Stabilization of the transverse instability of the periodic waves of the (2+1)D hyperbolic nonlinear Schrödinger equation

Stabilization of originally unstable states is a fundamental problem known since long ago. It is well known that one-dimensional periodic waves suffer from transverse instability in deep water and some electromagnetic systems, where there are two significant transverse dimensions. In this paper, we show that the transverse instability of periodic waves can be stabilized by introducing the periodic potential based on the (2 + 1)D nonlinear Schr & ouml;dinger equation. This conclusion is based on the calculation of the eigenvalue spectra for subharmonic perturbations, and corroborated by direct simulations of the perturbed evolution of the cnoidal waves. Besides, we find that, in comparison to the long-wavelength instabilities, we require a stronger periodic external field to stabilize short-wavelength instabilities.