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

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 $n=0\mathrm{}$ mode are very good fits. The maximum of the $n=1\mathrm{}$ mode, however, is off by about 30%. (The nonlinear Schrödinger 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.)