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Localized wave structures: Solitons and beyond

Ostrovsky, L.; Pelinovsky, E.; Shrira, V.; Stepanyants, Y.


L. Ostrovsky

E. Pelinovsky

Y. Stepanyants


The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg–de Vries (KdV) equation. Among the topics we focus upon are “radiating solitons,” the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV equations with the modular and “sublinear” nonlinearities. Such equations have an interesting class of solutions, called compactons, solitary waves defined on a finite spatial interval. Both the properties of single solitons and the interactions between them are discussed. We show that even minor non-elastic effects in the soliton–soliton collisions can accumulate and result in a qualitatively different asymptotic behavior. A statistical description of soliton ensembles (“soliton gas”), which emerges as a major theme, has been discussed for several models. We briefly outline the recent progress in studies of ring solitons and lumps within the framework of the cylindrical KdV equation and its two-dimensional extension. Ring solitons and lumps (2D solitons) are of particular interest since they have many features in common with classical solitons and yet are qualitatively different. Particular attention is paid to interactions between the objects of different geometries, such as the interaction of ring solitons and shear flows, ring solitons and lumps, and lumps and line solitons. We conclude our review with views of the future developments of the selected lines of studies of localized wave structures in the theory of weakly nonlinear, weakly dispersive waves.


Ostrovsky, L., Pelinovsky, E., Shrira, V., & Stepanyants, Y. (2024). Localized wave structures: Solitons and beyond. Chaos, 34(6), Article 062101.

Journal Article Type Article
Acceptance Date May 9, 2024
Online Publication Date Jun 10, 2024
Publication Date Jun 15, 2024
Deposit Date Jun 21, 2024
Publicly Available Date Jun 25, 2024
Journal Chaos: An Interdisciplinary Journal of Nonlinear Science
Print ISSN 1054-1500
Electronic ISSN 1089-7682
Publisher AIP Publishing
Peer Reviewed Peer Reviewed
Volume 34
Issue 6
Article Number 062101
Keywords Wave mechanics, Soliton solutions, Partial differential equations, Plasma waves, Korteweg-de Vries equation, Fluid flows, Internal waves, Surface waves, Gravity wave
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