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

¹ Blekinge Institute of Technology, 371 79 Karlskrona, Sweden
² Department of Physics, Moscow State University, 119991 Moscow, Russia
³ Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow, Russia
⁴ Schmidt Earth Physics Institute, Russian Academy of Sciences, Moscow, Russia

^* Corresponding author: claes.hedberg@gmail.com

Received: 10 October 2017
Accepted: 10 October 2017

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.