Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

In this chapter, we report the discovery of analytical and numerical butterfly-shaped and dromion-like optical waves in a tapered graded-index waveguide (GRIN). Specifically, we obtain these waves in a generalized nonlinear Schrödinger equation (GNLSE), which describes self-similar wave propagation in GRIN waveguide with variable group-velocity dispersion (GVD), nonlinearity, and gain, despite the fact that dromion-like structures, which have generally been investigated in the (2 + 1) or higher dimensional systems, are reported in the (1+1)-dimensional GNLSE. In what follows, we delineate the concept of soliton management for specific modulations of variable group-velocity dispersion and Kerr nonlinearity parameters. For a particular choice of the GVD parameter, we notice the emergence of vibration of dromion-like structures. Eventually, this oscillation structure is transformed into two dromion-like structures, indicating that the dromion-like structure is unstable. Toward the end of the chapter, we illustrate the phenomenon of unbreakable \(\mathcal{P}\mathcal{T}\) symmetry of these exotix nonlinear waves for three specific cases.