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Discontinuous change and superposition in nonlinear waves, asymptotics and bifurcation. (English) Zbl 0568.35067

The paper deals with basic ideas and recent developments in the field of nonlinear wave propagation. The topics covered include hyperbolic equations and waves, with a discussion of simple waves, entropy conditions, gradient catastrophe and shocks, blow-up of solutions, as well as dispersive waves for general evolution equations, with a discussion of solitons and solitary waves for KdV, nonlinear string and Boussinesq equations.
As an example of a recently discovered rather unusual soliton, the Ichikawa loop soliton arising from transverse oscillations of an elastic beam subject to an end thrust is reviewed. Furthermore, periodicity of solutions, bifurcation into chaos and stochastic behaviour for Hamiltonian dynamical systems at a threshold energy are discussed. As general methods for superposing nonlinear waves, the Hirota method and the Bäcklund transformation are analysed with reference to the sine- Gordon equation. The paper is mostly based on the author’s own research work.
Reviewer: P.Bassanini

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B32 Bifurcations in context of PDEs
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