v - in t / 9 60 30 12 v 1 2 A pr 1 99 6 Non - perturbative Non - integrability of Non - homogeneous Nonlinear Lattices Induced by Non - resonance Hypothesis ∗

We have succeeded in applying Ziglin’s test on non-integrability to nonhomogeneous nonlinear lattices (Fermi-Pasta-Ulam lattices). By explicit calculations of the eigenvalues of the monodromy matrices concerning the normal variational equations of Lamé type and with the use of the non-resonance hypothesis about the eigenvalues, we obtained a theorem proving the nonexistence of additional analytic conserved quantities other than the Hamiltonian itself for FPU lattices in the low energy limit. Furthermore, after introducing a concept of degree of non-integrability, we have investigated the classification of non-homogeneous nonlinear lattices using a transformation R from a non-homogeneous nonlinear lattice to another non-homogeneous nonlinear lattice, which preserves their degree of non-integrability. key words: non-integrability, nonlinear lattices, singularity analysis, monodromy matrices, non-resonance condition To appear in PhysicaD E-mail: [email protected]