123,766 research outputs found
The Schrodinger-like Equation for a Nonrelativistic Electron in a Photon Field of Arbitrary Intensity
The ordinary Schrodinger equation with minimal coupling for a nonrelativistic
electron interacting with a single-mode photon field is not satisfied by the
nonrelativistic limit of the exact solutions to the corresponding Dirac
equation. A Schrodinger-like equation valid for arbitrary photon intensity is
derived from the Dirac equation without the weak-field assumption. The
"eigenvalue" in the new equation is an operator in a Cartan subalgebra. An
approximation consistent with the nonrelativistic energy level derived from its
relativistic value replaces the "eigenvalue" operator by an ordinary number,
recovering the ordinary Schrodinger eigenvalue equation used in the formal
scattering formalism. The Schrodinger-like equation for the multimode case is
also presented.Comment: Tex file, 13 pages, no figur
Persistency of Analyticity for Nonlinear Wave Equations: An Energy-like Approach
We study the persistence of the Gevrey class regularity of solutions to
nonlinear wave equations with real analytic nonlinearity. Specifically, it is
proven that the solution remains in a Gevrey class, with respect to some of its
spatial variables, during its whole life-span, provided the initial data is
from the same Gevrey class with respect to these spatial variables. In
addition, for the special Gevrey class of analytic functions, we find a lower
bound for the radius of the spatial analyticity of the solution that might
shrink either algebraically or exponentially, in time, depending on the
structure of the nonlinearity. The standard theory for the Gevrey class
regularity is employed; we also employ energy-like methods for a generalized
version of Gevrey classes based on the norm of Fourier transforms
(Wiener algebra). After careful comparisons, we observe an indication that the
approach provides a better lower bound for the radius of analyticity
of the solutions than the approach. We present our results in the case of
period boundary conditions, however, by employing exactly the same tools and
proofs one can obtain similar results for the nonlinear wave equations and the
nonlinear Schr\"odinger equation, with real analytic nonlinearity, in certain
domains and manifolds without physical boundaries, such as the whole space
, or on the sphere
On the backward behavior of some dissipative evolution equations
We prove that every solution of a KdV-Burgers-Sivashinsky type equation blows
up in the energy space, backward in time, provided the solution does not belong
to the global attractor. This is a phenomenon contrast to the backward behavior
of the periodic 2D Navier-Stokes equations studied by
Constantin-Foias-Kukavica-Majda [18], but analogous to the backward behavior of
the Kuramoto-Sivashinsky equation discovered by Kukavica-Malcok [50]. Also we
study the backward behavior of solutions to the damped driven nonlinear
Schrodinger equation, the complex Ginzburg-Landau equation, and the
hyperviscous Navier-Stokes equations. In addition, we provide some physical
interpretation of various backward behaviors of several perturbations of the
KdV equation by studying explicit cnoidal wave solutions. Furthermore, we
discuss the connection between the backward behavior and the energy spectra of
the solutions. The study of backward behavior of dissipative evolution
equations is motivated by the investigation of the Bardos-Tartar conjecture
stated in [5].Comment: 34 page
Interior dynamics of neutral and charged black holes in f(R) gravity
In this paper, we explore the interior dynamics of neutral and charged black
holes in gravity. We transform gravity from the Jordan frame into
the Einstein frame and simulate scalar collapses in flat, Schwarzschild, and
Reissner-Nordstr\"om geometries. In simulating scalar collapses in
Schwarzschild and Reissner-Nordstr\"om geometries, Kruskal and Kruskal-like
coordinates are used, respectively, with the presence of and a physical
scalar field being taken into account. The dynamics in the vicinities of the
central singularity of a Schwarzschild black hole and of the inner horizon of a
Reissner-Nordstr\"om black hole is examined. Approximate analytic solutions for
different types of collapses are partially obtained. The scalar degree of
freedom , transformed from , plays a similar role as a physical
scalar field in general relativity. Regarding the physical scalar field in
case, when is negative (positive), the physical scalar field
is suppressed (magnified) by , where is the coordinate time. For dark
energy gravity, inside black holes, gravity can easily push to .
Consequently, the Ricci scalar becomes singular, and the numerical
simulation breaks down. This singularity problem can be avoided by adding an
term to the original function, in which case an infinite Ricci
scalar is pushed to regions where is also infinite. On the other hand, in
collapse for this combined model, a black hole, including a central
singularity, can be formed. Moreover, under certain initial conditions,
and can be pushed to infinity as the central singularity is approached.
Therefore, the classical singularity problem, which is present in general
relativity, remains in collapse for this combined model.Comment: 35 pages, 22 figures. (Special Issue. Modified Gravity Cosmology:
From Inflation to Dark Energy). Minor change. arXiv admin note: substantial
text overlap with arXiv:1507.0180
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