Abstract

Butterfly-shaped and dromion-like optical waves in a tapered graded-index waveguide (GRIN) with an external source are reported for the first time, to our knowledge. More pertinently, we obtain these waves both analytically and numerically in a generalized nonlinear Schrödinger equation (GNLSE), which describes self-similar wave propagation in GRIN with variable group-velocity dispersion (GVD), nonlinearity, gain, and source. The proposed GNLSE appertains to the study of similariton propagation through asymmetric twin-core fiber amplifiers. Dromion-like structures, which have generally been investigated in the (${2 + 1}$) or higher dimensional systems, are reported in the (${1 + 1}$) dimensional GNLSE with an external source. Herein, we introduce the concept of soliton management when the variable group-velocity dispersion and Kerr nonlinearity functions are suggested. For example, when the GVD parameter is perturbed, we observe the emergence of vibration of dromion-like structures. Then the dromion-like structure is transformed into oscillation by the modulation instability of the modified coefficient of the Gaussian GVD function, exhibiting interference based on two dromion-like structures. Additionally, the phenomenon of unbreakable ${\cal P}{\cal T}$ symmetry of these nonlinear waves has been demonstrated for three explicit examples.

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