Abstract
Existence, stability, and dynamics of $\mathcal{PT}$-symmetric fundamental bright solitons supported by localized super-Gaussian potentials in a focusing Kerr medium are investigated theoretically. We address how the shape and the magnitude of the transverse profile of the loss-gain distribution affect soliton stability. We find the stability region for nonlinear wave packets via a linear stability analysis, interpreting the insurgence of instability as an unbalanced flow of energy on the transverse plane. We confirm our results via numerical simulations, showing that an unstable soliton first undergoes longitudinal oscillations in propagation due to the interference between the soliton and the exponentially growing perturbation modes, eventually forming a highly localized single peak in the gain region.
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