Solitons are ubiquitous phenomena that appear, among others, in the description of tsunami waves, fiber-optic communication and ultracold atomic gases. The latter systems turned out to be an excellent playground for investigations of matter-wave solitons in a quantum world. This tutorial provides a general overview of the ultracold contact interacting Bose and Fermi systems in a one-dimensional space that can be described by the renowned Lieb–Liniger and Yang–Gaudin models. Both the quantum many-body systems are exactly solvable by means of the Bethe ansatz technique, granting us a possibility for investigations of quantum nature of solitonic excitations. We discuss in details a specific class of quantum many-body excited eigenstates called yrast states and show that they are strictly related to quantum dark solitons in the both considered Bose and Fermi systems.

孤子是无处不在的现象，其中出现在海啸波、光纤通信和超冷原子气体的描述中。后一种系统被证明是研究量子世界中物质波孤子的绝佳场所。本教程提供了一维空间中超冷接触相互作用的 Bose 和 Fermi 系统的一般概述，可以用著名的 Lieb-Liniger 和 Yang-Gaudin 模型来描述。这两个量子多体系统都可以通过 Bethe ansatz 技术精确求解，这为我们研究孤子激发的量子性质提供了可能性。