General explicit phase-plane trajectory for Fisher–Stefan travelling waves

Develop a general explicit expression for the phase-plane trajectory V(U) satisfying V(U) dV/dU = −c V(U) − U(1 − U) with V(1) = 0, which characterizes Fisher–Stefan travelling-wave solutions, for arbitrary wave speeds c.

Background

For the Fisher–Stefan model, the travelling-wave analysis leads to the phase-plane system U' = V and V' = −cV − U(1 − U). The authors reduce this to a single first-order ODE for the trajectory V(U) to facilitate analysis of invading and receding waves.

They note that a general closed-form for V(U) is not available; instead, they construct perturbation solutions around special cases (e.g., c ≈ 0) to approximate the trajectory and deduce relationships between c and the Stefan parameter κ.