On the solitary wave pulse in a chain of beads

English, J.; Pego, R.

doi:10.1090/S0002-9939-05-07851-2

On the solitary wave pulse in a chain of beads
HTML articles powered by AMS MathViewer

by J. M. English and R. L. Pego PDF

Proc. Amer. Math. Soc. 133 (2005), 1763-1768 Request permission

Abstract:

We study the shape of solitary wave pulses that propagate in an infinite chain of beads initially in contact with no compression. For this system, the repulsive force between two adjacent beads is proportional to the $p^\textrm {th}$ power of the distance of approach of their centers with $p=\frac 32$. It is known that solitary wave solutions exist for such a system when $p>1$. We prove extremely fast, double-exponential, asymptotic decay for these wave pulses. An iterative method of solution is also proposed and is seen to work well numerically.

References

T. B. Benjamin, J. L. Bona, and D. K. Bose, Solitary-wave solutions of nonlinear problems, Philos. Trans. Roy. Soc. London Ser. A 331 (1990), no. 1617, 195–244. MR 1062564, DOI 10.1098/rsta.1990.0065
A. Chatterjee, Asymptotic solution for solitary waves in a chain of elastic spheres, Phys. Rev. E 59 (1998) 5912–5918.
C. Coste, E. Falcon, S. Fauve, Solitary waves in a chain of beads under Hertz contact, Phys. Rev. E 56 (1997) 6104–6117.
G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit, Nonlinearity 12 (1999), no. 6, 1601–1627. MR 1726667, DOI 10.1088/0951-7715/12/6/311
Gero Friesecke and Jonathan A. D. Wattis, Existence theorem for solitary waves on lattices, Comm. Math. Phys. 161 (1994), no. 2, 391–418. MR 1266490
R. S. MacKay, Solitary waves in a chain of beads under Hertz contact, Phys. Lett. A 251 (1999) 191-192.
Jeong-Young Ji and Jongbae Hong, Existence criterion of solitary waves in a chain of grains, Phys. Lett. A 260 (1999), no. 1-2, 60–61. MR 1714633, DOI 10.1016/S0375-9601(99)00488-0