Exact solutions for the Porous–Fisher–Stefan travelling-wave phase-plane system

Find exact solutions, for arbitrary wave speeds c, to the Porous–Fisher–Stefan travelling-wave phase-plane system in variables φ = u^2 and ψ, given by dφ/dz = ψ and dψ/dz = −(c ψ)/√φ − 2√φ(1 − √φ), or equivalently an explicit expression for the trajectory ψ(φ) solving dψ/dφ = −c/√φ − 2√φ(1 − √φ)/ψ(φ) with ψ(1) = 0 and ψ(0) determined by the Stefan condition.

Background

Extending the Porous–Fisher model to a moving-boundary setting (Porous–Fisher–Stefan) yields sharp-fronted waves with a Stefan condition governing the front motion. By introducing φ = u^2, the travelling-wave reduction leads to a phase-plane system in (φ, ψ).

The authors cannot obtain exact closed-form solutions for the phase-plane system for arbitrary c and instead develop perturbative approximations in regimes such as |c| ≪ 1 or c approaching 1/√2 from below.