Transverse modulational instability of (2+1)-dimensional cnoidal waves in media with cubic nonlinearity

Abstract

We consider transverse modulational instability of (2+1)-dimensional cnoidal waves of cn, dn, and sn, types that are periodic in one direction and are uniform in the other direction. The new method of stability analysis of periodic waves presented here is based on the construction of a translation matrix for a perturbation vector and on the evolution of the eigenvalues of the matrix with changes in modulation frequency and Jacobi parameter that define the degree of energy localization of the corresponding cnoidal waves. We show that the dn wave is subject to the influence of both neck and snake instabilities, the cn wave is affected by neck instability, and the sn wave suffers from snake instability in (2+1) dimensions.