Compactons in Discrete Lattices

Abstract

As is well-known, solitons appear in a result of a balance between weak nonlinearity and dispersion. However, when the wave dispersion is purely nonlinear, some novel features may be observed and the most remarkable one is the existence of the so-called compactons, i. e. solitons with finite wavelength recently discovered by Rosenau and Hyman1 for a special class of the Korteweg-de Vries (KdV) type equations with nonlinear dispersion. These travelling-wave solutions have a remarkable property: Unlike the standard KdV soliton which narrows as the amplitude increases, the compacton’s width is independent of the amplitude. Having the constant width, such solutions can not be obtained, however, in a result of a proper continuum limit to discrete models. Indeed, soliton-bearing partial differential equations may be derived from discrete models of solids in a result of expansions in the wave amplitude and inverse pulse width which normally need a scaling procedure. In other words, the continuum limit approach yields the condition of the slowly varying wave envelope which is consistent with the effect of weak nonlinearity balanced by a weak dispersion. As soon as we deal with compactons instead of standard solitons, the continuum limit approximation cannot be properly justified because higher-order derivatives will be only numerically small.