Nonlinear coherent structures in two-component inhomogeneous nonlinear Schrödinger systems with inter-core coupling and four-wave mixing terms

We consider a two-component nonlinear Schrödinger system with cross-phase modulation, self-phase modulation, linear-coupling nonlinearities, four-wave mixing nonlinearities, and parabolic refractive index (external potential) which governs the dynamics of nonlinear waves in anisotropic graded index nonlinear dispersive media. Performing a modified lens-type transformation and identifying an appropriate similarity transformation, the integrability conditions are presented, and the model system is converted into two independent scalar nonlinear Schrödinger equations. Depending on the sign of the inter-core coupling, our physical model behaves either like a system without dissipation having “positive coherent coupling” or like a dissipative model “negative coherent coupling” in which the inter-core coupling plays the role of dissipation. We show that the linear superposition of various nonlinear wave solutions of the derived scalar nonlinear Schrödinger equations results in several kinds of nonlinear coherent structures. Those nonlinear coherent structures show the interaction of various nonlinear waves (interaction of two bright solitons, interaction of one periodic and one aperiodic breather, interaction of two aperiodic breathers, …) and the coexistence of various kinds of nonlinear waves such as, for example, coexisting rogue wave and breather, rogue wave and rogue wave, periodic breather and aperiodic breather. Focusing on kink-like and periodically modulated nonlinearity coefficients, we show how the nonlinearities affect nonlinear coherent structures in the physical model under consideration.