Single shock and periodic sawtooth-shaped waves in media with non-analytic nonlinearities

Abstract

The review of new mathematical models containing non-analytic nonlinearities is given. These equations have been proposed recently, over the past few years. The models describe strongly nonlinear waves of the first type, according to the classification introduced earlier by the authors. These models are interesting because of two reasons: (i) equations admit exact analytic solutions, and (ii) solutions describe the real physical phenomena. Among these models are modular and quadratically cubic equations of Hopf, Burgers, Korteveg-de Vries, Khokhlov-Zabolotskaya and Ostrovsky-Vakhnenko type. Media with non-analytic nonlinearities exist among composites, meta-materials, inhomogeneous and multiphase systems. Some physical phenomena manifested in the propagation of waves in such media are described on the qualitative level of severity.

Extinction of multiple shocks in the modular Burgers equation

“…which is different from the classical Burgers equation by the modular advection term. This model was previously used to describe inelastic dynamics of particles with piecewise interaction potentials [4,13]. Generalizations of this model with additional terms were also discussed in [10,11,12].…”