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 $L^2$ theory for the Gevrey class regularity is employed; we also employ energy-like methods for a generalized version of Gevrey classes based on the $\ell^1$ norm of Fourier transforms (Wiener algebra). After careful comparisons, we observe an indication that the $\ell^1$ approach provides a better lower bound for the radius of analyticity of the solutions than the $L^2$ 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 $\mathbb{R}^n$, or on the sphere $\mathbb{S}^{n-1}$

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 $f(R)$ gravity. We transform $f(R)$ 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 $f'$ 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 $\phi$, transformed from $f'$, plays a similar role as a physical scalar field in general relativity. Regarding the physical scalar field in $f(R)$ case, when $d\phi/dt$ is negative (positive), the physical scalar field is suppressed (magnified) by $\phi$, where $t$ is the coordinate time. For dark energy $f(R)$ gravity, inside black holes, gravity can easily push $f'$ to $1$. Consequently, the Ricci scalar $R$ becomes singular, and the numerical simulation breaks down. This singularity problem can be avoided by adding an $R^2$ term to the original $f(R)$ function, in which case an infinite Ricci scalar is pushed to regions where $f'$ 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, $f'$ and $R$ 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