Long-time dynamics of the modulational instability of deep water waves

In this paper, we experimentally and theoretically examine the long-time evolution of modulated periodic 1D Stokes waves which are described, to leading-order, by the nonlinear Schrِdinger (NLS) equation. The laboratory and numerical experiments indicate that under suitable conditions modulated periodic wave trains evolve chaotically. A Floquet spectral decomposition of the laboratory data at sampled times shows that the waveform exhibits bifurcations across standing wave states to left- and right-going modulated traveling waves. Numerical experiments using a higher-order nonlinear Schrِdinger equation (HONLS) are consistent with the laboratory experiments and support the conjecture that for periodic boundary conditions the long-time evolution of modulated wave trains is chaotic. Further, the numerical experiments indicate that the macroscopic features of the evolution can be described by the HONLS equation. Ultimately, these laboratory experiments provide a physical realization of the chaotic behavior previously established analytically for perturbed NLS systems.