标题： 关于尖线麦克斯韦-布洛赫方程的零背景孤子 显示英文标题

标题： On zero-background solitons of the sharp-line Maxwell-Bloch equations

摘要： 这项工作致力于系统地研究可能包含多个退化孤子组（DSGs）的一般$N$孤子解，在零背景下的尖线麦克斯韦-布洛赫方程的背景下。我们还表明，这些结果可以轻松迁移到其他可积系统中，这些系统具有相同的非自伴扎哈罗夫-沙巴特散射问题或类似的问题。 聚焦非线性薛定谔方程和复修改库尔特韦格-德弗里斯方程的结果作为明确的例子，用于演示目的。 一个 DSG 是由不可分离的具有相同速度的孤子组成的局域化相干非线性行波结构。 因此，DSGs 是单个孤子的推广（被视为$1$-DSGs），并构成了许多可积系统解的基本构建块。 我们提供了$N$-DSG 及其中心的显式公式。 借助 Deift-Zhou 的非线性最陡下降方法，我们证明了 DSG 的局域化，并计算了任意$N$-孤子解的长时间渐近行为。 结果表明，在遥远的过去和未来，解会成为多个 DSG 的线性组合，并给出了每个 DSG 渐近相位移的显式公式。 还讨论了单个孤子的其他推广形式，例如$N$阶孤子和孤子气体。 我们证明了每一个$N$阶孤子可以通过$N$孤子解的本征值融合得到，通过适当缩放归一化常数，并且证明孤子气体解可以视为$N$孤子解在$N\to+\infty$时的极限。 显示英文摘要

摘要： This work is devoted to systematically study general $N$-soliton solutions possibly containing multiple degenerate soliton groups (DSGs), in the context of the sharp-line Maxwell-Bloch equations with a zero background.We also show that results can be readily migrated to other integrable systems, with the same non-self-adjoint Zakharov-Shabat scattering problem or alike. Results for the focusing nonlinear Schr\"{o}dinger equation and the complex modified Korteweg-De Vries equation are obtained as explicit examples for demonstrative purposes. A DSG is a localized coherent nonlinear traveling-wave structure, comprised of inseparable solitons with identical velocities. Hence, DSGs are generalizations of single solitons (considered as $1$-DSGs), and form fundamental building blocks of solutions of many integrable systems. We provide an explicit formula for an $N$-DSG and its center. With the help of the Deift-Zhou's nonlinear steepest descent method, we prove the localization of DSGs, and calculate the long-time asymptotics for an arbitrary $N$-soliton solutions. It is shown that the solution becomes a linear combination of multiple DSGs in the distant past and future, with explicit formulae for the asymptotic phase shift for each DSG. Other generalizations of a single soliton are also discussed, such as $N$th-order solitons and soliton gases. We prove that every $N$th-order soliton can be obtained by fusion of eigenvalues of $N$-soliton solutions, with proper rescalings of norming constants, and demonstrate that soliton-gas solution can be considered as limits of $N$-soliton solutions as $N\to+\infty$.