Dubrovin's approach to the FPU Problem

In the study of perturbed Hamiltonian systems, there is a theorem (due to Poincarè) that says this: non degenerate integrable Hamiltonian systems, under generic perturbation, loose all the first integrals in the analytic class. Along such line, methods to extend solutions and first integrals of the unperturbed system have increased. In particular, in recent years, Boris Dubrovin developed new techniques, for perturbed Hamiltonian PDEs of hyperbolic type, to extend solutions and first integrals from the unperturbed system to the perturbed one. One of the relevant cases, treated in the work [1], is that of a continuum version of a particle chain with a pair interaction potential phi(r) (known as the generic FPU problem). Dubrovin showed that, under suitable dispersive perturbations, all the first integral of this unperturbed system admit a deformation at the second order iff and, for special choose of constants, we obtain the Toda potential. This means that the integrable Toda chain plays a kind of unique role among the FPU systems. In the thesis, we extend this results also for generic perturbation, adding a potential psi(r) at the second order of the perturbation, and apply this techniques to the actual FPU chains, regarded as a perturbation of the Toda chain, to see if it's possible to extend integral at the second order or further. A condition on this last target is given. 1. B. Dubrovin, ``On universality of critical behaviour in Hamiltonian PDEs'', Amer. Math. Soc. Transl. 224 (2008) 59-109.