Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.     

从一道错题谈教师对学生探究活动的引导

一、错题的探究题目如图,Rt△ABC中,∠ACB=90°,CD⊥AB,D为垂足,若AD,BD的长是方程2x2-12x+3m-1=0的两根,且△ABC的面积为12.(1)求CD的长;(2)求m的值.先让学习练习,学生A的解法是:解:(1)∵AD、BD的长是方程2x2-12x+3m-1=0的两根,∴AD+BD=6,即AB=6.又∵CD⊥AB且△ABC的面积为12,∴12AB·CD=12,12×6CD=12,∴CD=4.(2)∵∠ACB=90°,CD⊥AB,∴CD2=AB·BD.又∵AD·DB=3m-12,CD=4,∴3m-12=16,∴m=11.结果有二十个左右的学生的意见同上述解答一样,不同意见的也有二十个左右,没举手的有5个.学生B代表不同意见说:“上述(2…