An Integrable Approximation for the Fermi–Pasta–Ulam Lattice

Abstract

This contribution presents a review of results obtained from computations of approximate equations of motion for the Fermi-Pasta-Ulam lattice. These approximate equations are obtained as a finite-dimensional Birkhoff normal form. It turns out that in many cases, the Birkhoff normal form is suitable for application of the KAM theorem. In particular this proves Nishida's 1971 conjecture stating that almost all low-energetic motions of the anharmonic Fermi-Pasta-Ulam lattice with fixed endpoints are quasi-periodic. The proof is based on the formal Birkhoff normal form computations of Nishida, the KAM theorem and discrete symmetry considerations.

Elliptic tori in FPU non-linear chains with a small number of nodes

“…Let us focus on the plots in the left box of Figure 1. First, we remark that it was well expected that the values of |∆ω f ;1 | are increasing with respect the number N of nodes: on one hand, it is known that the variation of the local maxima of the function (47) is O(1/T 4 ) when quasi-periodic motions are studied and the Hanning filter is adopted (see [35]), on the other hand, ω f ;1 ≃ ν 1 that is defined in (3), then the fundamental period corresponding to the main oscillation (of the first normal mode) is O(N). Therefore, in relative terms, when N is increased, larger times are needed to let the angular velocities relax to their final values; this explains why the variation of |∆ω f ;1 | gets bigger with N, by keeping fixed the total time 2T .…”

“…Thus, the corresponding energy level E = H(y(0), x(0)) depends uniquely on the amplitude A of the initial excitation of the first mode, since we keep α and β as fixed parameters (recall definition (1)). We numerically determine the global maximum point ω f ;1 of the function υ → T 1 (υ) defined in (47). Of course, the calculation of the integral in ( 47) is made by a quadrature method that is applied after having replaced the function t → Y 1 (t), X 1 (t) (where t ∈ [0 , t]) with the finite sequence…”

“…(d1) As in the previous subsection, we set ω f ;1 = υ 1,1 that is the global maximum point of the function T 1 (υ) defined in (47).…”

“…First, the transition to the chaoticity is much sharper in the case of a quartic perturbation instead of a cubic one; therefore, it is natural to expect that a normal form method can approach better the breakdown threshold for any invariant manifold, when the β-model is considered. Moreover, there is an accurate integrable approximation of the α-model: the Toda lattice, because their expansions coincide up to terms of third order (see [18] and, e.g., the introduction of [47] in [20]). We think that this fact should play an important role in order to stabilize the dynamics when relatively high energies are considered, but our approach is completely blind to such an accurate approximation provided by a Toda-like Hamiltonian.…”

On Some Classes of New Solutions of Continuous β-FPU Chain

“…The Fermi-Pasta-Ulam (FPU) problem, as is well known, marks a true sea change in modern science. After the seminal paper appearing in 1955, [1] many localized models, solitons, [2] chaos, [3−5] discrete breathers, [6−9] biphonons [10] and chaotic breathers, [11] were discovered in the FPU system successively. However, most models are discrete.…”