422 research outputs found
Excitation Thresholds for Nonlinear Localized Modes on Lattices
Breathers are spatially localized and time periodic solutions of extended
Hamiltonian dynamical systems. In this paper we study excitation thresholds for
(nonlinearly dynamically stable) ground state breather or standing wave
solutions for networks of coupled nonlinear oscillators and wave equations of
nonlinear Schr\"odinger (NLS) type. Excitation thresholds are rigorously
characterized by variational methods. The excitation threshold is related to
the optimal (best) constant in a class of discr ete interpolation inequalities
related to the Hamiltonian energy. We establish a precise connection among ,
the dimensionality of the lattice, , the degree of the nonlinearity
and the existence of an excitation threshold for discrete nonlinear
Schr\"odinger systems (DNLS).
We prove that if , then ground state standing waves exist if
and only if the total power is larger than some strictly positive threshold,
. This proves a conjecture of Flach, Kaldko& MacKay in
the context of DNLS. We also discuss upper and lower bounds for excitation
thresholds for ground states of coupled systems of NLS equations, which arise
in the modeling of pulse propagation in coupled arrays of optical fibers.Comment: To appear in Nonlinearit
Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type
We prove local and global well-posedness for semi-relativistic, nonlinear
Schr\"odinger equations with
initial data in , . Here is a critical
Hartree nonlinearity that corresponds to Coulomb or Yukawa type
self-interactions. For focusing , which arise in the quantum theory of
boson stars, we derive a sufficient condition for global-in-time existence in
terms of a solitary wave ground state. Our proof of well-posedness does not
rely on Strichartz type estimates, and it enables us to add external potentials
of a general class.Comment: 18 pages; replaced with revised version; remark and reference on blow
up adde
Stable self-similar blow-up dynamics for slightly -supercritical generalized KdV equations
In this paper we consider the slightly -supercritical gKdV equations
, with the nonlinearity
and . We will prove the existence and
stability of a blow-up dynamic with self-similar blow-up rate in the energy
space and give a specific description of the formation of the singularity
near the blow-up time.Comment: 38 page
Orbital stability of periodic waves for the nonlinear Schroedinger equation
The nonlinear Schroedinger equation has several families of quasi-periodic
travelling waves, each of which can be parametrized up to symmetries by two
real numbers: the period of the modulus of the wave profile, and the variation
of its phase over a period (Floquet exponent). In the defocusing case, we show
that these travelling waves are orbitally stable within the class of solutions
having the same period and the same Floquet exponent. This generalizes a
previous work where only small amplitude solutions were considered. A similar
result is obtained in the focusing case, under a non-degeneracy condition which
can be checked numerically. The proof relies on the general approach to orbital
stability as developed by Grillakis, Shatah, and Strauss, and requires a
detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure
Solitary Wave Dynamics in an External Potential
We study the behavior of solitary-wave solutions of some generalized
nonlinear Schr\"odinger equations with an external potential. The equations
have the feature that in the absence of the external potential, they have
solutions describing inertial motions of stable solitary waves.
We construct solutions of the equations with a non-vanishing external
potential corresponding to initial conditions close to one of these solitary
wave solutions and show that, over a large interval of time, they describe a
solitary wave whose center of mass motion is a solution of Newton's equations
of motion for a point particle in the given external potential, up to small
corrections corresponding to radiation damping.Comment: latex2e, 41 pages, 1 figur
Stable self similar blow up dynamics for slightly L^2 supercritical NLS equations
We consider the focusing nonlinear Schr\"odinger equations in dimension and for slightly
supercritical nonlinearities p_c
with and 0<\e\ll 1. We prove the existence and stability in the energy space of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up tim
Relativistic Effects in the Scalar Meson Dynamics
A separable potential formalism is used to describe the and
interactions in the scalar-isoscalar states in the energy range
from the threshold up to 1.4 GeV. Introduction of relativistic
propagators into a system of Lippmann-Schwinger equations leads to a very good
description of the data ( per one degree of freedom). Three
poles are found in this energy region: fo(500) ( MeV,
MeV), fo(975) ( MeV, MeV) and
fo(1400) ( MeV, MeV). The fo(975) state can be
interpreted as a bound state. The fo(500) state may be
associated with the often postulated very broad scalar resonance under the
threshold (sometimes called or meson). The
scattering lengths in the and channels have also been
obtained. The relativistic approach provides qualitatively new results (e.g.
the appearance of the fo(500)) in comparison with previously used
nonrelativistic approach.Comment: 30 pages in LaTeX + 5 figures available on request. Preprint Orsay No
IPNO/TH 93-3
The reaction dynamics of the 16O(e,e'p) cross section at high missing energies
We measured the cross section and response functions (R_L, R_T, and R_LT) for
the 16O(e,e'p) reaction in quasielastic kinematics for missing energies 25 <=
E_miss <= 120 MeV at various missing momenta P_miss <= 340 MeV/c. For 25 <
E_miss < 50 MeV and P_miss \approx 60 MeV/c, the reaction is dominated by
single-nucleon knockout from the 1s1/2-state. At larger P_miss, the
single-particle aspects are increasingly masked by more complicated processes.
For E_miss > 60 MeV and P_miss > 200 MeV/c, the cross section is relatively
constant. Calculations which include contributions from pion exchange currents,
isobar currents and short-range correlations account for the shape and the
transversity but only for half of the magnitude of the measured cross section.Comment: 6 pages, 4 figures, submitted to Phys Rev Lett, formatting error
fixe
Strong-coupling expansion and effective hamiltonians
When looking for analytical approaches to treat frustrated quantum magnets,
it is often very useful to start from a limit where the ground state is highly
degenerate. This chapter discusses several ways of deriving {effective
Hamiltonians} around such limits, starting from standard {degenerate
perturbation theory} and proceeding to modern approaches more appropriate for
the derivation of high-order effective Hamiltonians, such as the perturbative
continuous unitary transformations or contractor renormalization. In the course
of this exposition, a number of examples taken from the recent literature are
discussed, including frustrated ladders and other dimer-based Heisenberg models
in a field, as well as the mapping between frustrated Ising models in a
transverse field and quantum dimer models.Comment: To appear as a chapter in "Highly Frustrated Magnetism", Eds. C.
Lacroix, P. Mendels, F. Mil
Virtual Compton Scattering and the Generalized Polarizabilities of the Proton at Q^2=0.92 and 1.76 GeV^2
Virtual Compton Scattering (VCS) on the proton has been studied at Jefferson
Lab using the exclusive photon electroproduction reaction (e p --> e p gamma).
This paper gives a detailed account of the analysis which has led to the
determination of the structure functions P_LL-P_TT/epsilon and P_LT, and the
electric and magnetic generalized polarizabilities (GPs) alpha_E(Q^2) and
beta_M(Q^2) at values of the four-momentum transfer squared Q^2= 0.92 and 1.76
GeV^2. These data, together with the results of VCS experiments at lower
momenta, help building a coherent picture of the electric and magnetic GPs of
the proton over the full measured Q^2-range, and point to their non-trivial
behavior.Comment: version 2: modified according to PRC Editor's and Referee's
recommendations. Archival paper for the E93-050 experiment at JLab Hall A. 28
pages, 23 figures, 5 cross-section tables. To be submitted to Phys.Rev.
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