Quantitive Theory of the Fermi-Pasta-Ulam Recurrence in the Nonlinear Schrödinger Equation

Abstract

By limiting attention to the lowest-order Fourier modes we obtain a theory of the Fermi-Pasta-Ulam recurrence that gives excellent agreement with recent numerical results. Both the predicted period of the recurrence and the temporal development of the $n=0$ mode are very good fits. The maximum of the $n=1$ mode, however, is off by about 30%. (The nonlinear Schr\"odinger equation governs the development of the envelope of the electric field of a nonlinear Langmuir wave in the plasma-physics context. It also describes gravity waves in deep water.)