Many signals in the real world are nonlinear and complex, and signal time-frequency analysis has been widely used in many fields. It remains a challenging task to accurately decompose amplitude modulation-frequency modulation (AM-FM) signals with complex variation laws. The existing methods still have room to improve the decomposition accuracy of complex AM-FM signals, such as signals with crossing instantaneous frequency (IF) or close IFs. This article presents a novel sparsity-assisted variational nonlinear component decomposition (SVNCD) for analyzing nonstationary multicomponent signal with complex variation laws. To better restrict the bandwidth of demodulated signal and accurately estimate IF, SVNCD establishes a new sparse optimization model with clear mathematical meaning for the demodulated signal of the original signal, considering the smoothness and Fourier spectrum constraints. Moreover, SVNCD establishes a sparse constraint optimization model on the IF of the signal and its increment, which is capable of effectively extracting the IF variation law of complex signals and addressing the mode aliasing problem. Besides, we build a unified framework for extracting the initial IFs of complex signals with uncrossing or crossing frequency trajectories. The decomposition experiments of various simulated and experimental complicated signals are carried out for verification. The results verify that SVNCD achieves better decomposition accuracy for complex nonstationary multicomponent signals with uncrossing or crossing frequency trajectories than existing signal decomposition and time-frequency transform methods. Moreover, SVNCD could accurately estimate the IF of the original signal and reconstruct the subsignals.