Multistability and forced nonlinear dynamics in a reduced model of pure-Kerr parametrically driven cavity solitons

Parametrically driven cavity solitons (PDCSs) in bichromatically driven Kerr resonators provide a setting in which cavity loss, dispersion, Kerr nonlinearity, and phase-sensitive parametric coupling interact. We study the deterministic nonlinear dynamics of the reduced pure-Kerr PDCS model obtained from the generalized bichromatically driven Lugiato–Lefever equation. The traveling-wave reduction is used only to analyze traveling-profile equilibria and profile-coordinate eigenvalues, whereas all long-time temporal dynamics are obtained directly from the slow-time reduced partial differential equation. For the representative parameters 𝛥 = 1 . 2 , 𝜇 = 1 . 3 7 , and 𝑑 2 = − 2 , the principal stationary localized solution has 𝛽 + = 1 . 4 6 1 7 , peak intensity 𝑃 m a x , + = 4 . 2 7 2 9 , and energy ℰ 𝑠 , + = 5 . 8 4 6 6 . The second exact stationary branch is also identified analytically and is distinguished from temporal stability. Fixed-parameter temporal integrations from distinct initial fields are used to investigate multistability. Periodic modulation of the effective detuning and parametric-drive coefficient is then used to investigate forced temporal dynamics through complementary time-series diagnostics. A central-band comparison with the parent bichromatically driven Lugiato–Lefever equation is used to quantify numerical convergence and model-reduction discrepancy. The resulting framework keeps traveling-profile geometry and temporal dissipative dynamics mathematically distinct and provides a consistent basis for numerical diagnostics.