Abstract:
The Fermi Pasta Ulam oscillator chain with periodic boundary conditions and n particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant manifolds that are investigated in this short note. For each k dividing n we find invariant k degree of freedom symplectic manifolds. They represent short wavelength solutions composed of k Fourier-modes and can be interpreted as embedded chains with periodic boundary conditions and only k particles. Inside these invariant symplectic manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasiperiodic solutions and standing and traveling waves. Some of these results have been found previously by other authors via a study of mode coupling coefficients. But we arrive at our results in a more systematic way and without any calculations. We show that the same invariant manifolds exist in the Klein-Gordon lattice and in the continuum limit. 1
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