Stability of nonlinear normal modes in the Fermi-Pasta-Ulamβchain in the thermodynamic limit

Abstract

All possible symmetry-determined nonlinear normal modes (also called simple periodic orbits, one-mode solutions, etc.) in both hard and soft Fermi-Pasta-Ulam β chains are discussed. A general method for studying their stability in the thermodynamic limit as well as its application for each of the above nonlinear normal modes are presented.

Stable and unstable periodic orbits in the one-dimensional latticeϕ4theory

“…All the modes are repetitions of these modes, except close to the boundaries, as explained below. Corresponding modes have been derived for the FPU model with periodic boundary conditions [18]. While the results are similar, it is interesting to note the the differences.…”

“…Similarly to the symmetric mode case, we need to consider the cases, ω 2 (t) = 3χ 2 0 , 3χ 2 0 + 4 for the equation Eq. (18). Let us keep the leading order term in χ 0 expansion and deduce what happens: Approximating χ 0 by E 1 /2 sin(2t), we find that ω 2 0 = 3χ 2 0 + 4 never satisfies the resonance condition, Eq.…”

“…In this section, we construct a class of periodic solutions in the φ 4 theory on the lattice, based on the normal modes of the harmonic theory. A more general study of periodic solutions using powerful group theoretical methods have been conducted [15,20,21], and have been applied to models such as the FPU model [18]. Here, we briefly explain a more elementary approach to the solutions which hopefully provides some different insight, and derive the results needed later.…”

“…While the physics of the φ 4 theory investigated in this work is classical, understanding of the classical theory is also important to the understanding of the quantum theory, and furthermore, classical solutions can be an important contributing factor in quantum theories.The periodic orbits we study are so-called "non-linear normal modes" of the class of non-linear models. Non-linear normal modes have been studied extensively for some time and various general properties have been established [10][11][12][13][14][15], and investigated in models such as the FPU model [16][17][18]. Most of the explicit work conducted so far seems to focus on theories with non-linear inter-site couplings.…”

Periodic orbits in nonlinear wave equations on networks

“…Another type of nonlinear periodic orbit can exist, which is a continuation of the linear normal modes [5]. For the Fermi-Pasta-Ulam system, this type of nonlinear periodic orbit has been found using group theoretical methods and Hamiltonian perturbation methods [6][7][8]. Also, in the theoretical mechanics community, such linear-nonlinear periodic orbits have been studied for some time, see the extensive review * caputo@insa-rouen.fr † imene.khames@insa-rouen.fr ‡ arnaud.knippel@insa-rouen.fr § panos@mym.iimas.unam.mx by [9].…”