Robustness and stability of doubly periodic patterns of the focusing nonlinear Schrödinger equation.

The nonlinear Schrödinger equation possesses doubly periodic solutions expressible in terms of the Jacobi elliptic functions. Such solutions can be realized through doubly periodic patterns observed in experiments in fluid mechanics and optics. Stability and robustness of these doubly periodic wave profiles in the focusing regime are studied computationally by using two approaches. First, linear stability is considered by Floquet theory. Growth will occur if the eigenvalues of the monodromy matrix are of a modulus larger than unity. This is verified by numerical simulations with input patterns of different periods. Initial patterns associated with larger eigenvalues will disintegrate faster due to instability. Second, formation of these doubly periodic patterns from a tranquil background is scrutinized. Doubly periodic profiles are generated by perturbing a continuous wave with one Fourier mode, with or without the additional presence of random noise. Effects of varying phase difference, perturbation amplitude, and randomness are studied. Varying the phase angle has a dramatic influence. Periodic patterns will only emerge if the perturbation amplitude is not too weak. The growth of higher-order harmonics, as well as the formation of breathers and repeating patterns, serve as a manifestation of the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence.