957 research outputs found
Phase-driven interaction of widely separated nonlinear Schr\"odinger solitons
We show that, for the 1d cubic NLS equation, widely separated equal amplitude
in-phase solitons attract and opposite-phase solitons repel. Our result gives
an exact description of the evolution of the two solitons valid until the
solitons have moved a distance comparable to the logarithm of the initial
separation. Our method does not use the inverse scattering theory and should be
applicable to nonintegrable equations with local nonlinearities that support
solitons with exponentially decaying tails. The result is presented as a
special case of a general framework which also addresses, for example, the
dynamics of single solitons subject to external forces
Fast soliton scattering by attractive delta impurities
We study the Gross-Pitaevskii equation with an attractive delta function
potential and show that in the high velocity limit an incident soliton is split
into reflected and transmitted soliton components plus a small amount of
dispersion. We give explicit analytic formulas for the reflected and
transmitted portions, while the remainder takes the form of an error. Although
the existence of a bound state for this potential introduces difficulties not
present in the case of a repulsive potential, we show that the proportion of
the soliton which is trapped at the origin vanishes in the limit
Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection
In this work we continue the description of soliton-like solutions of some
slowly varying, subcritical gKdV equations.
In this opportunity we describe, almost completely, the allowed behaviors:
either the soliton is refracted, or it is reflected by the potential, depending
on its initial energy. This last result describes a new type of soliton-like
solution for gKdV equations, also present in the NLS case.
Moreover, we prove that the solution is not pure at infinity, unlike the
standard gKdV soliton.Comment: 51 pages, submitte
Scattering for the L2supercritical point NLS
We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point"means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blowup profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ±∞ in the L2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation
On the 2d Zakharov system with L^2 Schr\"odinger data
We prove local in time well-posedness for the Zakharov system in two space
dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the
space of optimal regularity in the sense that the data-to-solution map fails to
be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev
scale. Moreover, it is a natural space for the Cauchy problem in view of the
subsonic limit equation, namely the focusing cubic nonlinear Schroedinger
equation. The existence time we obtain depends only upon the corresponding
norms of the initial data - a result which is false for the cubic nonlinear
Schroedinger equation in dimension two - and it is optimal because
Glangetas-Merle's solutions blow up at that time.Comment: 30 pages, 2 figures. Minor revision. Title has been change
Fast solitons on star graphs
We define the Schr\"odinger equation with focusing, cubic nonlinearity on
one-vertex graphs. We prove global well-posedness in the energy domain and
conservation laws for some self-adjoint boundary conditions at the vertex, i.e.
Kirchhoff boundary condition and the so called and boundary
conditions. Moreover, in the same setting we study the collision of a fast
solitary wave with the vertex and we show that it splits in reflected and
transmitted components. The outgoing waves preserve a soliton character over a
time which depends on the logarithm of the velocity of the ingoing solitary
wave. Over the same timescale the reflection and transmission coefficients of
the outgoing waves coincide with the corresponding coefficients of the linear
problem. In the analysis of the problem we follow ideas borrowed from the
seminal paper \cite{[HMZ07]} about scattering of fast solitons by a delta
interaction on the line, by Holmer, Marzuola and Zworski; the present paper
represents an extension of their work to the case of graphs and, as a
byproduct, it shows how to extend the analysis of soliton scattering by other
point interactions on the line, interpreted as a degenerate graph.Comment: Sec. 2 revised; several misprints corrected; added references; 32
page
Study of stability and control moment gyro wobble damping of flexible, spinning space stations
An executive summary and an analysis of the results are discussed. A user's guide for the digital computer program that simulates the flexible, spinning space station is presented. Control analysis activities and derivation of dynamic equations of motion and the modal analysis are also cited
On the Ground State Quantum Droplet for Large Chemical Potentials
In the present work we revisit the problem of the quantum droplet in atomic
Bose-Einstein condensates with an eye towards describing its ground state in
the large density, so-called Thomas-Fermi limit. We consider the problem as
being separable into 3 distinct regions: an inner one, where the Thomas-Fermi
approximation is valid, a sharp transition region where the density abruptly
drops towards the (vanishing) background value and an outer region which
asymptotes to the background value. We analyze the spatial extent of each of
these regions, and develop a systematic effective description of the rapid
intermediate transition region. Accordingly, we derive a uniformly valid
description of the ground state that is found to very accurately match our
numerical computations. As an additional application of our considerations, we
show that this formulation allows for an analytical approximation of excited
states such as the (trapped) dark soliton in the large density limit.Comment: 7 pages, 3 figure
Fast soliton scattering by delta impurities
We study the Gross-Pitaevskii equation (nonlinear Schroedinger equation) with
a repulsive delta function potential. We show that a high velocity incoming
soliton is split into a transmitted component and a reflected component. The
transmitted mass (L^2 norm squared) is shown to be in good agreement with the
quantum transmission rate of the delta function potential. We further show that
the transmitted and reflected components resolve into solitons plus dispersive
radiation, and quantify the mass and phase of these solitons.Comment: 32 pages, 3 figure
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