Stability of low-dimensional bushes of vibrational modes in the Fermi-Pasta-Ulam chains

Bushes of normal modes represent the exact excitations in nonlinear physical systems with discrete symmetries [Physica D117 (1998) 43]. The present paper is the continuation of our previous paper [Physica D166 (2002) 208], where these dynamical objects of a 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-alpha and FPU-beta models, in particular, of those revealed recently in [Physica D175 (2003) 31]

On the dark matter's halo theoretical description

We argued that the standard field scalar potential couldn't be widely used for getting the adequate galaxies' curve lines and determining the profiles of dark matter their halo. For discovering the global properties of scalar fields that can describe the observable characteristics of dark matter on the cosmological space and time scales, we propose the simplest form of central symmetric potential celestial - mechanical type, i.e. U(\phi) = -\mu/\phi. It was shown that this potential allows get rather satisfactorily dark matter profiles and rotational curves lines for dwarf galaxies. The good agreement with some previous results, based on the N-body simulation method, was pointed out. A new possibility of dwarf galaxies' masses estimation was given, also.Comment: 10p., 18 re

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) beta-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 N-particle 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 for mechanical systems with discrete symmetry. The integro-differential equation describing the FPU-alfa dynamics in the modal space is derived. The loss of stability of the bushes of modes for the FPU-alfa model, in particular, for the limiting case N >> 1 for the dynamical regime with displacement pattern having period twice the lattice spacing (Pi-mode) is studied. Our results for the FPU-alfa chain are compared with those by Poggi and Ruffo for the FPU-beta chain.Comment: To be published in Physica

Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice

The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and $n$ particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note. For each $k$ dividing $n$ we find $k$ degree of freedom invariant manifolds. They represent short wavelength solutions composed of $k$ Fourier-modes and can be interpreted as embedded lattices with periodic boundary conditions and only $k$ particles. Inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi-periodic solutions and standing and traveling waves. Some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating `bushes of normal modes'. The method of this paper is similar to the latter method and much more systematic than the former. We arrive at previously unknown results without any difficult computations. It is shown moreover that similar invariant manifolds exist also in the Klein-Gordon lattice and in the thermodynamic and continuum limits.Comment: 14 pages, 1 figure, accepted for publication in Physica