Bushes of vibrational modes for Fermi–Pasta–Ulam chains

Some exact solutions and multimode invariant submanifolds were found for the Fermi–Pasta–Ulam (FPU)-β model by Poggi and Ruffo [Physica D 103 (1997) 251]. 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-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.