Recurrent nonlinear modulational instability in the β-FPUT chain

We address the fully nonlinear stage of seeded modulational instability in the Fermi-Pasta–Ulam-Tsingou chain with quartic interaction potential (β-FPUT) subject to periodic boundary conditions. In particular, we investigate quantitatively the validity of the continuous approximation that describes the evolution of a narrow band of normal modes in terms of the ubiquitous nonlinear Schrödinger equation (NLSE) or its generalizations. By injecting three normal modes comprising a pair of unstable sidebands, we find that the FPUT chain exhibits, for weak enough nonlinear interaction, recurrent evolutions (though of different nature compared with the original work by FPUT). Such recurrences generally preserve the homoclinic structure of nonlinear modulational instability ruled by the NLSE, with generated higher order-modes being essentially enslaved to the unstable pair. Under some circumstance, we find that pseudo-random separatrix crossing events may occur even for a very weak interaction strength.