Suppression of the modulational instability in the ϕ^{4} model by periodic potentials.

Abstract

In the framework of the one- and two-dimensional (1D and 2D) ϕ^{4} equations, we elaborate a possibility to control the modulational instability (MI) of extended states by means of spatially periodic potentials, built in terms of the Jacobian elliptic functions. We demonstrate that both the 1D (longitudinal) and 2D (transverse) MIs of exact periodic snoidal states, found in the form of the elliptic sine, can be enhanced or completely suppressed by the snoidal squared potential, depending on the sign of the coefficient in front if it. The enhanced MI can support the generation of rogue waves, while the suppression of MI can open a way to create stable crystal structures (in particular, in ferroelectrics). The consideration is based on the calculation of the spectrum for small perturbations added to the stationary states, and confirmed by direct simulations of the perturbed evolution. In particular, a threshold (minimum) amplitude of the periodic potential, which provides the stabilization, is identified as a function of other parameters. In the 2D setting, the stabilization threshold is higher for the longitudinal MI than for its transverse counterpart.