Lindstedt Series Solutions of the Fermi-Pasta-Ulam Lattice

TL;DR

This paper uses the Lindstedt method to derive general solutions for the Fermi-Pasta-Ulam β lattice and confirms their accuracy through comparison with numerical results.

Contribution

It introduces a perturbative approach using the Lindstedt method to solve the FPU lattice equations analytically.

Findings

Analytical solutions closely match numerical simulations.

The perturbative scheme effectively handles the nonlinear coupling.

The method provides a general framework for solving similar lattice models.

Abstract

We apply the Lindstedt method to the one dimensional Fermi-Pasta-Ulam $\beta$ lattice to find fully general solutions to the complete set of equations of motion. The pertubative scheme employed uses $\epsilon$ as the expansion parameter, where $\epsilon$ is the coefficient of the quartic coupling between nearest neighbors. We compare our non-secular perturbative solutions to numerical solutions and find striking agreement.