Isochronism and tangent bifurcation of band edge modes in Hamiltonian lattices

• DOI: 10.1016/j.physd.2005.04.006
• Publication Date: 2005-05-11T11:51:09Z
• Authors: J. Dorignac; S. Flach
• Keywords: Condensed Matter - Other Condensed Matter; 0103 physical sciences; FOS: Physical sciences; Pattern Formation and Solitons (nlin.PS); Nonlinear Sciences - Pattern Formation and Solitons; 01 natural sciences; Other Condensed Matter (cond-mat.other)

In {\em Physica D} {\bf 91}, 223 (1996), results were obtained regarding the tangent bifurcation of the band edge modes ($q=0,��$) of nonlinear Hamiltonian lattices made of $N$ coupled oscillators. Introducing the concept of {\em partial isochronism} which characterises the way the frequency of a mode, $��$, depends on its energy, $��$, we generalize these results and show how the bifurcation energies of these modes are intimately connected to their degree of isochronism. In particular we prove that in a lattice of coupled purely isochronous oscillators ($��(��)$ strictly constant), the in-phase mode ($q=0$) never undergoes a tangent bifurcation whereas the out-of-phase mode ($q=��$) does, provided the strength of the nonlinearity in the coupling is sufficient. We derive a discrete nonlinear Schr��dinger equation governing the slow modulations of small-amplitude band edge modes and show that its nonlinear exponent is proportional to the degree of isochronism of the corresponding orbits. This equation may be seen as a link between the tangent bifurcation of band edge modes and the possible emergence of localized modes such as discrete breathers.<br/>23 pages, 1 figure<br/>