We propose an adaptive dimension-wise Cauchy perturbation mechanism to enhance the performance of differential evolution (DE). While traditional Cauchy perturbation improves solution diversity, it uses a fixed jumping rate and fails to address dimension-specific premature convergence. To overcome these limitations, the proposed method dynamically estimates the convergence level of each dimension in every generation and adaptively adjusts the jumping rate accordingly. This dimension-specific adaptive perturbation, applied during the crossover phase, mitigates premature convergence and strengthens the algorithm's ability to locate high-quality solutions. The proposed method was embedded into the Linear population size reduction Success RaTe-based Differential Evolution (L-SRTDE) algorithm, winner of the Institute of Electrical and Electronics Engineers Congress on Evolutionary Computation (IEEE CEC) 2024 competition. Extensive experiments on challenging benchmark optimization problems from the IEEE CEC 2017 test suite demonstrate that our method significantly outperforms the original L-SRTDE and several state-of-the-art DE variants in both convergence speed and solution accuracy.