Instabilities of periodic patterns for coherently coupled nonlinear Schrödinger systems

Coherently coupled nonlinear Schrödinger equations arise for wave interactions in media where the relative phase of the components is critical. An example is the propagation of electric fields in an optical waveguide in the weak birefringence limit. The competing factors are second-order dispersion and four-wave mixing. Doubly periodic structures of the nonlinear Schrödinger equation returning to their initial states after complex evolution have been previously studied. Recurrence for a coherently coupled system is studied here via approaches of spectral and linear instabilities. A doubly periodic solution for coherently coupled systems is established in terms of Jacobi elliptic functions, and its robustness is investigated. For spectral instability, the eigenvalues of the associated matrix are computed. For linear instability, direct numerical simulations are performed for slightly perturbed doubly periodic patterns. These patterns generally display various degrees of instability. Special disturbances favorable for recurrence phenomena arising from a continuous wave background and singly periodic solutions are identified. The agreement between spectral and linear instabilities on the trends of growth of disturbances is excellent. Knowledge gained here will be useful for studying wave evolution and instabilities in fluids and optics.