The paper presents a comprehensive study of the nonlinear evolution of matter power spectra using large-scale $N$-body simulations with power-law initial spectra. The authors compare the nonlinear evolution with existing analytic scaling formulae based on the "stable clustering" hypothesis, which assumes that highly nonlinear structures are frozen in proper coordinates. The results show that the scale-free spectra defined by the self-similarity scaling do not follow the predicted stable clustering locus, indicating that stable clustering may not be a valid assumption for understanding small-scale power spectrum evolution. The small-scale nonlinear power increases as both the power spectrum index $n$ and the density parameter $\Omega$ decrease, which is not well accounted for by previous scaling formulae. To address this issue, the authors propose a new method based on the "halo model," which does not assume stable clustering. This method is shown to fit both the scale-free results and previous CDM data more accurately than the Peacock–Dodds formula. The paper also discusses the numerical simulations, the measurement of power spectra, and the comparison with current nonlinear fitting formulae. Finally, it introduces a new globally optimized formula for fitting power spectra and compares it with the scale-free results and CDM data.The paper presents a comprehensive study of the nonlinear evolution of matter power spectra using large-scale $N$-body simulations with power-law initial spectra. The authors compare the nonlinear evolution with existing analytic scaling formulae based on the "stable clustering" hypothesis, which assumes that highly nonlinear structures are frozen in proper coordinates. The results show that the scale-free spectra defined by the self-similarity scaling do not follow the predicted stable clustering locus, indicating that stable clustering may not be a valid assumption for understanding small-scale power spectrum evolution. The small-scale nonlinear power increases as both the power spectrum index $n$ and the density parameter $\Omega$ decrease, which is not well accounted for by previous scaling formulae. To address this issue, the authors propose a new method based on the "halo model," which does not assume stable clustering. This method is shown to fit both the scale-free results and previous CDM data more accurately than the Peacock–Dodds formula. The paper also discusses the numerical simulations, the measurement of power spectra, and the comparison with current nonlinear fitting formulae. Finally, it introduces a new globally optimized formula for fitting power spectra and compares it with the scale-free results and CDM data.