Waves, Solids, and Nonlinearities

In this article nonlinearity is taken as a basic property of continua or any other wave-bearing system. The analysis includes the conventional wave propagation problems and also the wave phenomena that are not described by traditional hyperbolic mathematical models. The basic concepts of continuum mechanics and the possible sources of nonlinearities are briefly discussed. It is shown that the technique of evolution equations leads to physically well-explained results provided the basic models are hyperbolic. Complicated constitutive behavior and complicated geometry lead to mathematical models of different character and, as shown by numerous examples, other methods are then used for the analysis. It is also shown that propagating instabilities possess wave properties and in this case the modeling of energy redistribution has a great importance. Finally, some new directions in the theory and applications are indicated.