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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. [ABSTRACT FROM AUTHOR]

Several threads of the last 25 years' developments in nonlinear wave theory that stem from the classical Korteweg–de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a nonlocal integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors' view of the future development of the chosen lines of nonlinear wave theory. [ABSTRACT FROM AUTHOR]

Nonlinear internal solitary waves observed in laboratory experiments are discussed from the standpoint of their relation to different soliton theories, from the classical integrable models such as the Korteweg–de Vries, Gardner, Benjamin–Ono, and Joseph–Kubota–Ko–Dobbs equations and their modifications, through the nonintegrable models describing higher-order nonlinear effects, viscosity, rotation, and cylindrical spreading, to the strongly nonlinear models. First, these theoretical models are briefly described and, then, laboratory data and their comparison with the theory are presented. [ABSTRACT FROM AUTHOR]

Studies the dynamics of particles in the steady flows of an inviscid fluid. Equations of motion and some exact solutions; Dynamics of a particle in axisymmetric flows; Dynamics of particles in a cellular flow; Symmetry of Lyapuniv exponents.