#### Nonlinear waves in granular chains

##### Guillaume James
The purpose of this talk is to give a review of some nonlinear wave phenomena occuring in granular chains (in particular the propagation of solitons) and describe some mathematical techniques useful in this context.

A classical example of granular chain is the Newton's cradle, which consists of a chain of touching beads suspended from a bar by inelastic strings. This system is widely used to illustrate the laws of energy and momentum conservation, since the release of a bead at one end of the cradle generates a series of collisions and the ejection of a bead at the opposite side. Despite its apparent simplicity, this phenomenon is delicate to understand because it involves many-body collisions and the propagation of a highly-localized wave along the granular chain, known as a solitary wave or "soliton". Solitary waves are ubiquitous in nonlinear physical systems, and their understanding is essential to describe phenomena such as the propagation of pulses in optical fibers, Tsunamis or localized plasma waves. They can be generated in granular systems due to the nonlinear character of Hertzian contact interactions.

The first part of the talk will concern wave propagation in classical linear and nonlinear models of granular chains. Topics will include : linear dispersion, the analysis of solitons via the Korteweg-de Vries (KdV) equation, highly-localized solitons with compact support, nonlinear periodic waves.

The second part will address more complex granular systems in which beads are trapped in strong confining potentials (due e.g. to stiff attachements or to an elastic matrix), so that bead collisions and local oscillations occur on similar time scales, and more complex nonlinear waves can be generated. For example, an initial impact can result in the propagation of a "travelling breather", i.e. a solitary wave having an internal oscillation.

References :
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• J.M. English and R.L. Pego. On the solitary wave pulse in a chain of beads, Proc. Amer. Math. Soc. 133, n. 6 (2005), 1763-1768.
• E. J. Hinch and S. Saint-Jean. The fragmentation of a line of ball by an impact, Proc. R. Soc. London, Ser. A 455 (1999), 3201-3220.
• G. James, Nonlinear waves in Newton's cradle and the discrete p-Schrödinger equation, Math. Models Meth. Appl. Sci. 21 (2011), 2335-2377.
• V.F. Nesterenko, Dynamics of heterogeneous materials, Springer Verlag, 2001.
• S. Sen, J. Hong, J. Bang, E. Avalos and R. Doney. Solitary waves in the granular chain, Physics Reports 462 (2008), 21-66.