Abstract
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.
Highlights
This is not a review article per se with the careful collection of all the references available on the topic, but rather a viewpoint of the author explained by numerous examples
Nonlinear wave motion as a physical phenomenon is a rich area of contemporary science where mechanics is spiced with mathematics, physics of materials interwoven with technology, and new problems form a driving force for developing new methods of analysis
A rather general statement, advocated in rational mechanics describes a wave as a state moving into another state with a finite speed
Summary
This is not a review article per se with the careful collection of all the references available on the topic, but rather a viewpoint of the author explained by numerous examples. Dealing with continua of complicated properties, besides the conventional observable variables, the notion of internal variables is useful (Kestin, 1992) This leads directly to nonhyperbolic governing equations that still may exhibit wave-type solutions. Hyperbolicity may be lost in deriving the governing equations but still certain wave phenomena could be described by these mathematical models. This list gives the main idea of the study: starting from traditional mathematical models we would like to describe nonlinear waves in complicated situations characteristic to contemporary science and technology. The section deals with traditional problems on the basis of hyperbolic or asymptotically hyperbolic mathematical models In this case the evolution equations for single waves are derived. Some new directions in the theory and applications are indicated
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