5,167 research outputs found
Waves of maximal height for a class of nonlocal equations with homogeneous symbols
We discuss the existence and regularity of periodic traveling-wave solutions
of a class of nonlocal equations with homogeneous symbol of order , where
. Based on the properties of the nonlocal convolution operator, we apply
analytic bifurcation theory and show that a highest, peaked, periodic
traveling-wave solution is reached as the limiting case at the end of the main
bifurcation curve. The regularity of the highest wave is proved to be exactly
Lipschitz. As an application of our analysis, we reformulate the steady reduced
Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator
with symbol . Thereby we recover its unique highest
-periodic, peaked traveling-wave solution, having the property of being
exactly Lipschitz at the crest.Comment: 25 page
One-dimensional fluids with second nearest-neighbor interactions
As is well known, one-dimensional systems with interactions restricted to
first nearest neighbors admit a full analytically exact statistical-mechanical
solution. This is essentially due to the fact that the knowledge of the first
nearest-neighbor probability distribution function, , is enough to
determine the structural and thermodynamic properties of the system. On the
other hand, if the interaction between second nearest-neighbor particles is
turned on, the analytically exact solution is lost. Not only the knowledge of
is not sufficient anymore, but even its determination becomes a
complex many-body problem. In this work we systematically explore different
approximate solutions for one-dimensional second nearest-neighbor fluid models.
We apply those approximations to the square-well and the attractive two-step
pair potentials and compare them with Monte Carlo simulations, finding an
excellent agreement.Comment: 26 pages, 12 figures; v2: more references adde
Arnold maps with noise: Differentiability and non-monotonicity of the rotation number
Arnold's standard circle maps are widely used to study the quasi-periodic
route to chaos and other phenomena associated with nonlinear dynamics in the
presence of two rationally unrelated periodicities. In particular, the El
Nino-Southern Oscillation (ENSO) phenomenon is a crucial component of climate
variability on interannual time scales and it is dominated by the seasonal
cycle, on the one hand, and an intrinsic oscillatory instability with a period
of a few years, on the other. The role of meteorological phenomena on much
shorter time scales, such as westerly wind bursts, has also been recognized and
modeled as additive noise. We consider herein Arnold maps with additive,
uniformly distributed noise. When the map's nonlinear term, scaled by the
parameter , is sufficiently small, i.e. , the map is
known to be a diffeomorphism and the rotation number is a
differentiable function of the driving frequency . We concentrate on
the rotation number's behavior as the nonlinearity becomes large, and show
rigorously that is a differentiable function of ,
even for , at every point at which the noise-perturbed map is
mixing. We also provide a formula for the derivative of the rotation number.
The reasoning relies on linear-response theory and a computer-aided proof. In
the diffeomorphism case of , the rotation number
behaves monotonically with respect to . We show, using again a
computer-aided proof, that this is not the case when and the
map is not a diffeomorphism.Comment: Electronic copy of final peer-reviewed manuscript accepted for
publication in the Journal of Statistical Physic
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