Bianchi transformation between the real hyperbolic Monge-Ampère equation and the Born-Infeld equation

By casting the Born-Infeld equation and the real hyperbolic Monge-Ampère equation into the form of equations of hydrodynamic type, we find that there exists an explicit transformation between them. This is Bianchi transformation.     

■-dressing method for the complex modified KdV equation

The dressing method based on the 2 × 2 matrix ■-problem is generalized to study the complex modified KdV equation(cmKdV). Through two linear constraint equations, the spatial and time spectral problems related to the cmKdV equation are derived. The gauge equivalence between the cmKdV equation and the Heisenberg chain equation is obtained. Using a recursive operator,a hierarchy of cmKdV with source is proposed. On the basis of the ■-equation, the N-solition solutions of the cmKdV equation are obt...     

Application of coupled equation method on resonance processes of atomic lithium

The coupled equation method (CEM) has been applied to investigating the resonance structures for the ground state ls~22s~2S of the neutral lithium from the first threshold up to 64.5 eV.Resonance structures of atomic lithium due to single excitations of the ls and 2s electrons are studied by infinite-order calculations in detail.The effect of spin-orbit splitting is also included for some of the low-lying ls2snp(↑↓) resonance, and the influence of the interference between ls2s ~3Snp↓and ls2s ~1Snp↑states on the resonance structure has been confirmed theoretically.The results show that the presented technique can give the reasonable resonance structures very well in photoionization processes.     

自由曲面光束调控Monge-Ampère方法研究进展

自由曲面具有灵活的面形结构,用于光束调控可获得高性能、轻小型的系统,可创造新的结构形式和实现新的光束调控功能。自由曲面光束调控是一个根据输入和目标反求光学自由曲面的逆问题。Monge-Ampère(MA)方法基于理想光源近似,将自由曲面光束调控逆问题转化成一个带有非线性边界条件的MA方程。MA方法无需预先给定光线落点位置,而是通过控制曲面的高斯曲率分布来实现对光传输的高效灵活调控,被认为是当前最有效的可自动满足曲面连续性可积条件的自由曲面设计方法。对MA方法的研究进展进行了概述,详细介绍了自由曲面光束强度调控模型,以及自由曲面光束强度和波前同时调控模型的构建过程与求解方法,并通过三个设计实例充分展示了各类光束调控模型的有效性和MA方法的优势。     

Application of h-adaptive,high order finite element method to solve radial Schrödinger equation

The h-adaptive, high order finite element method is applied to solve a second order one dimension eigenvalue problem. The finite element formulation for the Lobatto basis is given, for which basis functions of arbitrary order can be constructed. The adaptive algorithm is simple, yet very efficient and straightforward to implement. The algorithm is based on the observation that the expansion coefficients of Lobatto basis functions decay rapidly. It allows evaluating the smallest eigenvalues simultaneously with the comparable accuracy for all eigenvalues. The presented algorithm is applied to solve the radial Schrödinger equation with the Coulomb and the Woods–Saxon potentials. For both potentials the convergence rate is presented. After seven adaptive iterations nine-digit accuracy was obtained.     

Application of the method of static fluctuational approach to the Bogolyubov–Kolesnikov–Shelah model

A method of calculating the equilibrium correlation functions of any arbitrary order for the Baldwin– Kolesnikov–Shelah (BKSh) model is suggested based on the static fluctuational approach. The method based on only one controllable approach allows the so-called equations of long-range coupling to be obtained which contain all information on the sought-after equilibrium correlation functions within the scope of the BKSh model. Calculations of the sought-after equilibrium correlation functions allow one to go beyond the scope of the conventional molecular field approach and to take into account the effect of field fluctuations on the gap behavior and the heat capacity to the left and right of the critical point. For the simplest case disregarding a dependence of the potential on the wave vector, temperature dependences of the energy gap and heat capacity with allowance for the fluctuations are presented. It is demonstrated that in this case, the fluctuations are small for three-dimensional systems, but sharply increase with decreasing dimensionality of the system.     

Using the complex WKB method for studying the evolution of initial pulses obeying the nonlinear Schrödinger equation

Using the complex WKB method, we found an asymptotic solution of the associated Zakharov-Shabat problem in the limit of a small coefficient h 0 multiplying the derivative of the potential that has a single hump. The obtained formulas can be used in describing the evolution of optical pulses of such shape obeying the nonlinear Schrödinger equation (NSE). Several examples are considered.V. V. Kuibyshev State University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 19–25, May, 1993.     

Fast finite difference solvers for singular solutions of the elliptic Monge–Ampère equation

The elliptic Monge–Ampère equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail.In this article we build a finite difference solver for the Monge–Ampère equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton’s method.Computational results in two and three-dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities.     

Application of a recent solution of the Navier–Stokes equation to flow on the surface of a globe

We reduce an exact solution of the 3D Navier–Stokes equation (Muriel, 2011) [1] to two dimensions to model flow on the surface of a globe, producing the following results: (a) an analytic discovery of the time evolution of two streams, one each above and below the equator, (b) analytic speed-up of modeling bypassing iterative numerical simulation.     

A conservative numerical method for the Cahn–Hilliard equation in complex domains

We propose an efficient finite difference scheme for solving the Cahn–Hilliard equation with a variable mobility in complex domains. Our method employs a type of unconditionally gradient stable splitting discretization. We also extend the scheme to compute the Cahn–Hilliard equation in arbitrarily shaped domains. We prove the mass conservation property of the proposed discrete scheme for complex domains. The resulting discretized equations are solved using a multigrid method. Numerical simulations are presented to demonstrate that the proposed scheme can deal with complex geometries robustly. Furthermore, the multigrid efficiency is retained even if the embedded domain is present.     

Application of Sobolev gradient method to Poisson–Boltzmann system

The idea of a weighted Sobolev gradient, introduced and applied to singular differential equations in [1], is extended to a Poisson–Boltzmann system with discontinuous coefficients. The technique is demonstrated on fully nonlinear and linear forms of the Poisson– Boltzmann equation in one, two, and three dimensions in a finite difference setting. A comparison between the weighted gradient and FAS multigrid is given for large jump size in the coefficient function.     

Application of the multireference equation of motion coupled cluster method,including spin–orbit coupling,to the atomic spectra of Cr,Mn, Fe and Co

The recently introduced multireference equation of motion (MR-EOM) approach is combined with a simple treatment of spin–orbit coupling, as implemented in the ORCA program. The resulting multireference equation of motion spin–orbit coupling (MR-EOM-SOC) approach is applied to the first-row transition metal atoms Cr, Mn, Fe and Co, for which experimental data are readily available. Using the MR-EOM-SOC approach, the splittings in each L-S multiplet can be accurately assessed (root mean square (RMS) errors of about 70 cm^?1). The RMS errors for J-specific excitation energies range from 414 to 783 cm^?1 and are comparable to previously reported J-averaged MR-EOM results using the ACESII program. The MR-EOM approach is highly efficient. A typical MR-EOM calculation of a full spin–orbit spectrum takes about 2 CPU hours on a single processor of a 12-core node, consisting of Intel XEON 2.93 GHz CPUs with 12.3 MB of shared cache memory.     

Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev-Petviashvili equation

The infinite-dimensional symmetry group of the potential Kadomtsev-Petviashvili (PKP) equation is found and used to obtain a Bäcklund transformation, involving two arbitrary functions of time. This transformation is then used to generate several different types of solutions from the zero solution of the PKP equation.     

Developments of parabolic equation method in the period of 2000–2016

Parabolic equation(PE) method is an efficient tool for modelling underwater sound propagation, particularly for problems involving range dependence. Since the PE method was first introduced into the field of underwater acoustics,it has been about 40 years, during which contributions to extending its capability has been continuously made. The most recent review paper surveyed the contributions made before 1999. In the period of 2000–2016, the development of PE method basically focuses on seismo-acoustic problems, three-dimensional problems, and realistic applications. In this paper, a review covering the contribution from 2000 to 2016 is given, and what should be done in future work is also discussed.     

Exact solutions to complex Ginzburg–Landau equation

In this paper, the trial equation method and the complete discrimination system for polynomial method are applied to retrieve the exact travelling wave solutions of complex Ginzburg–Landau equation. Both the Kerr and power laws of nonlinearity are considered. All the possible exact travelling wave solutions consisting of the rational function-type solutions, solitary wave solutions, triangle function-type periodic solutions and Jacobian elliptic functions solutions are obtained, and some of them are new solutions. In addition, concrete examples are presented to ensure the existence of obtained solutions. Moreover, four types of representative solutions are depicted to present the nature of the obtained solutions.     

Fisher information and the complex nature of the Schrödinger wave equation

We show that the minimum Fisher information (MFI) approach to estimating the probability law p(x) on particle position x, over the class of all two-component laws p(x), yields the complex Schrödinger wave equation. Complexity, in particular, traces from an efficiency scenario (demanded by MFI) where the two components of p(x) are so separated that their informations add.     

Application of movable collimators to the BEPC Ⅱ

BEPC Ⅱ is a double ring e+e-collider with high beam currents and luminosity,so the high beamrelated backgrounds may disturb the detector.In order to have a good quality of data taking,backgrounds should be kept at a level as low as possible.A series of collimators are designed and installed in both the e+and e-rings.Two of the collimators are horizontally movable,each for one ring,about 8 m upstream from the interaction point.Experiments have been done to identify the effectiveness of the movable collimators with different apertures and beam currents.The results show that the movable collimators are very effective and can reduce as much as about 50% of beam-related backgrounds.     

Analytical solution of the Korteweg–de Vries equation and microtubule equation using the first integral method

In this paper, the first integral method is applied to solve the Korteweg–de Vries equation with dual power law nonlinearity and equation of microtubule as nonlinear RLC transmission line. This method is manageable, straightforward and a powerful tool to find the exact solutions of nonlinear partial differential equations.     

Application of movable collimators to the BEPCⅡ

BEPCⅡ is a double ring e＋ecollider with high beam currents and luminosity,so the high beamrelated backgrounds may disturb the detector.In order to have a good quality of data taking,backgrounds should be kept at a level as low as possible.A series of collimators are designed and installed in both the e＋ and erings.Two of the collimators are horizontally movable,each for one ring,about 8 m upstream from the interaction point.Experiments have been done to identify the e-ectiveness of the movable collimators with different apertures and beam currents.The results show that the movable collimators are very effective and can reduce as much as about 50% of beam-related backgrounds.