Bushes of vibrational modes for Fermi-Pasta-Ulam chains

Some exact solutions and multimode invariant submanifolds were found for the Fermi-Pasta-Ulam (FPU)-beta model by Poggi and Ruffo [Physica D 103 (1997) 25 1]. In the present paper we demonstrate how results of such a type can be obtained for an arbitrary N-particle chain with periodic boundary conditions with the aid of our group-theoretical approach [Physica 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-alpha dynamics in the modal space is derived. The loss of stability of the bushes of modes for the FPU-alpha model, in particular, for the limiting case N --> infinity for the dynamical regime with displacement pattern having period twice the lattice spacing (pi-mode) is Studied. Our results for the FPU-alpha chain are compared with those by Poggi and Ruffo for the FPU-beta chain. (C) 2002 Elsevier Science B.V. All rights reserved.