Fermi-Pasta-Ulam-Tsingou Recurrence Phenomena for the Sine-Gordon Equation

Abstract

The Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) generally refers to the property of a nonlinear system to return to its initial states after complex stages of evolution. For the nonlinear Schrödinger equation, exact solutions modeling recurrence have been derived and a few cycles of FPUT are observed in optical experiments. Consideration is extended here to the sine-Gordon equation (sG). An exact doubly periodic solution for sG is derived. This new solution is robust to small disturbances, and can emerge from oscillating initial and boundary conditions. A cascading instability scenario is studied. Despite being small initially, higher order modes grow more rapidly than the fundamental. Breather occurs when all modes attain the same order of magnitude. Breather subsequently decays, but instability resumes at small amplitude. This cyclic pattern is repeated, generating one scenario of FPUT. Predictions from the cascading mechanism match results from numerical simulations extremely well.