Classical solution of the nonlinear Boltzmann equation in allR 3: Asymptotic behavior of solutions

Proof is given of the existence of a classical solution to the nonlinear Boltzmann equation in allR ³. The solution, which is global in time, exists if the initial data go to zero fast enough at infinity and the mean free path is sufficiently large. The solution is smooth in the space variable if the initial value is smooth. The asymptotic behavior of solutions is also given. It is shown that ast the solution to the Boltzmann equation can be approximated by the solution to the free motion problem.     

On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity

The Bardeen-Cooper-Schrieffer integral equation with a positive kernel is studied in full generality. It is shown that, there exists a unique finite transition temperature, T _c so that, if T _c ,the equation possesses a positive solution, representing the onset of the superconducting phase, while if T>T _c ,the only solution of the equation is the trivial one, indicating the occurrence of the normal phase. Moreover, it is demonstrated that such a positive solution may be approximated by a sequence of solutions of the equation restricted on bounded domains. This latter result provides a useful computational scheme for the problem.     

Inflationary solutions in general scalar tensor theory of gravitation

Extended inflation solution in Brans-Dicke theory given by Mathiazhagan and Johri (MJ) is shown as the unique solution only if the scale factor is assumed to be a power function of the scalar field. Only the consistent solution amongst the set of solutions given by Patra, Roy and Ray is found identical to the MJ solution. Both exponential inflation and power function inflation are studied in general scalar tensor theory where the parameter to is a function of the scalar, field. It is noted that exponential inflation is forbidden in Brans-Dicke theory wherew is a constant.     

Optimization of salt concentration in PEG-based crystallization solutions

Although polyethylene glycol (PEG) is the most widely used precipitant in protein crystallization, the concentration of co-existing salt in the solution has not been well discussed. To determine the optimum salt concentration range, several kinds of protein were crystallized in a 30% PEG 4000 solution at various NaCl concentrations with various pH levels. It was found that, if crystallization occurred, the lowest effective salt concentration depended on the pH of the protein solution and the pI of the protein molecule; that is, higher salt concentrations were required for crystal growth if the difference between pH and pI was increasing. The linear relationship between the charge density of the protein and the ionic strength of the crystallization solution was further verified. These results suggested that the lowest effective concentration of salt in a crystallization solution can be predicted before performing a crystallization experiment. Our results can be a tip for tuning crystallization conditions by the vapor-diffusion method.     

On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame

Analytic solution for unsteady magnetohydrodynamic (MHD) flow is constructed in a rotating non-Newtonian fluid through a porous medium. Constitutive equations for a Maxwell fluid have been taken into consideration. The hydromagnetic flow in the uniformly rotating fluid is generated by a suddenly moved infinite plate in its own plane. Analytic solution of the governing flow problem is obtained by means of the Fourier sine transform. It is shown that the obtained solution satisfies both the associate partial differential equation and the initial and boundary conditions. The solution for a Navier-Stokes fluid is recovered if λ→0. The steady state solution is also obtained for t→∞.     

A set of invariant quality factors measuring the deviation from the Kerr metric

A number of scalar invariant characterizations of the Kerr solution are presented. These characterizations come in the form of quality factors defined in stationary space-times. A quality factor is a scalar quantity varying in the interval $[0,1]$ with the value 1 being attained if and only if the space-time is locally isometric to the Kerr solution. No knowledge of the Kerr solution is required to compute these quality factors. A number of different possibilities arise depending on whether the space-time is Ricci-flat and asymptotically flat, just Ricci-flat, or Ricci non-flat. In each situation a number of quality factors are constructed and analysed. The relevance of these quality factors is clear in any situation where one seeks a rigorous formulation of the statement that a space-time is “close” to the Kerr solution, such as: its non-linear stability problem, the asymptotic settlement of a radiating isolated system undergoing gravitational collapse, or in the formulation of some uniqueness results.     

Supports of Measure Solutions for Spatially Homogeneous Boltzmann Equations

We prove that the support of the unique measure solution for the spatially homoge-neous Boltzmann equation in R3 is the whole space, if the initial distribution is not a Dirac measure and has 4-order moment. More precisely, we obtain the lower bound of exponential type for the probability of any small ball in ℝ³ relative to the measure solution.     

Cylindrically Symmetric Scalar Field and it’s Lyapunov stability in General Relativity

In this paper we found an Exact solution for massless scalar field with cosmological constant. This exact solution generalized the Levi-Civita vacuum solution Levi-Civita (Rend. Acc. Lincei 27:183, 1917) to a massless scalar field, with a cosmological constant term.This solution in the absence of the Cosmological constant recovers the spacetime of a massless scalar field with cylindrical symmetry (Buchdahl metric (Buchdahl in Phys. Rev. 115:1325, 1959)). Also if the scalar field disappears, the spacetime will be a representation of de-Sitter space.We prove that the form of the metric’s function which was purposed in Momeni and Miraghaei (Int. J. Mod. Phys. A 24(31):5991, 2009) is valid even if we assume a general form. Furthermore we show that in which conditions this solution satisfies energy conditions. Finally the credibility of focusing theorem is proved.     

A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system

We show that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The main tool is an estimate which shows that a solution is global if all the matter remains away from the center of symmetry.Research supported in part by NSF DMS 9101517     

Invariance of the Massless Field Equations under Changes of the Metric

It is shown that the Maxwell equations with sources, expressed in terms of the covariant tensor field F_ijand the current density four-vector Jⁱ, are invariant under the change of the metric g_ijby g_ij = g_ij+ l_il_jif l_iis a principal null direction of F_ijand that an analogous result holds in the case of the massless Klein-Gordon equation if l_iis null and orthogonal to the gradient of the field and in the case of the null dust equations if l_iis parallel to the dust four-velocity. An elementary proof of the following generalization of the Xanthopoulos theorem is also given: Let (g_ij, F_ij) be an exact solution of the Einstein-Maxwell equations and let l_ibe a principal null direction of F_ij, then (g_ij+ l_il_j, F_ij) is also an exact solution of the Einstein-Maxwell equations if and only if (l_il_j, 0) satisfies the Einstein-Maxwell equations linearized about the background solution (g_ij, F_ij). Furthermore, analogous theorems, where the source of the gravitational field is a massless Klein-Gordon field or null dust, are presented.     

Stability of Nonlinear Wave of Kadomtsev-Petviashvili-Type Equation

For ion-acoustic waves in a plasma with non-isothermal electrons,the MKP equation is its governing equation.The instability of a soliton solution of MKP equation to two-dimensional long-wavelength perturbations is investigated up to the third order.It indicates that the one-soliton solution of MKP equation is unstable if v = -1wheras it is stable if v = 1 until the third order approximation has been considered.     

Polaron stability in molecular crystals

A semi-empirical Peierls–Holstein model is applied to studies of the stability of polarons in two-dimensional molecular crystal systems. Calculations for a broad range of intra- and inter-molecular parameters within this model were performed in order to obtain detailed knowledge concerning the stability of the polaron solution with respect to a rigid lattice band solution. For realistic values of the parameters the polaron solution is stable with a polaron energy in the range 50–100 meV. A metastable polaron solution is also identified. The polarons that result from our model are highly localized and it is questionable if adiabatic polaron transport can occur in the system.     

An Example of Nonuniqueness for Solutions to the Homogeneous Boltzmann Equation

The paper deals with the spatially homogeneous Boltzmann equation for hard potentials. An example is given which shows that, even though it is known that there is only one solution that conserves energy, there may be other solutions for which the energy is increasing; uniqueness holds if and only if energy is assumed to be conserved.     

On local solutions of the initial value problem for the Vlasov-Maxwell equation

The initial value problem of the Vlasov-Maxwell equation has a unique solution in a time interval [0,T] for each initial data in some function space.T is estimated by the size of the initial data. The solution is classical, if the initial data is smooth.     

关于干涉反演光谱技术能否扩大F-P干涉仪光谱范围的探讨

本文从物理学的角度提出了Fredholm第一类积分方程数值解的可靠性概念,证明了在被称为Fabry-Perot干涉反演光谱技术中,当△σ=2/x,△σ=2/σ时,若取样点数为一个适当的奇数,那么积分方程的数值解是稳定的.但是进一步的计算机模拟实验表明,该数值解不是原积分方程的可靠解,因此,干涉反演光谱技术不能扩大Fabry-Perot干涉仪的光谱范围.理论分析表明,在前述条件下,积分方程数值解不可靠的根本原因在于该积分方程本身没有唯一解.     

Generation and propagation of pulse trains with ultrashort pulse separation

We investigate the nonlinear Schrödinger equation with variable coefficients by employing perturbation method. The analysis solution of the harmonic form is presented. The solution is one of forms to describe pulse trains with ultrashort pulse separation, which is about two orders of magnitude shorter than one of sech-type solitons considered before. And we could systematically adjust the perturbation parameter to obtain different pulse separation. As an example, we consider a nonlinear dispersive system with spatial parameter variations, and the results show that, the pulse train with ultrashort pulse separation presented by analysis solution may keep its shape even if the velocity is changed. The stability of the solution is discussed numerically, and the results reveal that the finite initial perturbations, such as white noise could not influence the main character of the solution. In addition, the stability of the solution is also discussed under more general conditions.     

Scattering and bound state solutions for a class of nonlocal potentials (s-wave)

Thes-wave scattering solution is discussed for a class of nonlocal (non-separable) potentials. Existence and uniqueness theorems are given and the analyticity domain in thek-variable (k = wave number in the C.M. system) is determined. Furthermore it is proved that solutions of the bound state problem exist and a discussion of the square-integrable solutions, which can occur for a real positive value of the energy, is given. In this last case the scattering solution also exists but it is not unique. Finally theS-matrix is introduced and it is proved that it is unambigously defined even if the scattering solution is not unique.     

On the mathematical conditions for the existence of periodic fluctuations in non-uniform media

In many areas of mathematical physics where one is interested in the propagation of waves through non-uniform media, it is often assumed that periodic excitations result in periodic responses. This assumption is examined by rigorously investigating the existence of periodic solutions of linear hyperbolic differential equations whose coefficients vary with position and whose solution must satisfy periodic boundary or source data. It is shown that the nature of the coefficients of the undifferentiated terms of the differential system is crucial in determining whether or not the solution is periodic. In physical applications, these coefficients usually depend on the gradients of media properties as well as on the media properties themselves. In particular, it is shown that for a general hyperbolic system of two equations in one space dimension, the solution is not periodic. Moreover, this can remain true even if the media gradients are assumed small. However, if the media gradients vanish, or if they vanish except for a bounded region of space, the solution is shown to be periodic for a large enough time. Furthermore, if these gradients vanish asymptotically at large distances, then the disturbances will be asymptotically periodic for increasing time. Special attention is given to the propagation of infinitesimal pressure disturbances through non-uniform steady flows of a lossless fluid.     

Quasiclassical solutions of the Klein-Gordon equation in a space-time with closed time-like curves

A formulation of the Cauchy problem for the Klein-Gordon equation in a space-time with a wormhole is studied. It is shown that if closed time-like curves pass through the Cauchy surface, then a global solution exists in the high-frequency (quasiclassical) limit only if the initial data satisfy nonlocal consistency conditions. In the simplest case these conditions determine a discrete spectrum of frequencies (energies of a quasiclassical scalar particle). Zh. éksp. Teor. Fiz. 113, 3–11 (January 1998)     

Complete Characterization of Convergence to Equilibrium for an Inelastic Kac Model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index ?? is determined by the so-called degree of inelasticity, p>0, of the particles: $\alpha=\frac{2}{1+p}$ . This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index ??. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.