Soliton Dynamics of NLS Equation at the Presence of Perturbations

A new direct approach based on the separation of variables for soliton perturbations is developed. With the aid of this approach, the effects of perturbation on a soliton of nonlinear Schrödinger (NLS) equation is obtained. In comparison with other direct methods,our approach is very concise and easy to be understood. Besides, no more approximation is employed except for the linearization of the perturbed NLS equation.     

A Bohmian approach to the non-Markovian non-linear Schrödinger–Langevin equation

In this work, a non-Markovian non-linear Schrödinger–Langevin equation is derived from the system-plus-bath approach. After analyzing in detail previous Markovian cases, Bohmian mechanics is shown to be a powerful tool for obtaining the desired generalized equation.     

Number-resolved master equation approach to quantum measurement and quantum transport

In addition to the well-known Landauer–Büttiker scattering theory and the nonequilibrium Green’s function technique for mesoscopic transports, an alternative (and very useful) scheme is quantum master equation approach. In this article, we review the particle-number (n)-resolved master equation (n-ME) approach and its systematic applications in quantum measurement and quantum transport problems. The n-ME contains rich dynamical information, allowing efficient study of topics such as shot noise and full counting statistics analysis. Moreover, we also review a newly developed master equation approach (and its n-resolved version) under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.     

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Holm equation is obtained. One-loop soliton solution of the Degasperis-Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.     

分析完全电离等离子体扩散问题的边界单元法

一、扩散方程 完全电离等离子体的扩散问题,可归结为下面的微分方程定解问题。 在内; 在Γ_1上; 在Γ_2上; 式中,A=ηκB~Z/B~Z,n=n(r,t)是等离子体密度,κ是玻尔兹曼常数,T是绝对温度,B是磁感应强度,η是电导率,Ω是由Γ=Γ_1 Γ_2界定的区域,ω是边界的外法向方向,和是边界上的已知函数。     

Schrödinger Equation for An Extended Electron

A new quantum mechanical wave equation describing the dynamics of an extended electron is derived via Bohmian mechanics. The solution to this equation is found through a wave packet approach which establishes a direct correlation between a classical variable with a quantum variable describing the dynamics of the center of mass and the width of the electron wave packet. The approach presented in this paper gives a comparatively clearer picture than approaches using elaborative manipulation of infinite series of operators. It is shown that the new Schrödinger equation is free of any runaway solutions or any acausal responses.     

Contaminant Flow and Transport Simulation in Cracked Porous Media Using Locally Conservative Schemes

The purpose of this paper is to analyze some features of contaminant flow passing through cracked porous medium, such as the influence of fracture network on the advection and diffusion of contaminant species, the impact of adsorption on the overall transport of contaminant wastes. In order to precisely describe the whole process, we firstly build the mathematical model to simulate this problem numerically. Taking into consideration of the characteristics of contaminant flow, we employ two partial differential equations to formulate the whole problem. One is flow equation; the other is reactive transport equation. The first equation is used to describe the total flow of contaminant wastes, which is based on Darcy law. The second one will characterize the adsorption, diffusion and convection behavior of contaminant species, which describes most features of contaminant flow we are interested in. After the construction of numerical model, we apply locally conservative and compatible algorithms to solve this mathematical model. Specifically, we apply Mixed Finite Element (MFE) method to the flow equation and Discontinuous Galerkin (DG) method for the transport equation. MFE has a good convergence rate and numerical accuracy for Darcy velocity. DG is more flexible and can be used to deal with irregular meshes, as well as little numerical diffusion. With these two numerical means, we investigate the sensitivity analysis of different features of contaminant flow in our model, such as diffusion, permeability and fracture density. In particular, we study $K_d$ values which represent the distribution of contaminant wastes between the solid and liquid phases. We also make comparisons of two different schemes and discuss the advantages of both methods.     

Asymptotic equations for two-body correlations

An asymptotic equation for two-body correlations is proposed for a large number of particles in the framework of the integro-differential equation approach. The quality of this equation is discussed with examples.     

A new high-order spectral problem of the mKdV and its associated integrable decomposition

A new approach to formulizing a new high-order matrix spectral problem from a normal 2× 2 matrix modified Korteweg--de Vries (mKdV) spectral problem is presented. It is found that the isospectral evolution equation hierarchy of this new higher-order matrix spectral problem turns out to be the well-known mKdV equation hierarchy. By using the binary nonlinearization method, a new integrable decomposition of the mKdV equation is obtained in the sense of Liouville. The proof of the integrability shows that r-matrix structure is very interesting.     

A Modified Homogeneous Balance Method and Its Applications

A modified homogeneous balance method is proposed by improving some key steps in the homogeneous balance method. Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneous balance method. Generalized Boussinesq equation, KP equation, and mKdV equation are chosen as examples to llustrate our method. This approach is also applicable to a large variety of nonlinear evolution equations.     

Dirac equation in the presence of coulomb and linear terms in (1 + 1) dimensions; the supersymmetric approach

Using the concept of supersymmetry, we exactly solve the Dirac equation in (1 + 1) dimension for a potential containing both linear and coulomb terms. This potential, due to its physical interpretation, is of interest within many areas of theoretical physics. To do this, using the SUSY approach, we first find the Hamiltonian of the corresponding Schrödinger equation and then, using the idea of shape invariance, find the eigenfunctions and eigenvalues.     

Hierarchy of equations for the surface tension of solids

Of the four main equations in thermodynamics for the surface tension of condensed matter, i.e. the generalized and classical Lippmann equations and the Shuttleworth and Gokhshtein equations, only the classical Lippmann and Gokhshtein equations have been confirmed experimentally. The generalized Lippmann (Couchman–Davidson) equation is considered to be more universal, since three other equations could be derived from it. Although this fact has been widely accepted, it was recently reevaluated in two opposite ways. In the first approach, the experimental verification of the Gokhshtein equation should support the correctness of the generalized Lippmann and Shuttleworth equations. In the second approach, the incompatibility of the Shuttleworth equation with Hermann's mathematical structure of thermodynamics throws doubts upon all its corollaries, including the generalized Lippmann and Gokhshtein equations. However, both of these approaches are here shown to be erroneous, since the Gokhshtein equation cannot be correctly derived from any of the above-mentioned equations, and the opposite is also true: neither the generalized Lippmann nor Shuttleworth equations could be derived from the Gokhshtein equation.     

Augmented Langevin approach to fluctuations in nonlinear irreversible processes

A Fokker-Planck equation derived from statistical mechanics by M. S. Green [J. Chem. Phys. 20:1281 (1952)] has been used by Grabertet al. [Phys. Rev. A 21:2136 (1980)] to study fluctuations in nonlinear irreversible processes. These authors remarked that a phenomenological Langevin approach would not have given the correct reversible part of the Fokker-Planck drift flux, from which they concluded that the Langevin approach is untrustworthy for systems with partly reversible fluxes. Here it is shown that a simple modification of the Langevin approach leads to precisely the same covariant Fokker-Planck equation as that of Grabertet al., including the reversible drift terms. The modification consists of augmenting the usual nonlinear Langevin equation by adding to the deterministic flow a correction term which vanishes in the limit of zero fluctuations, and which is self-consistently determined from the assumed form of the equilibrium distribution by imposing the usual potential conditions. This development provides a simple phenomenological route to the Fokker-Planck equation of Green, which has previously appeared to require a more microscopic treatment. It also extends the applicability of the Langevin approach to fluctuations in a wider class of nonlinear systems.     

General Symmetry Approach to Solve Variable—Coefficient Nonlinear Equations

After considering the variable coefficient of a nonlinear equation as a new dependent variable,some special types of variable-coefficient equation can be solved from the corresponding constant-coefficient equations by using the general classical Lie approach.Taking the nonlinear Schrodinger equation as a concrete example,the method is recommended in detail.     

The study of sensitivity and precession of a single-source diffused tomography for detecting of target depth in biological phantom

Optical imaging can be used to study the cancerous stages of breast cancer; and this imaging is noninvasive and safe for healthy tissues. The key problem of optical imaging is the image reconstruction which depends on photon migration in biological tissues; because the study of photon migration in the biological tissues is a complicated problem. The diffusion equation is sometimes used to simulate the photon migration in the biological tissues. Due to limitation of diffusion equation and its approximated nature, we want to explore the accuracy and precision of this method. So in this study, we report the design of a single-source diffuse imaging system with simulating code based on finite element method (FEM) to detect the location of tumor in breast phantoms. The comparison between the reconstructed results and actual values can be considered as a criterion of accuracy of this diffused system. In this experimental setup, to reduce the expense of computational procedure, single source is applied, and the accuracy and precision of this single-source are investigated.     

Variational supersymmetric approach to evaluate Fokker-Planck probability

In this work we introduce a method to determine the time dependent probability density for the one-dimensional Fokker-Planck equation. The treatment is based in an analysis of the Schrödinger equation through the variational method associated to the formalism of supersymmetric quantum mechanics (SQM). The approach uses an ansatz for the superpotential which allows us to obtain the trial functions of the variational method. The hierarchy of effective Hamiltonians permits us to determine the variational eigenfunctions and energies of the excited states to the evaluation of the probability. The symmetric bistable potential is used to illustrate the approach whose results are compared with results obtained by the state-dependent diagonalization method and by direct numerical calculation.     

Analytical solutions for the slow neutron capture process of heavy element nucleosynthesis

In this paper, the network equation for the slow neutron capture process (s-process) of heavy element nucleosynthesis is investigated. Dividing the s-process network reaction chains into two standard forms and using the technique of matrix decomposition, a group of analytical solutions for the network equation are obtained. With the analytical solutions, a calculation for heavy element abundance of the solar system is carried out and the results are in good agreement with the astrophysical measurements.     

Soliton Solutions of Discrete Complex Ginzburg-Landau Equation via Extended Hyperbolic Function Approach

In this paper, we extend the hyperbolic function approach for constructing the exact solutions of nonlinear differential-difference equation (NDDE) in a unified way. Applying the extended approach and with the aid of Maple,we have studied the discrete complex Ginzburg-Landau equation (dCGLE). As a result, we find a set of exact solutions which include bright and dark soliton solutions.     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.``