论文标题
衍生物非线性schr $ \ ddot {o} $ dinger方程的长期渐近线
The long-time asymptotic of the derivative nonlinear Schr$\ddot{o}$dinger equation with step-like initial value
论文作者
论文摘要
本文中的考虑是对衍生物非线性schr $ \ ddot {o} $ dinger方程的长期渐近造型，具有阶梯式初始值\ begin \ begin {eqnarray} q（x，0）= q_ {0} = q_ {0} a_ {1} e^{i ϕ} e^{2ibx}，\ quad \ quad x <0，\\ a_ {2} e^{ - 2ibx}，\ quad \ quad \ quad ~~ x> 0。 \ end {split} \ nonumber \ end {cases} \ end {eqnarray}通过deift-zhou方法。矩阵Riemann-Hilbert问题描述了类似的初始问题。本文中使用的一种关键成分是引入$ g $功能机制，以将跳跃矩阵条目的问题呈指数增长为$ t \ rightarrow \ rightarrow \ infty $。结果表明，theta函数$θ$表达的dnls方程的渐近解决方案的主要顺序项涉及3属3属的riemann-surface和由抛物线缸和通风函数表达的二次顺序术语。
Consideration in this present paper is the long-time asymptotic of solutions to the derivative nonlinear Schr$\ddot{o}$dinger equation with the step-like initial value \begin{eqnarray} q(x,0)=q_{0}(x)=\begin{cases} \begin{split} A_{1}e^{iϕ}e^{2iBx}, \quad\quad x<0,\\ A_{2}e^{-2iBx}, \quad\quad~~ x>0. \end{split}\nonumber \end{cases} \end{eqnarray} by Deift-Zhou method. The step-like initial problem described by a matrix Riemann-Hilbert problem. A crucial ingredient used in this paper is to introduce $g$-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as $t\rightarrow\infty$. It is shown that the leading order term of the asymptotic solution of the DNLS equation expressed by the Theta function $Θ$ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.
