383 research outputs found
Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations
We consider a family of dispersion generalized Benjamin-Ono equations (dgBO)
which are critical with respect to the L2 norm and interpolate between the
critical modified (BO) equation and the critical generalized Korteweg-de Vries
equation (gKdV). First, we prove local well-posedness in the energy space for
these equations, extending results by Kenig, Ponce and Vega concerning the
(gKdV) equations. Second, we address the blow up problem in the spirit of works
of Martel and Merle on the critical (gKdV) equation, by studying rigidity
properties of the (dgBO) flow in a neighborhood of solitons. We prove that when
the model is close to critical (gKdV), solutions of negative energy close to
solitons blow up in finite or infinite time in the energy space. The blow up
proof requires in particular extensions to (dgBO) of monotonicity results for
localized versions of L2 norms by pseudo-differential operator tools.Comment: 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
I=3/2 Scattering in the Nonrelativisitic Quark Potential Model
We study elastic scattering to Born order using
nonrelativistic quark wavefunctions in a constituent-exchange model. This
channel is ideal for the study of nonresonant meson-meson scattering amplitudes
since s-channel resonances do not contribute significantly. Standard quark
model parameters yield good agreement with the measured S- and P-wave phase
shifts and with PCAC calculations of the scattering length. The P-wave phase
shift is especially interesting because it is nonzero solely due to
symmetry breaking effects, and is found to be in good agreement with experiment
given conventional values for the strange and nonstrange constituent quark
masses.Comment: 12 pages + 2 postscript figures, Revtex, MIT-CTP-210
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
Kaon-Nucleon Scattering Amplitudes and Z-Enhancements from Quark Born Diagrams
We derive closed form kaon-nucleon scattering amplitudes using the ``quark
Born diagram" formalism, which describes the scattering as a single interaction
(here the OGE spin-spin term) followed by quark line rearrangement. The low
energy I=0 and I=1 S-wave KN phase shifts are in reasonably good agreement with
experiment given conventional quark model parameters. For Gev
however the I=1 elastic phase shift is larger than predicted by Gaussian
wavefunctions, and we suggest possible reasons for this discrepancy. Equivalent
low energy KN potentials for S-wave scattering are also derived. Finally we
consider OGE forces in the related channels K, KN and K,
and determine which have attractive interactions and might therefore exhibit
strong threshold enhancements or ``Z-molecule" meson-baryon bound states.
We find that the minimum-spin, minimum-isospin channels and two additional
K channels are most conducive to the formation of bound states.
Related interesting topics for future experimental and theoretical studies of
KN interactions are also discussed.Comment: 34 pages, figures available from the authors, revte
A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
We consider the problem of identifying sharp criteria under which radial
(finite energy) solutions to the focusing 3d cubic nonlinear
Schr\"odinger equation (NLS) scatter,
i.e. approach the solution to a linear Schr\"odinger equation as . The criteria is expressed in terms of the scale-invariant quantities
and , where denotes the
initial data, and and denote the (conserved in time) mass and
energy of the corresponding solution . The focusing NLS possesses a
soliton solution , where is the ground-state solution to a
nonlinear elliptic equation, and we prove that if and
, then the
solution is globally well-posed and scatters. This condition is sharp in
the sense that the soliton solution , for which equality in these
conditions is obtained, is global but does not scatter. We further show that if
, then the solution blows-up in finite time. The
technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in
their study of the energy-critical NLS
Uniqueness and Nondegeneracy of Ground States for in
We prove uniqueness of ground state solutions for the
nonlinear equation in , where
and for and for . Here denotes the fractional Laplacian
in one dimension. In particular, we generalize (by completely different
techniques) the specific uniqueness result obtained by Amick and Toland for
and in [Acta Math., \textbf{167} (1991), 107--126]. As a
technical key result in this paper, we show that the associated linearized
operator is nondegenerate;
i.\,e., its kernel satisfies .
This result about proves a spectral assumption, which plays a central
role for the stability of solitary waves and blowup analysis for nonlinear
dispersive PDEs with fractional Laplacians, such as the generalized
Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page
Nondispersive solutions to the L2-critical half-wave equation
We consider the focusing -critical half-wave equation in one space
dimension where denotes the
first-order fractional derivative. Standard arguments show that there is a
critical threshold such that all solutions with extend globally in time, while solutions with may develop singularities in finite time.
In this paper, we first prove the existence of a family of traveling waves
with subcritical arbitrarily small mass. We then give a second example of
nondispersive dynamics and show the existence of finite-time blowup solutions
with minimal mass . More precisely, we construct a
family of minimal mass blowup solutions that are parametrized by the energy
and the linear momentum . In particular, our main result
(and its proof) can be seen as a model scenario of minimal mass blowup for
-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page
Using Nonlinear Response to Estimate the Strength of an Elastic Network
Disordered networks of fragile elastic elements have been proposed as a model
of inner porous regions of large bones [Gunaratne et.al., cond-mat/0009221,
http://xyz.lanl.gov]. It is shown that the ratio of responses of such
a network to static and periodic strain can be used to estimate its ultimate
(or breaking) stress. Since bone fracture in older adults results from the
weakening of porous bone, we discuss the possibility of using as a
non-invasive diagnostic of osteoporotic bone.Comment: 4 pages, 4 figure
NN Core Interactions and Differential Cross Sections from One Gluon Exchange
We derive nonstrange baryon-baryon scattering amplitudes in the
nonrelativistic quark model using the ``quark Born diagram" formalism. This
approach describes the scattering as a single interaction, here the
one-gluon-exchange (OGE) spin-spin term followed by constituent interchange,
with external nonrelativistic baryon wavefunctions attached to the scattering
diagrams to incorporate higher-twist wavefunction effects. The short-range
repulsive core in the NN interaction has previously been attributed to this
spin-spin interaction in the literature; we find that these perturbative
constituent-interchange diagrams do indeed predict repulsive interactions in
all I,S channels of the nucleon-nucleon system, and we compare our results for
the equivalent short-range potentials to the core potentials found by other
authors using nonperturbative methods. We also apply our perturbative
techniques to the N and systems: Some
channels are found to have attractive core potentials and may accommodate
``molecular" bound states near threshold. Finally we use our Born formalism to
calculate the NN differential cross section, which we compare with experimental
results for unpolarised proton-proton elastic scattering. We find that several
familiar features of the experimental differential cross section are reproduced
by our Born-order result.Comment: 27 pages, figures available from the authors, revtex, CEBAF-TH-93-04,
MIT-CTP-2187, ORNL-CCIP-93-0
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