Kink, breather and asymmetric envelope or dark solitons in nonlinear chains. I. Monatomic chain

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Abstract
The authors have examined and obtained analytic expressions for the basic nonlinear excitations in a monoatomic chain with cubic and quartic interatomic potentials and verified their stability under collision. They determined bounds on the potential parameters for which kink, breather, envelope and dark solitons exist. The kink and the envelope of a soliton are determined in the long-wavelength approximation, while the oscillations of the carrier in the envelope soliton are treated exactly. The introduction of second-neighbour interactions (SNI) changes drastically the character of the solutions. Kinks are supersonic if the dispersive term is positive and subsonic if it is negative. In a cubic potential the envelope solitons are asymmetric (there is a net displacement). There is a switch from a kink to a symmetric envelope as the potential parameters are changed; these parameters also determine the regions for which plane waves are unstable. The above excitations can be created by using arbitrary initial conditions which, however, have a net displacement for kinks and the correct Fourier components for an envelope soliton. The asymmetric envelope solitons explain earlier computer simulations using as initial conditions optical gaussian pulses.