Soliton models: Traditional and novel, one- and multidimensional

Abstract

This article offers an introduction to the vast area of experimental and theoretical studies of solitons. It is composed of two large parts. The first one provides a review of effectively one-dimensional (1D) settings. The body of theoretical and experimental results accumulated for 1D solitons is really large, the most essential among them being overviewed here. The second part of the article provides a transition to the realm of multidimensional solitons. These main parts are split into a number of sections, which clearly define particular settings and problems addressed by them. This article may be used by those who are interested in a reasonably short, but, nevertheless, sufficiently detailed introduction to the modern “soliton science”. It addresses, first, well-known “traditional” topics. In particular, these are the integrable Korteweg–de Vries, sine-Gordon, and nonlinear Schrödinger (NLS) equations in 1D, as well as the Kadomtsev–Petviashvili equations in 2D, and basic physical realizations of these classical equations. Then, several novel topics are addressed. Especially important between them are 2D and 3D solitons of the NLS type, which are stabilized against the collapse (catastrophic self-compression, which is the fundamental problem impeding the realization of multidimensional solitons) by the spin-orbit coupling or effects by quantum fluctuations in two-component Bose–Einstein condensates in ultracold atomic gases. This article introduces a part of the material which is represented in a systematic form in a new book, Multidimensional Solitons (B. A. Malomed, AIPP, 2022).