The modulational theory of nonlinear gravity-wave in fluid with varying depth and nonuniform current

By means of WKB expansions, new fourth order evolution equations are derived for two-dimensional Stokes waves over the bottom with arbitrary depth. The effects of slowly varying depthh=h(ε ²x) and currentU=U(ε ²x,ε²t,ε⁴z) on the evolution of a packet of Stokes waves are considered as well. In addition, numerical simulation is performed for the evolution of single envelope by finite-difference method. Project supported by National Natural Science Foundation of China and Centre of Advanced Academic Research of Zhongshan University.     

翻转的Varga材料球壳的渐近分支

众所周知,弹性球壳在翻转后可能不再保持球壳形状而出现起皱现象,也就是出现分支解。本文运用WKB方法,分析了各向同性且不可压缩的超弹性Varga材料球壳翻转后的变形问题。在大模数情形下,对于A-B=O(1),得到了球壳内、外径比的分支临界值的渐近表达式。对模数区域内几乎所有的模数,分析结果与数值结果吻合得很好。     

SWKB formalism for confined quantum systems

A formalism is developed to study spatially confined one‐dimensional quantum mechanical systems in the framework of the supersymmetric Wentzel–Kramers–Brillouin (SWKB) method. The approximation technique is applied to estimate the energy eigenvalues of two confined potentials—the harmonic oscillator V(x)=x² and the screened Coulomb potential V(x)=−V₀sech²x. The results thus obtained are found to be in better agreement with the exact numerical values than are those from the ordinary WKB approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 267–273, 2000     

超对称准经典近似和本征值问题

运用超对称准经典近似方法给出三维谐振子、氢原子的能谱,进而将该方法用于含角坐标的二阶微分方程,得到角动量平方L2的本征值和非中心势的角向本征值.     

Resurgent aspects of applied exponential asymptotics

In many physical problems, it is important to capture exponentially small effects that lie beyond-all-orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial-over-power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial-over-power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.     

An asymptotic expansion for energy eigenvalues of anharmonic oscillators

In the present contribution, we derive an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V(x)=κx^2q+ωx²,q=2,3,…$V\left(x\right)=\kappa x^{2q}+\omega x^2,q=2,3,\dots$ as the energy level n$n$ approaches infinity. The asymptotic expansion is obtained using the WKB theory and series reversion. Furthermore, we construct an algorithm for computing the coefficients of the asymptotic expansion for quartic anharmonic oscillators, leading to an efficient and accurate computation of the energy values for n≥6$n\ge 6$.     

The Aharonov–Bohm effect in scattering theory

The Aharonov–Bohm effect is considered as a scattering event with nonrelativistic charged particles of the wavelength which is less than the transverse size of an impenetrable magnetic vortex. The quasiclassical WKB method is shown to be efficient in solving this scattering problem. We find that the scattering cross section consists of two terms, one describing the classical phenomenon of elastic reflection and another one describing the quantum phenomenon of diffraction; the Aharonov–Bohm effect is manifested as a fringe shift in the diffraction pattern. Both the classical and the quantum phenomena are independent of the choice of a boundary condition at the vortex edge, providing that probability is conserved. We show that a propagation of charged particles can be controlled by altering the flux of a magnetic vortex placed on their way.     

On the exact WKB analysis of higher order simple-pole type operators

Higher order simple-pole type operators, that is, higher order linear ordinary differential operators with a large parameter η whose coefficients have simple poles at the origin, are discussed from the viewpoint of the exact WKB analysis. Making use of the technique of microdifferential operators, we clarify the singularity structure of the Borel transform of their WKB solutions.     

Sound modes in a nonequilibrium fluid with acoustic boundary conditions

The sound modes in a plane fluid layer with sound-absorbing walls at different temperatures are evaluated. The eigenvalue problem is solved by means of a singular perturbation theory and the WKB method. The bending of sound, the nonexponential damping, and the roie of the wall admittance are discussed.     

一维幂次势阱中粒子能谱规律的分析

分析了一维幂次势阱中粒子的能谱,讨论了能谱形状随幂次变化的规律．