Abstract

With a vanishing boundary condition, we consider a revised Riemann–Hilbert problem (RHP) for the derivative nonlinear Schrödinger equation (DNLS), where an integral factor is introduced such that the RHP satisfies the normalization condition. In the reflectionless situation, we construct the formulas for the \(N\)th-order solutions of the DNLS equation, including the solitons and positons that respectively correspond to \(N\) pairs of simple poles and one pair of \(N\)th-order poles of the RHP. According to the Cauchy–Binet formula, we show the expressions for \(N\)th-order solitons. Additionally, we give an explicit expression for the second-order positon and graphically describe evolutions of the third-order and fourth-order positons.

中文翻译：

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导数非线性薛定谔方程的修正黎曼-希尔伯特问题：消失边界条件

摘要

在边界条件消失的情况下，我们考虑导数非线性薛定谔方程 (DNLS) 的修正黎曼-希尔伯特问题 (RHP)，其中引入积分因子以使 RHP 满足归一化条件。在无反射情况下，我们构造了DNLS方程的\(N\)阶解的公式，其中包括分别对应于\(N\)对简单极点和一对\(N \) RHP 的三阶极点。根据柯西-比奈公式，我们给出了\(N\)个三阶孤子的表达式。此外，我们给出了二阶位置的显式表达式，并以图形方式描述了三阶和四阶位置的演化。