4,506 research outputs found
Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case
We study the periodic and the Neumann boundary value problems associated with
the second order nonlinear differential equation \begin{equation*} u'' + c u' +
\lambda a(t) g(u) = 0, \end{equation*} where is a
sublinear function at infinity having superlinear growth at zero. We prove the
existence of two positive solutions when and
is sufficiently large. Our approach is based on Mawhin's
coincidence degree theory and index computations.Comment: 26 page
Initial Value Problems for Integrable Systems on a Semi-Strip
Two important cases, where boundary conditions and solutions of the
well-known integrable equations on a semi-strip are uniquely determined by the
initial conditions, are rigorously studied in detail. First, the case of
rectangular matrix solutions of the defocusing nonlinear Schr\"odinger equation
with quasi-analytic boundary conditions is dealt with. (The result is new even
for a scalar nonlinear Schr\"odinger equation.) Next, a special case of the
nonlinear optics (-wave) equation is considered.Comment: Boundary conditions are recovered from the initial ones. The paper
supplements in this respect our previous article arXiv:1403.8111, where
initial conditions are recovered from the boundary condition
Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems
Many problems in Physics are described by dynamical systems that are
conformally symplectic (e.g., mechanical systems with a friction proportional
to the velocity, variational problems with a small discount or thermostated
systems). Conformally symplectic systems are characterized by the property that
they transform a symplectic form into a multiple of itself. The limit of small
dissipation, which is the object of the present study, is particularly
interesting.
We provide all details for maps, but we present also the modifications needed
to obtain a direct proof for the case of differential equations. We consider a
family of conformally symplectic maps defined on a
-dimensional symplectic manifold with exact symplectic form
; we assume that satisfies
. We assume that the family
depends on a -dimensional parameter (called drift) and also on a small
scalar parameter . Furthermore, we assume that the conformal factor
depends on , in such a way that for we have
(the symplectic case).
We study the domains of analyticity in near of
perturbative expansions (Lindstedt series) of the parameterization of the
quasi--periodic orbits of frequency (assumed to be Diophantine) and of
the parameter . Notice that this is a singular perturbation, since any
friction (no matter how small) reduces the set of quasi-periodic solutions in
the system. We prove that the Lindstedt series are analytic in a domain in the
complex plane, which is obtained by taking from a ball centered at
zero a sequence of smaller balls with center along smooth lines going through
the origin. The radii of the excluded balls decrease faster than any power of
the distance of the center to the origin
Probability & incompressible Navier-Stokes equations: An overview of some recent developments
This is largely an attempt to provide probabilists some orientation to an
important class of non-linear partial differential equations in applied
mathematics, the incompressible Navier-Stokes equations. Particular focus is
given to the probabilistic framework introduced by LeJan and Sznitman [Probab.
Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al.
[Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140,
2004, in press]. In particular this is an effort to provide some foundational
facts about these equations and an overview of some recent results with an
indication of some new directions for probabilistic consideration.Comment: Published at http://dx.doi.org/10.1214/154957805100000078 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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