一个广义导数非线性Schrodinger方程的Riemann-Hilbert问题:长时间渐近行为  

Riemann-Hilbert problem to a generalized derivative nonlinear Schrodinger equation:Long-time asymptotic behavior

作　　者：田守富 Shou-Fu Tian

机构地区：[1]中国矿业大学数学学院,徐州221116

出　　处：《中国科学：数学》2022年第5期505-542,共38页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11975306);江苏省自然科学基金(批准号:BK20181351);江苏省“六大人才高峰”高层次人才计划(批准号:JY-059);中国矿业大学重大项目培育专项(批准号:2019ZDPY07)和青年基金项目(批准号:2019QNA35)资助项目。

摘　　要：本文提出一个新的可积广义导数非线性Schrodinger (generalized derivative nonlinear Schrodinger, gDNLS)方程,并给出其Lax对表示.此g DNLS方程可以包含Chen-Lee-Liu方程和Gerjikov-Ivanov方程作为两种特殊情形.本工作首次建立了gDNLS方程关于初值问题在直线上的Riemann-Hilbert问题.有趣的是,原初值问题的解可以被表示成相关Riemann-Hilbert问题的解.基于建立的Riemann-Hilbert问题,本文借助Deift-Zhou非线性速降法成功地推导出gDNLS方程初值问题在直线上解的长时间渐近行为.A new integrable generalized derivative nonlinear Schrodinger(gDNLS) equation is introduced with its Lax pair. The gDNLS equation includes Chen-Lee-Liu equation and Gerjikov-Ivanov equation as two special cases. In this paper, a new Riemann-Hilbert problem formalism is presented for the initial value problem of the equation on the line. The solution of this initial value problem can be written in terms of the solution of some associated Riemann-Hilbert problem. Based on the new Riemann-Hilbert problem, the long-time asymptotic behavior for the gDNLS equation can be derived by using the Deift-Zhou nonlinear steepest-descent method on the line.

关 键 词：可积系统 RIEMANN-HILBERT问题 初值问题 长时间渐近行为 非线性速降法 

分 类 号：O175.29[理学—数学]