Elsevier

Physics Reports

Volume 18, Issue 1, May 1975, Pages 1-123
Physics Reports

Studies of a non-linear lattice

https://doi.org/10.1016/0370-1573(75)90018-6Get rights and content

Abstract

An exact treatment of the propagation of waves in one-dimensional non-linear lattices is presented. The lattice consists of particles with exponential nearest-neighbour coupling. Particular solutions of the equations of motion are given in analytic form, and are shown to have wide applicabilities in elucidating general features of non-linear waves. It is shown that the waves consist of stable pulses called “solutions”. Constants of motion besides the total energy and the total momentum are given, and the non-ergodic property of the lattice is discussed with special reference to computer experiments. The equations of motion can be written in a matrix formalism, and the initial value problem is treated by using the method of inverse scattering of a matrix equation. Reflections at boundaries are also discussed, and a possible mechanism of breadown of materials due to non-linearity of elasticity is indicated. In the continuum limit the equations of the non-linear lattice reduce to the Korteweg-de Vries equation, which is treated in parallel with the discrete lattice.

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