Nonlinear coherent wavetrains and bifurcations in a class of deformable models for one-dimensional nonlinear discrete Klein-Gordon systems

Abstract

The discrete static properties of a class of deformable double-well potential models are investigated. The Peierls-stress potential of the models is explicitely given. Numerical analysis of the equation of motion reveal different soliton wavetrain profiles, most of which are periodic. Soliton wavetrains are also found analytically in terms of continuum nonlinear periodic wavefunction solutions then called periodons. The periodon stability in a lattice phonon bath is discussed. Looking at bifurcation phenomena and routes to chaos for a representative model of this class, the return map is derived in terms of a two-dimensional, two-parameter map. This map appears to be area preserving, possesses three characteristic fixed points with one elliptic, and displays complex bifurcation diagrams with hopf-like singularities. The routes to chaos also show complexities all due to the interplay of two control parameters.