Using the Uncharged Kerr Black Hole as a Gravitational Mirror

We extend the study of the possible use of the Schwarzschild black hole as a gravitational mirror to the more general case of an uncharged Kerr black hole. We use the null geodesic equation in the equatorial plane to prove a theorem concerning the conditions the impact parameter has to satisfy if boomerang photons are to exist. We derive an equation for these boomerang photons and an equation for the emission angle. Finally, the radial null geodesic equation is integrated numerically in order to illustrate boomerang photons.     

Theory of nonlinear external noise

We investigate a stochastic differential equation (SDE) with general nonlinearity in the noise. We derive an integro-differential equation for the probability density (PD) and an approximate equation for the mean first-passage time (MFPT). The approximate Fokker-Planck equation (AFPE) and MFPT are obtained for some examples.     

Evolution equation for short surface waves on water of finite depth

We address the question of determining the evolution equation for surface waves propagating in water whose depth is much larger than the typical wavelength of the surface disturbance. We avoid making the usual approximation of supposing the evolution to be given in the form of a modulated wave-packet. We treat the problem by means of a conformal transformation allowing to explicitly find the Dirichlet-to-Neumann operator for the problem together with asymptotic expansions in parameters measuring the nonlinearity and depth. This allows us to obtain an equation in physical variables valid in the weakly nonlinear, deep-water regime. The equation is an integro-differential equation, which reduces to known cases for infinite depth. We discuss solutions in a perturbative setting and show that the evolution equation describes Stokes-like waves.     

Non-Markovian relaxation of a three-level system: quantum trajectory approach

The non-Markovian dynamics of a three-level quantum system coupled to a bosonic environment is a difficult problem due to the lack of an exact dynamic equation such as a master equation. We present for the first time an exact quantum trajectory approach to a dissipative three-level model. We have established a convolutionless stochastic Schr?dinger equation called the time-local quantum state diffusion (QSD) equation without any approximations, in particular, without Markov approximation. Our exact time-local QSD equation opens a new avenue for exploring quantum dynamics for a higher dimensional quantum system coupled to a non-Markovian environment.     

Quasilinear theory of the 2D euler equation

We develop a quasilinear theory of the 2D Euler equation and derive an integrodifferential equation for the evolution of the coarse-grained vorticity omega;(r,t). This equation respects all of the invariance properties of the Euler equation and conserves angular momentum in a circular domain and linear impulse in a channel. We show under which hypothesis we can derive an H theorem for the Fermi-Dirac entropy and make the connection with statistical theories of 2D turbulence.     

Nonlinear Integral Equations and the Parameter Imbedding Method

The parameter imbedding method is applied to the solution of the nonlinear Schwinger-Dyson equation. A new parameter isintroduced which interpolates between a linear equation and the original nonlinear equation. We have thus an initial value problem which can be solved numerically. We present these solutions and verify that they sat&fy the nonlinear equation rather accurately.     

Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2 + 1

We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that leads to an integrable non-linear partial differential equation. This equation is an analogue of the Kadomtsev–Petviashvili (of type B) equation.     

Local solutions to high-frequency 2D scattering problems

We consider the solution of high-frequency scattering problems in two dimensions, modeled by an integral equation on the boundary of a smooth scattering object. We devise a numerical method to obtain solutions on only parts of the boundary with little computational effort. The method incorporates asymptotic properties of the solution and can therefore attain particularly good results for high frequencies. We show that the integral equation in this approach reduces to an ordinary differential equation.     

The Ernst equation as a motion on a universal Grassmann manifold

We find a linearization of the Ernst equation by means of universal Grassmann manifold (UGM) techniques. All local analytic solutions defined at the origin are obtained by solving an initial value problem for a linear differential equation on a UGM. We give an explicit formula which represents solutions of the Ernst equation. By using this formula, we generate several special solutions.     

A Fractional Schrödinger Equation and Its Solution

This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.     

The electromagnetic coupling in Kemmer-Duffin-Petiau theory

We analyze the electromagnetic coupling in the Kemmer-Duffin-Petiau (KDP) equation. Since the KDP equation which describes spin-0 and spin-1 bosons is of Dirac type, we examine some analogies with and differences from the Dirac equation. The main difference with the Dirac equation is that the KDP equation contains redundant components. We will show that as a result certain interaction terms in the Hamilton form of the KDP equation do not have a physical meaning and will not affect the calculation of physical observables. We point out that a second order KDP equation derived by Kemmer as an analogy to the second order Dirac equation is of limited physical applicability as (i) it belongs to a class of second order equations which can be derived from the original KDP equation and (ii) it lacks a back-transformation which would allow one to obtain solutions of the KDP equation out of solutions of the second order equation.     

Baeicklund Transformation and Multiple Soliton Solutions for （3＋1）-Dimensional Potential-YTSF Equation

We study an approach to constructing multiple soliton solutions of the (3 1)-dimensional nonlinear evolution equation. We take the (3 1)-dimensional potential- YTSF equation as an example. Using the extended homogeneous balance method, one can find a Backlund transformation to decompose the (3 1)-dimensional potential-YTSF equation into a set of partial differential equations. Starting from these partial differential equations, some multiple soliton solutions for the (3 1)-dimensional potential-YTSF equation are obtained by introducing a class of formal solutions.     

Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on ℝ3

We study the positive solutions of a semilinear equation with a Coulomb potential on ?³. We give a new uniqueness theorem for the positive radial solutions of such an equation and we apply these results to the Thomas-Fermi-Dirac-von Weizsäcker equation without electrostatic repulsion.     

B?cklund transformation and multiple soliton solutions for the (3+1)-dimensional Jimbo-Miwa equation

We study an approach to constructing multiple soliton solutions of the (3+1)-dimensional nonlinear evolution equation. We take the (3+1)-dimensional Jimbo－Miwa (JM) equation as an example. Using the extended homogeneous balance method, one can find a B?cklund transformation to decompose the (3+1)-dimensional JM equation into a linear partial differential equation and two bilinear partial differential equations. Starting from these linear and bilinear partial differential equations, some multiple soliton solutions for the (3+1)-dimensional JM equation are obtained by introducing a class of formal solutions.     

Relativistic wave equation for a boson-fermion system

We present a two-body relativistic wave equation for a system composed of a boson and a fermion. One-body equations such as the Dirac and the Klein-Gordon equations are often used as an approximate equation for relativistic two-body systems. However, when the masses of two particles are not very different, the use of one-body equations comes into question. We use the Feshbach-Villars formalism for the boson so that the wave equation can be given in the form of an eigenvalue equation for the Hamiltonian. Differences between our equation and the one-body equations are examined and illustrated in a numerical example of a two-body system with scalar and vector potentials.Communicated by: W. Weise     

Euler-Poincaré Formalism of (Two Component) Degasperis-Procesi and Holm-Staley type Systems

Abstract

In this paper we propose an Euler-Poincaré formalism of the Degasperis and Procesi (DP) equation. This is a second member of a one-parameter family of partial differential equations, known as b-field equations. This one-parameter family of pdes includes the integrable Camassa-Holm equation as a first member. We show that our Euler-Poincaré formalism exactly coincides with the Degasperis-Holm-Hone (DHH) Hamiltonian framework. We obtain the DHH Hamiltonian structues of the DP equation from our method. Recently this new equation has been generalized by Holm and Staley by adding viscosity term. We also discuss Euler-Poincaré formalism of the Holm-Staley equation. In the second half of the paper we consider a generalization of the Degasperis and Procesi (DP) equation with two dependent variables. we study the Euler-Poincaré framework of the 2-component Degasperis-Procesi equation. We also mention about the b-family equation.     

The （l＋l）-dimensional spatial solitons in media with weak nonlinear nonlocality

We study the propagation of (1+1)-dimensional spatial soliton in a nonlocal Kerr-type medium with weak nonlocality. First, we show that an equation for describing the soliton propagation in weak nonlocality is a nonlinear Schr?dinger equation with perturbation terms. Then, an approximate analytical solution of the equation is found by the perturbation method. We also find some interesting properties of the intensity profiles of the soliton.     

Asymptotic of Grazing Collisions and Particle Approximation for the Kac Equation without Cutoff

The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence.     

Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems

We derive asymptotically an order parameter equation in the limit where nascent bistability and long-wavelength modulation instabilities coalesce. This equation is a variant of the Swift-Hohenberg equation that generally contains nonvariational terms of the form psinabla(2)psi and /nablapsi/(2). We briefly review some of the properties already derived for this equation and derive it on three examples taken from chemical, biological, and optical contexts. Finally, we derive the equation on a general class of partial differential systems.     

Analytical solution of the equations of the physical theory of meteors for a single (non-fragmenting) body with mass loss in a non-isothermal (arbitrary) atmosphere with a variable ablation parameter

We have obtained an analytical solution of two simultaneous ordinary differential equations of the physical theory of meteors: the equation of motion for the center of mass of a meteoroid (deceleration equation), the thermal balance equation (ablation equation), the luminosity equation, and the ionization trail equation. The solution has been obtained by assuming a straight-line trajectory and a power-law dependence of the ablation parameter on the meteoroid velocity for an arbitrary atmosphere in the continuous flow regime.