Butterfly-shaped and dromion-like optical waves in a tapered graded-index waveguide with variable group-velocity dispersion

Butterfly-shaped and dromion-like optical waves in a tapered graded-index waveguide (GRIN) are reported for the first time. The generalized nonlinear Schrödinger equation, which describes wave propagation in GRIN with variable group-velocity dispersion (GVD), nonlinearity, PT symmetric optical potentials, is investigated and analytical solutions for this dynamical system are obtained. The physical effects affecting these waves are explicated in detail. The stability of dromion-like structures is analyzed when the GVD parameter is perturbed. We have observed oscillation structure exhibiting strong interference due to this applied perturbation. For a particular value of the modulation of the GVD parameter, the oscillation structure is transformed into two dromion-like structures. It indicates that the dromion-like structure is unstable, and the coherence intensity is affected by the modified perturbation parameter. We further demonstrate the phenomenon of unbreakable PT symmetry of these novel nonlinear waves for three explicit examples.