328 research outputs found
Travelling waves for the Gross-Pitaevskii equation II
The purpose of this paper is to provide a rigorous mathematical proof of the
existence of travelling wave solutions to the Gross-Pitaevskii equation in
dimensions two and three. Our arguments, based on minimization under
constraints, yield a full branch of solutions, and extend earlier results,
where only a part of the branch was built. In dimension three, we also show
that there are no travelling wave solutions of small energy.Comment: Final version accepted for publication in Communications in
Mathematical Physics with a few minor corrections and added remark
Traveling waves for nonlinear Schr\"odinger equations with nonzero conditions at infinity, II
We prove the existence of nontrivial finite energy traveling waves for a
large class of nonlinear Schr\"odinger equations with nonzero conditions at
infinity (includindg the Gross-Pitaevskii and the so-called "cubic-quintic"
equations) in space dimension . We show that minimization of the
energy at fixed momentum can be used whenever the associated nonlinear
potential is nonnegative and it gives a set of orbitally stable traveling
waves, while minimization of the action at constant kinetic energy can be used
in all cases. We also explore the relationship between the families of
traveling waves obtained by different methods and we prove a sharp nonexistence
result for traveling waves with small energy.Comment: Final version, accepted for publication in the {\it Archive for
Rational Mechanics and Analysis.} The final publication is available at
Springer via http://dx.doi.org/10.1007/s00205-017-1131-
Stability of solitary waves for the generalized higher-order Boussinesq equation
This work studies the stability of solitary waves of a class of sixth-order
Boussinesq equations.Comment: 32 pages. Submitte
Global well-posedness of the KP-I initial-value problem in the energy space
We prove that the KP-I initial value problem is globally well-posed in the
natural energy space of the equation
Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
We present an abstract framework for analyzing the weak error of fully
discrete approximation schemes for linear evolution equations driven by
additive Gaussian noise. First, an abstract representation formula is derived
for sufficiently smooth test functions. The formula is then applied to the wave
equation, where the spatial approximation is done via the standard continuous
finite element method and the time discretization via an I-stable rational
approximation to the exponential function. It is found that the rate of weak
convergence is twice that of strong convergence. Furthermore, in contrast to
the parabolic case, higher order schemes in time, such as the Crank-Nicolson
scheme, are worthwhile to use if the solution is not very regular. Finally we
apply the theory to parabolic equations and detail a weak error estimate for
the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic
heat equation
An effective mass theorem for the bidimensional electron gas in a strong magnetic field
We study the limiting behavior of a singularly perturbed
Schr\"odinger-Poisson system describing a 3-dimensional electron gas strongly
confined in the vicinity of a plane and subject to a strong uniform
magnetic field in the plane of the gas. The coupled effects of the confinement
and of the magnetic field induce fast oscillations in time that need to be
averaged out. We obtain at the limit a system of 2-dimensional Schr\"odinger
equations in the plane , coupled through an effective selfconsistent
electrical potential. In the direction perpendicular to the magnetic field, the
electron mass is modified by the field, as the result of an averaging of the
cyclotron motion. The main tools of the analysis are the adaptation of the
second order long-time averaging theory of ODEs to our PDEs context, and the
use of a Sobolev scale adapted to the confinement operator
The phase shift of line solitons for the KP-II equation
The KP-II equation was derived by [B. B. Kadomtsev and V. I.
Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of
line solitary waves of shallow water. Stability of line solitons has been
proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi,
Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the
local phase shift of modulating line solitons are not uniform in the transverse
direction. In this paper, we obtain the -bound for the local phase
shift of modulating line solitons for polynomially localized perturbations
Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation
The aim of this paper is the accurate numerical study of the KP equation. In
particular we are concerned with the small dispersion limit of this model,
where no comprehensive analytical description exists so far. To this end we
first study a similar highly oscillatory regime for asymptotically small
solutions, which can be described via the Davey-Stewartson system. In a second
step we investigate numerically the small dispersion limit of the KP model in
the case of large amplitudes. Similarities and differences to the much better
studied Korteweg-de Vries situation are discussed as well as the dependence of
the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at
http://www.mis.mpg.de/preprints/index.html
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