Wave patterns of the coupled nonlinear Schrödinger equations in phononic crystal fibers with four-wave mixing

Abstract In this paper, we examine the behavior of modulation instability within photonic crystals. The model employed is the coherent coupled nonlinear Schrödinger equation, incorporating weak birefringence and four-wave mixing, which arises at the edge of the optical mode. By analyzing the modulation instability spectrum, we identify both stable and unstable modes, thereby confirming the breakdown of the plane wave. Specifically, numerical simulations high-light the occurrence of Benjamin-Feir instability, where wave patterns emerge under the influence of four-wave mixing. Additionally, solitonic waves are generated, demonstrating the presence of Akhmediev breathers and other modulated structures, confirming that photonic crystals with four-wave mixing are conducive to these formations. The findings from this study could inform future research in the development of nonlinear photonic waveguides.