The structure of Gelfand-Levitan-Marchenko type equations for Delsarte transmutation operators of linear multi-dimensional differential operators and operator pencils. Part 1

通过 Gel'fand--Levitan--Marchenko 方程对若干负阶非线性波动方程的反散射方法

Inverse scattering method via Gel'fand--Levitan--Marchenko equation for some negative order nonlinear wave equations

本文利用三方程 Lax 层级构造负阶 AKNS 方程，并借助 Gel'fand--Levitan--Marchenko 方程的反散射方法，给出耦合标量场的非线性 Klein--Gordon 方程与负阶 mKdV 方程的 N 孤子解。

A survey of the spectral and differential geometric aspects of the De Rham-Hodge-Skrypnik theory related with Delsarte transmutation operators in multidimension and its applications to spectral and soliton problems. Part 2

The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions are studied making use of the De Rham-Hodge-Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multidimensional differential operators are done including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills eq…

非线性负一阶耦合克莱因-戈登方程的逆散射方法

Inverse scattering method for nonlinear negative first order coupled Klein-Gordon equation

本文运用逆散射方法推导了耦合负一阶克莱因-戈登方程的N孤子解，构造了其守恒律等结构，建立了核函数与势的对应关系，为该类耦合非线性方程的求解提供了新途径。

Positivity of Some Integral Transforms, and Generalization of Bochner's Theorem on Functions of Positive Type

Using the integral representations of the solutions of Schrödinger equation, which are the essential ingredients of the Gel'fand-Levitan and Marchenko integral equations of inverse scattering theory, we obtain a general theorem on the positivity of some integral transforms, and extend the theorem of Bochner on Fourier transforms of functions of positive type to more general transforms. The present study is restricted to the positive half-axis. We then obtain a theorem on the positivity of Fourier cosine transform of the phase-shifts.

Multi-indexed Extensions of Soliton Potential and Extended Integer Solitons of KdV Equation

We calculate infinite set of initial profiles of higher integer KdV solitons, which are both exactly solvable for the Schrodinger equation and for the Gel'fand-Levitan-Marchenko equation in the inverse scattering transform method of KdV equation. The calculation of these higher integer soliton solutions is based on the recently developed multi-indexed extensions of the reflectionless soliton potential.

NMR脉冲设计的离散逆散射理论

Discrete Inverse Scattering Theory for NMR Pulse Design

本文引入Zakharov-Shabat系统的离散散射理论，推导了用于核磁共振RF脉冲生成的高效递归算法，并证明该算法产生的脉冲在某些方面优于标准SLR脉冲。