Complete characterization of flocking versus nonflocking of Cucker–Smale model with nonlinear velocity couplings

We consider the Cucker–Smale model with a regular communication rate and nonlinear velocity couplings, which can be understood as the parabolic equations for the discrete p-Laplacian (p ≥ 1) with nonlinear weights involving a parameter β( > 0). For this model, we study the initial data and the ranges of p and β to characterize when flocking and nonflocking occur. Specifically, we analyze the nonflocking case, subdividing it into semi-nonflocking (only velocity alignment holds) and full nonflocking (group formation and velocity alignment do not hold). More precisely, we show that if β ∈ (0, 1], p ∈ [1, 3), then flocking occurs for any initial data. If β ∈ (0, 1], p ∈ [3, ∞), then semi-nonflocking occurs for any initial data. If β ∈ (1, ∞), p ∈ [1, 3), then flocking occurs for some initial data. In the case β ∈ (1, ∞) and p ∈ [3, ∞), we observe alternative states. Finally, we have numerically verified the conclusions obtained by analytical calculations.