Mode analysis and modeling of soliton explosion: based on singular-value decomposition

Based on singular-value decomposition, we analyze the dynamics of soliton explosion in the complex cubic-quintic Ginzburg-Landau equation. Soliton explosion involves two modes: a stable dissipating soliton and a growing perturbation mode. During the explosion stage, soliton explosion lacks primary or stable modes. We present the effects of linear gain-loss, dispersion, and nonlinearity on the perturbation mode, revealing the significance of dispersion in dark soliton generation. The transition process from symmetric soliton explosion to asymmetric soliton explosion is closely related to the perturbation mode. We provide approximate expressions for the staged soliton explosion, aiding for explaining various nonlinear characteristics associated with soliton explosion.