Quasicontinuum approximation and iterative method for envelope solitons in anharmonic chains

Abstract

For solitary waves on a monoatomic chain with nearest neighbor interactions the continuum approximation has a limited validity range and exhibits certein mathematical problems. For pulse solitons these problems are overcome by the Quasicontinuum Approach (QCA), and the validity range is considerably extended. We generalize the QCA to oscillatory excitations and derive analytic expressions for bright and dark envelope solitons, limiting ourselves to a polynomial interaction potential with harmonic, cubic and quartic terms. Moreover we describe and apply a numerical iteration procedure in Fourier space in order to take into account discreteness effects in a systematic way. This procedure yields envelope solitons with a width in the order of the lattice constant. In the case of zero velocity these solutions can be compared with intrinsic localized modes derived by other authors. The stability and accuracy of all our solutions are tested by numerical simulations.