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.     

Scattering States of Schrdinger Equation under the Modified Cusp Potential

We solve the one-dimensional time-independent Schrdinger equation in the presence of the modified Cusp potential and report the solutions in terms of the Whittaker functions. We obtain the reflection and transmission coefficients as well as the bound-state solutions in terms of the Whittaker functions. We comment on the solutions and discuss them in terms of the engaged parameters.     

Pion-Nucleus Scattering and the Interacting Boson Model

A procedure of combining the medium energy scattering theory in the Eikonal approximation with the interacting boson model is proposed to study the pion-nucleus scattering. The channel coupling effects in the excitation of nuclear collective state with ρ^- and ρ⁺ elastic and inelastic scattering are analyzed. The calculated cross sections for the excitation of 2₁⁺ state of ²¹⁸Sn agree with experimental data quite well.     

Pion-Rotational Nucleus Scattering and the Interacting Boson Model

A procedure of combining the medium energy scattering theory in the eikonal approximation with the interacting boson model (IBM) is proposed to study the pion-rotational nucleus scattering. The channel coupling effects in the excitation of nuclear collective state with π^- and π⁺ elastic and inelastic scattering are analyzed. The calculated cross sections for the excitation of 2₁⁺ state of ¹⁵²Sm agmree with the experimental data quite well.     

Scattering of Solitons of Modified KdV Equation with Self-consistent Sources

This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate case of two solitons and ＂ghost＂ solitons, etc. Co-moving coordinate frames are employed in asymptotic analysis.     

Scattering States of Schrödinger Equation under the Modified Cusp Potential

We solve the one-dimensional time-independent Schrödinger equation in the presence of the modified Cusp potential and report the solutions in terms of the Whittaker functions. We obtain the reflection and transmission coefficients as well as the bound-state solutions in terms of the Whittaker functions. We comment on the solutions and discuss them in terms of the engaged parameters.     

Scattering States of l-Wave Schrödinger Equation with Modified Rosen-Morse Potential

Within a Pekeris-type approximation to the centrifugal term, we examine the approximately analytical scattering state solutions of the l-wave Schrödinger equation with the modified Rosen-Morse potential. The calculation formula of phase shifts is derived, and the corresponding bound state energy levels are also obtained from the poles of the scattering amplitude.     

Explicit N-Soliton Solution of the Modified Nonlinear SchrSdinger Equation by Means of the Inverse Scattering Transformation

The Zakharov-Shabat equations o: inverse scattering transformation for the modified nonlinear Schr6dinger equation in the reflectionless case are solved by means of the technique of determinant calculation. An explicit expression of the N-soliton solution is obtained. Its expected asymptotic behavior is derived.     

Inverse Scattering Transform for the Derivative Nonlinear Schrodinger Equation

Since the Jost solutions of the derivative nonlinear Schrodinger equation do not tend to the free Jost solutions, when the spectral parameter tends to infinity（｜A｜→∞）, the usual inverse scattering transform （IST） must be revised. If we take the parameter κ = λ^-1 as the basic parameter, the Jost solutions in the limit of ｜κ｜→∞ do tend to the free Jost solutions, hence the usual procedure to construct the equations of IST in κ-plane remains effective. After we derive the equation of IST in terms of κ, we can obtain the equation of IST in λ-plane by the simple change of parameters λ = κ^-1. The procedure to derive the equation of IST is reasonable, and attention is never paid to the function W（x） introduced by the revisions of Kaup and Newell. Therefore, the revision of Kaup and Newell can be avoided.     

Modified Structure-Preserving Schemes for the Degasperis-Procesi Equation

We investigate the structure-preserving numerical algorithm of the Degasperis-Procesi equation which can be transformed into a bi-Hamiltonian form using the discrete variational derivative method.Based on two different space discretization methods,the Fourier pseudospectral method and the wavelet collocation method,we develop two modified structure-preserving schemes under the periodic boundary condition.These proposed schemes are proved to preserve the Hamiltonian invariants theoretically and numerically.Meanwhile,the numerical results confirm that they can simulate the propagation of solitons effectively for a long time.     

On the Inverse Scattering Method for Integrable PDEs on a Star Graph

We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schrödinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is introduced. For such conditions, the method is shown to be as efficient as the ISM on the full-line.     

Demonstration of Inverse Scattering Transform for DNLS Equation

Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as ｜λ｜→∞, the usual inverse scattering transform （IST） must be revised. Beside the Kaup and Newell＇s approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying λ^-2 or λ^-1. To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.     

Demonstration of Inverse Scattering Transform for DNLS Equation

Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as |λ| →∞, the usual inverse scattering transform (IST) must be revised. Beside the Kaup and Newell's approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying λ-2 or λ-1. To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.     

致密物质状态方程:中子星与奇异星

中子星结构一直是核物理、粒子物理和天体物理共同关注的热点难题,双中子星并合事件GW170817的发现更是掀起这一研究的高潮。致密物质的状态方程是决定中子星结构的关键输入量,但是到目前为止,高密度的核物质状态方程行为依然很难确定。如今国内外已有许多运行或规划的大型核实验装置和天文观测设备,有望帮助我们很快解开致密物质状态方程的谜团。本文系统地阐述了基于微观多体理论和唯象模型对脉冲星类天体状态方程的研究现状,也讨论了奇异相变和奇异物质。结合理论计算和核物理实验及天文观测数据,致密物质状态方程的研究已取得相当多进展,但是也面临不少挑战,比如从实验和观测数据提取状态方程信息时的模型依赖,中子星各部分模型的不自洽以及各种依赖热密物质复杂动力学性质的实验和观测量。随着LIGO即将再运行而发现更多双中子星甚至中子星-黑洞等并合事件,多信使天文观测可望最终揭开中子星结构之谜。The matter state inside neutron stars (NSs) is an exciting problem in nuclear physics, particle physics and astrophysics. The equation of state (EOS) of NSs plays a crucial role in the present multimessenger astronomy, especially after the event of GW170817. Thanks to accruing studies with advanced telescopes and radioactive beam facilities, the unknown EOS of supranuclear matter could soon be understood. We review the current status of the EOS for pulsar-like compact objects, that have been studied with both microscopic many-body approaches and phenomenological models. The appearance of strange baryonic matter and strange quark matter are also discussed. We compare the theoretical predictions with different data coming from both nuclear physics experiments and astrophysical observations. Despite great progresses obtained in dense nuclear matter properties, there are various challenges ahead, such as the model dependence of the constraints extracted from either experimental or observational data, the lack of a consistent and rigorous many-body treatment of all parts of the star, the dependence of many observables on the turbulent dynamics of relevant hot dense system. As LIGO is about to run again and discover more NS merger events, multimessenger observations are expected to finally unravel the mystery of NS structure.     

Modified Transition Temperature Equation for Superconductors

We present the replacement and modification of the Debye temperature θD by the average phonon frequency （ω） in the Rowell [Solid State Commun. 19 （1976） 1131] linear transition-temperature equation for superconducting materials. We not only improve Rowell＇s results but also describe the experimental results accurately and consistently for the various superconducting systems which cover the entire range of λ between 0.72 and 2.59. The proposed linear equation of the transition temperature Tc is found to be better and more accurate than those of dain and Kachhava [Can. d. Phys. 58 （1980） 1614].     

Binary Darboux Transformation for the Modified Kadomtsev--Petviashvili Equation

Via the elementary Darboux transformation (DT) of the modified Kadomtsev--Petviashvili (mKP) equation, a binary Darboux transformation (BDT) of the mKP equation is constructed.     

Equation of State of Protoneutron Star Matter

We have calculated the equation of state of protoneutron star matter by using the lowest order constrained variational method. In our calculations, the modern Argonne potential (AV₁₈) together with its older model potential (AV₁₄) are used. It is found that the equation of state for high lepton fraction is stiffer than for low lepton fraction. It is seen that the increasing effect of pressure due to high lepton fraction and due to entropy are comparable. It is shown that the temperature and adiabatic index depend on the values of both entropy and lepton fraction.     

On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation

In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as ²0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.     

On the Inverse Scattering Approach to the Camassa-Holm Equation

Abstract

A simple algoritm for the inverse scattering approach to the Camassa-Holm equation is presented.     

Characteristic Properties of the Scattering Data for the mKdV Equation on the Half-Line

In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip or in the quarter plane (0<x<)×[0,T), T. We suppose that this solution has a C initial function vanishing as x, and C boundary values, vanishing as t when T=. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for x0 and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result of the paper is the collection A–B–C of characteristic properties of the scattering functions given below.