Dynamics, stability analysis and quantization of β-Fermi–Pasta–Ulam lattice

Abstract

We study the well-known one-dimensional problem of N particles with nonlinear interaction. The β-Fermi-Pasta-Ulam model is the special case of quadratic and quartic interaction potential among nearest neighbours. We enumerate and classify the simple periodic orbits for this system and find the stability zones, employing Floquet theory. We quantize the nonlinear normal modes and construct a wavefunction for what we believe is a primitive nonlinear analogue of a 'phonon'.

Nonlinear normal modes in the β-Fermi-Pasta–Ulam-Tsingou chain

“…The disagreement between the two solutions is less than 1.4 × 10 −5 for all particle motions, indicating that the proposed solution is indeed a mode of the system. These modes are similar to those discussed in [13] except that the constraints on the amplitudes are much more relaxed and the wavelength of the mode can be many particles long.…”

“…The predicted relaxation profile and that shown by the numerical integration agree fairly well for the four simulations shown: following the functional form of Eq. (13) and only deviating as the system approaches its quasiequilibrium state where the noise term η becomes more important.…”

“…This constraint appears since the number of algebraic equations for a N particle system grows as 2N while the number of free parameters grows as N + 2. This means that for systems larger than two oscillators, the system of equations is generally inconsistent, and does not have a non-trivial solution unless a specific pattern of amplitudes is found [13,14] or the value of the masses and spring constants are all precisely tuned. We therefore focus on the purely nonlinear system, sometimes called an acoustic vacuum; presenting various forms of the similar NNMs, some of which lack a proper analogue in the purely linear system.…”

“…There is already significant literature contrasting the unique behavior of NNMs with linear normal modes [2][3][4][5][6][7][8] as well as their analytic descriptions in discrete systems [1,[9][10][11][12][13][14]. In [10], the form of a recursive NNM solution was discussed for a power law potential, but the exact solution was not developed.…”

“…In [10], the form of a recursive NNM solution was discussed for a power law potential, but the exact solution was not developed. In [13], five non-trivial NNM solutions where found for the β-Fermi-Pasta-Ulam-Tsingou (β-FPUT) system in terms of the Jacobi cn function. Here, we will extend the set of Jacobi function NNM solutions for the β-FPUT chain and characterize their relaxation towards the quasi-equilibrium state [15,16].…”