ABSTRACT
Extreme waves can have profiles that resemble harmonic waves, shock waves, solitons, and oscillons. On the other hand, they can have multivalued profiles, that is, profiles that have folds. These profiles are reminiscent of Euler’s figures, so we called them elastica-like waves. At the same time, a few times we got wave profiles that follow from Euler’s figures but were not included in the traditional list of them (see, for example, Figures 2.17, 2.19–2.21, 5.7, 5.8, 9.17, and 11.3). They are formed by almost harmonic or cnoidal waves and particles soaring above them. It is possible to define this type of solution as particle-wave. It is important that all these waves are described by nonlinear equations.
Here, we explore approximate solutions of nonlinear wave equations that can be interpreted as periodic or localized particle-waves. We think that during fast dynamics, scalar fields should generate a configuration like a particle-wave. Thus, in this chapter, we continue to study a new class of extreme nonlinear waves, namely, particle-waves.
