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In this study, we investigate the kink soliton dynamics for power-law nonlinear systems. Based on the F-expansion method, we first derive the novel kink soliton solution of the nonlinear Schrödinger equation (NLSE) with third-order dispersion term, power-law dependent nonlinearity term, linear attenuation term, and self-steepness term under appropriate parameter settings. With pictorial demonstration, we show that the obtained kink soliton solution not only has the soliton features of the classical NLSE, but also has power-law features. The theoretical results presented in our work can be used to guide the observation of soliton behavior in power-law dependent media.