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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.

Bushes of normal modes represent the exact excitations in the nonlinear physical systems with discrete symmetries [Physica D 117 (1998) 43]. The present paper is the continuation of our previous paper [Physica D 166 (2002) 208], where these dynamical objects of new type were discussed for the monoatomic nonlinear chains. Here, we develop a simple crystallographic method for finding bushes in nonlinear chains and investigate stability of one-dimensional and two-dimensional vibrational bushes for both FPU-α and FPU-β models, in particular, of those revealed recently in [Physica D 175 (2003) 31].

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