Modified Scattering for the Boson Star Equation

We consider the question of scattering for the boson star equation in three space dimensions. This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type. We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data. We describe the behavior of such solutions at infinity by identifying a suitable nonlinear asymptotic correction to scattering. As a byproduct of the weighted energy estimates alone, we also obtain global existence and (linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb.     

On the canonical character of the Sawada-Kotera equation

Recursion operators for solutions of two linear equations, associated with the SK equation, are given. A method to construct constants on the motion is described. The hamiltonian structure and the relation with the inverse scattering problem is indicated.     

A dynamical formulation of one-dimensional scattering theory and its applications in optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrödinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.     

The Heisenberg equation for phonon operators and soliton solutions in nonlinear lattices

The Heisenberg equation for phonon operators in nonlinear lattices is derived establishing the interaction Hamiltonian included higher powers of particle-hole pairs in nonlinear lattices. A phonon operator consists of a particle-hole pair in the harmonic potential approximation in the two band model; it represents an up or down transition of atoms between two levels. Applying the boson transformation method to the Heisenberg equation for phonon operators, we obtain the classical dynamical equation and a linear equation with the self-consistent potential created by the extended objects in nonlinear lattices. The boson transformation leads to soliton solutions in the long wavelength limit. The linear equation can be used to obtain scattering states, bound states and translational modes for phonons.     

Nonlinear effects in interference of bose-einstein condensates

Nonlinear effects in the interference of Bose-Einstein condensates are studied using exact solutions of the one-dimensional nonlinear Schrodinger equation, which is applicable when the lateral motion is confined or negligible. With the inverse scattering method, the interference pattern is studied as a scattering problem with the linear Schrodinger equation, whose potential is profiled by the initial density distribution of the condensates. Our theory not only provides an analytical framework for quantitative predictions for the one-dimensional case, it also gives an intuitive understanding of some mysterious features of the interference patterns observed in experiments and numerical simulations.     

Probability distributions for photon exit

A linear integral equation is formulated for the probability that a photon in a spectral line formed at a specified depth in a scattering atmosphere eventually escapes at a particular frequency in the spectral line. Numerical solutions are obtained for an isothermal atmosphere and their physical meaning discussed under a variety of conditions.     

MOL solution of DOM for transient radiative transfer in 3-D scattering media

A methodology based on the method of lines solution of discrete ordinates method for solution of the 3-D transient radiative transfer equation is introduced. The method is applied to the prediction of transient and steady state transmittances in a cubical enclosure containing purely scattering medium and validated against Monte Carlo solutions from the literature. The flexibility of the method for implementation of linear spatial differencing schemes, flux limiters and weighted essentially non-oscillatory methods is demonstrated. Van Leer flux limiter is found to provide stable, accurate and efficient solutions.     

Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.     

Quantal description of inelastic heavy-ion scattering

A model Schrödinger equation for scattering with energy loss is discussed. The equation is linear and is closely related to the coupled-channels approach to elastic and inelastic scattering. A possible parametrization of the model for applications to heavy-ion scattering is considered.     

Solitary wave dynamics in shallow water over periodic topography

The problem of long-wave scattering by piecewise-constant periodic topography is studied both for a linear solitary-like wave pulse, and for a weakly nonlinear solitary wave [Korteweg-de Vries (KdV) soliton]. If the characteristic length of the topographic irregularities is larger than the pulse length, the solution of the scattering problem is obtained analytically for a leading wave in the framework of linear shallow-water theory. The wave decrement in the case of the small height of the topographic irregularities is proportional to delta2, where delta is the relative height of the topographic obstacles. An analytical approximate solution is also obtained for the weakly nonlinear problem when the length of the irregularities is larger than the characteristic nonlinear length scale. In this case, the Korteweg-de Vries equation is solved for each piece of constant depth by using the inverse scattering technique; the solutions are matched at each step by using linear shallow-water theory. The weakly nonlinear solitary wave decays more significantly than the linear solitary pulse. Solitary wave dynamics above a random seabed is also discussed, and the results obtained for random topography (including experimental data) are in reasonable agreement with the calculations for piecewise topography.     

Linear alternative to the Boltzmann equation without linearization: III. Impurity scattering and electric field

Our previously reported projection formalism leading to a linear alternative to the Boltzmann (-Uehling-Uhlenbeck) kinetic equation is here applied to standard elastic scattering on impurities in solids and external electric field. In the former case in which the Boltzmann equation is linear, we prove that our formalism reproduces this equation. In the dc as well as ac electric field and for any scattering mechanism, it is verified that inclusion of the field reduces (upon neglecting of the field dependence of scattering processes) in the lowest order to inclusion of just the standard field driving term.     

Discrete Time and Intrinsic Length in Quantum Mechanics

We consider the possibility that simultaneously time and intrinsic length can be regarded as discrete real parameters. We study the dynamics of the free particle. For both scattering and bound states there are configurations where the energy is bounded from above and from below even for positive wave-function solutions. For the case of continuous evolution we show that the wave equation with a linear scalar coupling describes an oscillator that has built-in hidden supersymmetry.     

Elastic scattering of hadrons outside the diffraction cone

The model-independent solution of the s-channel unitarity condition for the imaginary part of the hadronic elastic scattering amplitude outside the diffraction peak allows to make conclusions about the shape of the differential cross sections in this region. The eigensolution of the homogeneous linear integral equation and the iterative solutions of the inhomogeneous equation with different assumptions about the behavior of the overlap function and the real part of the amplitude are presented. The properties of the overlap function and of the ratio of the real-to-imaginary part of the amplitude are discussed. In particular, it is explicitly shown that the ratio changes its sign at a definite value of the transferred momentum. Some comments concerning the present day experimental results about the behavior of the differential cross section of elastic scattering outside the diffraction cone are given.     

The soliton connection

It is pointed out that the linear scattering problem for a non-linear evolution equation which admits soliton solutions may be described in terms of a linear connection on a principal SL(2, ). The equation in question is satisfied if and only if the curvature of this connection vanishes. Some other properties of the curvature are identified. The sine-Gordon, Korteweg-de Vries and modified Korteweg-de Vries equations are treated explicitly.On leave of absence. Visit supported by NSF Grant 36217 and by the Norman Foundation.     

Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.     

Time Asymptotics for Solutions of Vlasov–Poisson Equation in a Circle

We prove that there exists a class of solutions of the nonlinear Vlasov–Poisson equation (VPE) on a circle that converges weakly, as t , to a stationary homogeneous solution of VPE. This behavior is called, in the linear case, Landau damping. The result is obtained by constructing a suitable scattering problem for the solutions of the Vlasov–Poisson problem. A consequence of this result is that a class of stationary solutions of the Vlasov–Poisson equation is unstable in a weak topology.     

A parametric study of radiative transfer with anisotropic scattering in a one-dimensional system

Radiative heat transfer in an absorbing, emitting, anistropically-scattering, one-dimensional medium is analyzed. Unlike many of the existing works, the present analysis does not require a known temperature distribution within the medium. Assuming a model of linear anistropic scattering, the transfer equation and the energy equation are solved simultaneously by utilizing a recently developed successive approximation technique. Closed-form approximate solutions and accurate higher-order results are both presented. Calculations show that the relative importance of the anistropic scattering effect generally decreases with decreasing wall emissivity and decreasing optical thickness. For radiative equilibrium without internal heat generation, it is demonstrated that the anistropic-scattering heat-transfer results can be approximated quite adequately by the isotropic-scattering result with the introduction of the concept of an effective optical thickness. For media with internal heat generation, an interesting effect of the scattering albedo is observed. It is established that, in the limit of a large scattering albedo, the temperature of the medium approaches a constant value that is independent of anistropic-scattering effects and wall emissivity. The exact limiting expressions for the temperature and apparent emissivity of an isothermal slab are found.     

On the Diagonalization of the Bethe–Salpeter Equation for the Imaginary Part of the Scattering Amplitude

The Bethe–Salpeter (BS) equation for the imaginary part of the scattering amplitude is examined in the ladder approximation of scalar quantum electrodynamics (QED). Asymptotic solutions of the BS equation derived for the imaginary part of the scattering amplitude are shown to exhibit the Regge amplitude behavior for small momentum transfer and scattering in the forward direction and at arbitrary angles.     

密集波分复用石英光纤通信系统中受激Raman散射的稳态分析模型

假设石英光纤的Raman增益谱为线性谱,并给出了拟合直线.以此为基础,得到了前向N信道 受激Raman散射稳态耦合波方程的解析解.这个解析解是在考虑了N个信号光之间串话下得到 的,它适用于任意功率大小的信号光和任意信道间隔排列的情况.N个信号光在石英光纤中经 过受激Raman散射作用后,具有以下特点：在传输过程中,任意两信道的信号光光子通量的比 值随光纤的有效互作用长度、总的输入光子通量和两信道频率间隔按指数规律变化.解析解 与数值解进行了比较,两者取得了很好的一致. 关键词： 受激Raman散射 密集波分复用 石英光纤 Raman放大