Soliton Solutions of DNLS Equation Found by IST Anew and its Verification in Marchenko Formalism

A general procedure is proposed to derive the multi-soliton solutions of DNLS equation with vanishing boundary value, and the two-soliton solutions of it is given as an example. Furthermore, the verification of multi-soliton solutions is done through Marchenko formalism. PACS numbers: 05.45.Yv; 02.30.-f; 11.10.Ef     

Green's Function Method for Perturbed sine-Gordon Equation

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.     

Inverse Scattering Transform in Squared Spectral Parameter for DNLS Equation under Vanishing Boundary Conditions

After a transformation, the inverse scattering transform for the derivative nonlinear Schr6dinger （DNLS） equation is developed in terms of squared spectral parameter. Following this approach, we obtain the orthogonality and completeness relations of free Jost solutions, which is impossibly constructed with usual spectral parameter in the previous works. With the help these relations, the Zakharov-Shabat equations as well as Marchenko equations of IST are derived in the standard way.     

Soliton Asymptotics of Solutions of the Sine-Gordon Equation

An asymptotic analysis of the Marchenko integral equation for the sine-Gordon equation is presented. The results are used for a construction of soliton asymptotics of decreasing and some non-decreasing solutions of the sine-Gordon equation. The soliton phases are shown to have an additional shift with respect to the reflectionless case caused by the non-zero reflection coefficient of the corresponding Dirac operator. Explicit formulas for the phases are also obtained. The results demonstrate an interesting phenomenon of splitting of non-decreasing solutions into an infinite series of asymptotic solitons.     

Demonstration of Inverse Scattering Transform in G-L-M Formalism for NLS Equation in Normal Dispersion with Non-vanishing Boundary

Using the integral representation of the Jost solution, we deduce some conditions as the kernel function N（x, y, t） if the Jost solution satisfies the two Lax equations. Then we verify the multi-soliton solution of NLS equation with non-vanishing boundary conditions if we prove that these conditions can be demonstrated by the GLM equation, which determines the kernel function N （x, y, t） in according to the inverse scattering method.     

An Inverse Scattering Transformation for the Modified Nonlinear Schrijdinger Equation

A new Lax pair of the modified nonlinear Schrödinger equation is introduced in terme of the variable of the Fourier transform λ. The Lax pair has no usual symmetries between 12 and 21 elements and avoids the factor λ^1/2. The basic equation of inverse scattering transformation is deduced in the Zakharov-Shabat form as well as in the Marchenko form.     

Zakharov-Shabat Equations for Dark Soliton Solutions to the NLS Equation

For dark soliton solutions of the NLS equation, an inverse scattering transform is redeveloped. Deductions are essentially simplified in terms of an auxiliary spectral parameter from the beginning. Equations of inverse scattering transform in the form of Zakharov-Shabat are found to be simpler than those in the form of Marchenko. An explicate expression for the dark N-soliton solution and its asymptotic behaviors in the limits as t →±∞ are simply derived.     

The series solution of the Marchenko equation

The convergence of a Neumann-series solution of the Marchenko equation in L ₂[x, ∞) is proved and an explicit expression of the series solution is given.     

Exactly Solvable Combinations of Scalar and Vector Potentials for the Dirac Equation Interrelated by Riccati Equations

New classes of solvable scalar and vector potentials for the Dirac equation are obtained, together with the associated exact Dirac spinors. The method of derivation is based on an a priori constraint between the solutions, leading to an interrelation between the scalar and vector potential in the form ofa Riccati equation. The present note generalizes a series of former articles.     

Exact Three-Dimensional Relativistic Equation for qq Bound states

Starting with the QCD generating functional, a new derivation and a new form of the Dirac-Schriidinger (D-S) equation,catisfied by the equal-time Bethe-Salpeter amplitude of quark-antiquark bound states have been given. In this equation, the effective interaction kernel has a compact expression and can be directly calculated by means of the conventional QCD Feynman rules. F'urthermore, an equivalent reduction of the above equation to a Pauli-Schriidinger equation has also been achieved and a closed form of the effective interaction Hamiltonian appearing in the latter equation has been explicitly written out.     

CLOSED FORM SOLUTIONS TO THE INTEGRABLE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using a matrix triplet (A, B, C). Here A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N-soliton and breather solutions as special cases. We also prove that these solutions reduce to known continuous matrix NLS solutions as the discretization step vanishes.     

A new approach to the inverse scattering and spectral problems for the Sturm-Liouville equation

New proofs of the known uniqueness theorems for the one-dimensional inverse spectral and scattering problems are given. Proof of the invertibility of all of the steps in the inversion procedures of Gelfand-Levitan and Marchenko is given. The proposed method of investigation yields some new results, for example, a Marchenko-type equation at x = 0 which holds on the whole axis, rather than on a half-axis, as usual for the scattering theory on half-axis. It also yields a new method, shorter and simpler than earlier published, for proving that the potential in the class L_1,1, obtained by the Marchenko reconstruction procedure, generates the scattering data from which it was reconstructed.     

Master Equation and Fokker–Planck Equation: Comparison of Entropy and of Rate Constants

A usual approximation of the master equation is provided by the Fokker–Planck equation. For chemical systems with one species, we prove generally that the prediction of the rate constant of the metastable state given by the Master equation and the Fokker–Planck approximation differ exponentially with respect to the size of the system. We show that this is related to the fact that the entropy of the metastable state is not described correctly by the Fokker–Planck equation. We prove that the rate given by the Fokker–Planck equation overestimates that rate given by the Master equation.     

Singular Boundaries in the Forward Chapman-Kolmogorov Differential Equation

The forward Chapman-Kolmogorov differential equation is used to model the time evolution of the Probability Density Function of fluctuations. This equation may be restricted to either Master, Fokker-Planck or Liouville equations. A derivation of the Liouville equation with possible singular boundary conditions has already been presented in a previous publication (Valiño and Hierro in Phys. Rev. E 67:046310, 2003). In this paper, that derivation is extended to the full Chapman-Kolmogorov differential equation.     

Low-Energy Proton-Deuteron Scattering with a Coulomb-Modified Faddeev Equation

A modified version of the Faddeev three-body equation to accommodate the Coulomb interaction, which was used in the study of three-nucleon bound states, is applied to the proton-deuteron scattering problem at energies below the three-body breakup threshold. A formal derivation of the equation in a time-independent scattering theory is given. Numerical results for phase-shift parameters are presented to be compared with those of other methods and results of the phase-shift analysis. Differential cross sections and nucleon analyzing powers are calculated with the effects of three-nucleon forces, and these results are compared with recent experimental data. The difference between the nucleon analyzing power in proton-deuteron scattering and that in neutron-deuteron scattering is discussed.Received March 14, 2002; accepted September 29, 2002 Published online June 27, 2003     

COMPLETENESS OF THE JOST SOLUTIONS IN THE CASE OF THE NLS+ EQUATION WITH NONVANISHING BOUNDARY CONDITION

In the case of the NLS⁺ equation with nonvanishing boundary condition, a complete set of the Jost solutions is chosen, and its completeness is shown by means of the Marchenko inverse scattering equation.     

量子噪音与量子Langevin方程

介绍了量子噪音和量子Langevin方程 ,并与经典噪音和经典Langevin方程进行了比较 .量子噪音来源于两种途径 ,第一种与经典噪音相似 ,第二种则起源于Heisengberg测不准原理 .同时 ,也简略给出了量子Langevin方程的推导.The properties of quantum noise and Langevin equation are discussed. Comparisons between the quantum noise and Langevin eqution and the classic one are presented. A brief derivation for quantum Langevin equation is showed. The quantum noise comes from two ways, namely, the way as same as that of classic noise and the Heisenberg uncertainty.     

A Search on Dirac Equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.     

Multisoliton Solutions for the Isospectral and Nonisospectral BKP Equation with Self-Consistent Sources

The isospectral and nonisospectral BKP equation with self-consistent sources is derived to study the linear problem of the BKP system. The bilinear form of the nonisospeetral BKP equation with self-consistent sources is given and the N-soliton solutions are obtained with the Hirota method and Pfaffian technique, respectively.