求解DNLS方程的反散射法的基本问题

对DNLS方程, 反散射法应选取k²=p为基本谱参数, 其中k为通常谱参数. 并引入一个新的谱参数及一个规范变换, 由此得到自由Jost解的规范正交系, 这就是反散射法的基本问题. 同时还得到了基于谱参数p的Marchenko方程和Zakharov-Shabat反散射方程. 关键词： DNLS方程 反散射法 自由Jost解 规范正交系     

A novel method for solving the three-body scattering problem

A numerical scheme for solving a three-body scattering problem within the framework of the configuration space Faddeev equations in three-dimension, i.e., without resort to explicit partial wave expansion, is presented. The method is applied to calculate the low-energy n-d observables.     

Perturbation method for solving the inverse vibrational problem in dependent coordinates

Solution of the inverse vibrational problem (IVP) when the dependent coordinates are excluded leads to a linear combination of the molecular force constants, of very limited usefulness for a series of related compounds. In this work we consider the question of separating these combinations during the solution of the IVP in a system of dependent coordinates. Such a solution is obtained by introducing a perturbation into the matrix of kinematic coefficients with the help of a certain self-consistent matrix. We obtain intrinsic force constants for tetrahedral hydrides (methane, silane, and germane) which reproduce experimental frequency spectra well.Three Hundredth Anniversary of the Reunion of the Ukraine with Russia State University, DnepropetrovskTranslated from Izvestiya Uchebnykh Zavedenii, Fizika, No. 8, pp. 26–30, August, 1987.     

The inverse scattering problem and the equivalent local potential

We examine the role of the non-locality of the potential in the two-body scattering by taking the nucleon-Σ system as an example. We employ a non-local potential for the nucleon-Σ channel which has the characteristic features of the quark model and reproduces the phase shifts of widely used local potentials. We use inverse scattering methods to obtain the equivalent local potential from the non-local potential, and show that the obtained local potential has a strong short range repulsion.     

A Zakharov-Shabat inverse scattering problem with a polynomial spectral dependence in the potential

The inverse scattering problem for the Zakharov-Shabat system , is considered. It is solved by reduction to the inverse scattering problem for two coupled Zakharov-Shabat systems in a 4n-dimensional space: . This work has been done as part of the program ‘Recherche Coopérative sur Programme No. 264: Etude interdisciplinaire des problèmes inverses’. Physique Mathématique et Théorique, Equipe de recherche associée au CNRS, No. 154.     

带源参数的热传导反问题的无网格方法

利用无网格有限点法求带有源参数的一维热传导反问题，推导了相应的离散方程.与其他基于网格的方法相比，有限点法采用移动最小二乘法构造形函数，只需要节点信息，不需要划分网格，用配点法离散求解方程，可以直接施加边界条件，不需要在区域内部求积分，减小了计算量.用有限点法求解热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点.最后通过算例验证了该方法的有效性.     

On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations

Supersymmetric Liouville and sine-Gordon equations are studied. We write down for these models the system of linear equations for which the method of inverse scattering should be applicable. Expressions for an infinite set of conserved currents are explicitly given. Supersymmetric Bäcklund transformations and generalized conservation laws are constructed.     

A meshless method for solving an inverse spacewise-dependent heat source problem

In this paper an effective meshless and integration-free numerical scheme for solving an inverse spacewise-dependent heat source problem is proposed. Due to the use of the fundamental solution as basis functions, the method leads to a global approximation scheme in both spatial and time domains. The standard Tikhonov regularization technique with the generalized cross-validation criterion for choosing the regularization parameter is adopted for solving the resulting ill-conditioned system of linear algebraic equations. The effectiveness of the algorithm is illustrated by several numerical examples.     

Finite-difference approximation for inverse scattering problem

For the first time potentials are reconstructed in a finite-difference approximation using a genuine inverse scattering method instead of multiple repeated solutions of a direct problem with iterative fitting of scattering data. Up to now a fundamental difference between spectral properties of the Schrödinger operator and its discrete analog hindered from doing this.     

Three-body inverse scattering problem

Two methods are suggested to reconstruct three-body potentials from three-body scattering data. This was achieved by using the reduction of the corresponding Schrödinger equation to a system of ordinary differential equations (not integro-differential equations as usual in the direct problem). Exactly solvable three-body models are presented. A new simple method for solving the multi-dimensional inverse problem in a finite-difference approximation is considered in the Appendix.     

Factorization method for inverse obstacle scattering problem in three-dimensional planar acoustic waveguides

In this paper, we consider the inverse scattering problem of reconstructing a bounded obstacle in a three-dimensional planar waveguide from the scattered near-field data measured on a finite cylindrical surface containing the obstacle and corresponding to infinitely many incident point sources also placed on the measurement surface. The obstacle is allowed to be an impenetrable scatterer or a penetrable scatterer. We establish the validity of the factorization method with the nearfield data to characterize the obstacle in the planar waveguide by constructing an outgoing-to-incoming operator which is an integral operator defined on the measurement surface with the kernel given in terms of an infinite series.     

A second degree Newton method for an inverse obstacle scattering problem

A regularized second degree Newton method is proposed and implemented for the inverse problem for scattering of time-harmonic acoustic waves from a sound-soft obstacle. It combines ideas due to Johansson and Sleeman [18] and Hettlich and Rundell [8] and reconstructs the obstacle from the far field pattern for scattering of one incident plane wave.     

Nanoparticle shape reconstruction by solving the direct and inverse small-angle scattering problems for a unit potential localized inside a torus

The simplest doubly connected surface of revolution (torus) is used as an example to demonstrate the possibility of stable reconstruction a three-dimensional homogeneous body defined by a unit potential U(r) using a spherically averaged small-angle scattering (SAS) curve I(s). Annealing Monte Carlo simulations are performed without using prior information about nanoparticle shape and size. Exact and approximate expressions are obtained for the form factor of a torus. It is shown graphically that the exact and approximate SAS form-factor curves agree for an experimentally accessible scattering region. Examples are given.     

Method of solving the inverse spectral problem

We describe a method of solving the inverse spectral problem which is applicable to free molecules as well as to molecules in a medium. We give a criterion for the uniqueness of the definition of the force matrix. This criterion links the necessary number of normal proper vibrations of a molecule with a number of necessary form parameters. We calculate the force matrix of tetrahedral molecules CH₄, CD₄, and CT₄. When we use zero-point values for frequencies of normal vibrations the relative discrepancy in the corresponding force-matrix elements does not exceed 0.3%.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 52–57, April, 1987.     

Optimized method for solving the inverse problem for a triangular plasmonic waveguide using a response of scanning heterodyne microscope

A method proposed previously to solve the inverse problem for triangular plasmonic waveguides using a TE-polarized response of scanning differential heterodyne microscope has been optimized. The use of the phase contrast of microscope response at two wavelengths as initial data made it possible to reduce significantly the solution error and eliminate the instability domains of initial data, in which the solution error dramatically increased. It is demonstrated that the accuracy of determining the waveguide parameters significantly improves in comparison with the case where the amplitude and phase contrast of response at one wavelength are used as initial data. It is shown that the solution of the inverse problem can be further optimized using contrasts of amplitude responses as additional initial data. The dependences of the error of inverse problem solution on the error in determining the initial data are calculated.     

On relativistic-invariant formulation of the inverse scattering transform method

A manifestly relativistic-invariant formulation of the method of inverse scattering transform for relativistic-invariant equations is proposed. The sine-Gordon model and the massive Thirring model are considered.     

Relevance vector machine technique for the inverse scattering problem

A novel method based on the relevance vector machine(RVM) for the inverse scattering problem is presented in this paper.The nonlinearity and the ill-posedness inherent in this problem are simultaneously considered.The nonlinearity is embodied in the relation between the scattered field and the target property,which can be obtained through the RVM training process.Besides,rather than utilizing regularization,the ill-posed nature of the inversion is naturally accounted for because the RVM can produce a probabilistic output.Simulation results reveal that the proposed RVM-based approach can provide comparative performances in terms of accuracy,convergence,robustness,generalization,and improved performance in terms of sparse property in comparison with the support vector machine(SVM) based approach.