具有非零背景的经典与非局域非线性薛定谔方程的解：双线性化与约化方法
The solutions of classical and nonlocal nonlinear Schrödinger equations with nonzero backgrounds: Bilinearisation and reduction approach
AI总结：
本文提出双线性化-约化方法，从二阶 AKNS 耦合方程出发，构造具有非零背景的经典及反向空间、反向时间、反向时空非局域 NLS 方程的解，并给出聚焦 NLS 方程任意阶怪波公式。
摘要：
本文发展了一种双线性化-约化方法，用于推导具有非零背景的经典和非局域非线性薛定谔（NLS）方程的解。我们从作为未约化系统的二阶 Ablowitz-Kaup-Newell-Segur 耦合方程出发。借助一对解 $(q_0,r_0)$，我们对未约化系统进行双线性化，并以拟双 Wronskian 的形式得到解。然后，通过对 Wronskian 的列向量引入约束来实现约化，最终得到约化方程的解，分别包括经典 NLS 方程以及具有反向空间、反向时间和反向时空的非局域 NLS 方程。以一组平面波解 $(q_0,r_0)$ 作为背景解，我们给出了这些列向量的显式公式。作为例子，我们分析并展示了聚焦 NLS 方程和反向空间非局域 NLS 方程的解。特别地，我们给出了聚焦 NLS 方程任意阶怪波的公式。
英文摘要：
In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schrödinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur coupled equations as an unreduced system. With a pair of solutions $(q_0,r_0)$ we bilinearize the unreduced system and obtain solutions in terms of quasi double Wronskians. Then we implement reductions by introducing constraints on the column vectors of the Wronskians and finally obtain solutions to the reduced equations, including the classical NLS equation and the nonlocal NLS equations with reverse-space, reverse-time and reverse-space-time, respectively. With a set of plane wave solution $(q_0,r_0)$ as a background solution, we present explicit formulae for these column vectors. As examples, we analyze and illustrate solutions to the focusing NLS equation and the reverse-space nonlocal NLS equation. In particular, we present formulae for the rouge waves of arbitrary order for the focusing NLS equation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
