Reflection and Transmission of Lattice Solitons

Abstract

The scattering of lattice solitons by a mass interface is studied through the use of reductive perturbation method. In the lowest order approximation, the incident soliton and the reflected and transmitted waves propagate independently of each other and each of them is governed by the generalized Korteweg-de Vries equation. The boundary condition at the mass interface is the same as that for linear waves. Reflection and transmission of solitons through the interface are qualitatively discussed. The results are compared with those of the recent numerical experiments. · § I. Introduction Since Todan found the existence of pulse-like waves passing through one another without losing their identities in an anharr~J.Onic one-dimensional lattice system with the nearest neighbor interaction of the exponential type, much work on the lattice solitons has been made.J Recently, an interesting numerical experi­ ment was carried out by Yoshida, Nakayama and Sakuma. 3J They investigated reflection of solitons in an anharmonic lattice system with the cubic and quartic interaction potential. It is seen from their results that the solitons are reflected as stable entities from a fixed boundary, preserving their identities. However, the reflection of solitons from a free boundary is quite different; for the lattice system with only quartic anharmonicity the incident soliton reflects at the boundary transforming its type, for example, from compressive type into rarefactive one, and vice versa, whilst, for a system in which there exists cubic nonlinearity the incident soliton decays with time into a train of ripples after the reflection. They further studied the scattering of lattice solitons from a mass interface. The soliton disintegrates into reflected and transmitted waves, in a way such that their ampli­ tudes are determined by the reflection coefficient calculated by a linear wave theory. This implies that a soliton, when passing through the mass interface, behaves like a linear wave. With regard to the reflection problem of lattice solitons, Toda has made the following conjecture :4l Reflection of a soliton at a fixed boundary can be generally treated by the method of image because the fixed boundary condition can be replaced by an image soliton beyond the boundary. However, reflection at a free boundary provides a quite new problem to be worked out. On the analogy of a harmonic lattice, it is anticipated that a compressive (rarefactive) __ soliton will be reflected