the propagation of pulses in optical fibers including the
effects of second, third and fourth order dispersion. The approach is based on
the reduction of the generalized nonlinear Schr\"{o}dinger equation to an
ordinary nonlinear differential equation. The periodic solutions obtained form
one-parameter family which depend on an integration constant . The solitary
wave solution with shape is the limiting case of this family
with . The solutions obtained describe also a train of soliton-like pulses
with shape. It is shown that the bounded solutions arise only
for special domains of integration constant.Comment: We consider in this paper also the case with negative parameter
(defocusing nonlinearity
Global entrainment of transcriptional systems to periodic inputs
This paper addresses the problem of giving conditions for transcriptional
systems to be globally entrained to external periodic inputs. By using
contraction theory, a powerful tool from dynamical systems theory, it is shown
that certain systems driven by external periodic signals have the property that
all solutions converge to a fixed limit cycle. General results are proved, and
the properties are verified in the specific case of some models of
transcriptional systems. The basic mathematical results needed from contraction
theory are proved in the paper, making it self-contained
Zero-Hopf bifurcation in the FitzHugh-Nagumo system
We characterize the values of the parameters for which a zero--Hopf
equilibrium point takes place at the singular points, namely, (the origin),
and in the FitzHugh-Nagumo system. Thus we find two --parameter
families of the FitzHugh-Nagumo system for which the equilibrium point at the
origin is a zero-Hopf equilibrium. For these two families we prove the
existence of a periodic orbit bifurcating from the zero--Hopf equilibrium point
. We prove that exist three --parameter families of the FitzHugh-Nagumo
system for which the equilibrium point at and is a zero-Hopf
equilibrium point. For one of these families we prove the existence of , or
, or periodic orbits borning at and
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