Bushes of Vibrational Modes for Fermi-pasta-ulam Chains

Some exact solutions and multi-mode invariant submanifolds were found for the Fermi-Pasta-Ulam (FPU) β-model by Poggi and Ruffo in Phys. D 103 (1997) 251. In the present paper we demonstrate how results of such a type can be obtained for an arbitrary Nparticle chain with periodic boundary conditions with the aid of our group-theoretical approach [Phys. D 117 (1998) 43] based on the concept of bushes of normal modes in mechanical systems with discrete symmetry. The integro-differential equation describing the FPU-α dynamics in the modal space is derived. The loss of stability of the bushes of modes for the FPU-α model, in particular, for the limiting case N → ∞ for the dynamical regime with displacement pattern having period twice the lattice spacing (π-mode) is studied. Our results for the FPU-α chain are compared with those by Poggi and Ruffo for the FPU-β chain. PACS: 05.45.-a; 45.90.+t; 63.20.Ry; 63.20.Dj