Non-constant exact solutions of the travelling-wave ODE (39) for positive A

Construct non-constant exact solutions of the second-order ordinary differential equation ϕ''(ω) + (γ/d1) ϕ'(ω) + A ϕ(ω) (ϕ(ω) − 1) = 0 that arises from the travelling-wave reduction u(t,x) = ϕ(x − γ t) of the reaction–diffusion system u_t = d1 u_xx − A u(1 − u), v_t = d2 v_xx + A u v, specifically in the case A > 0. Determine explicit closed-form expressions (if they exist) for ϕ(ω) that are non-constant.

Background

In Section 3.2, the authors apply a Q-conditional symmetry to the reaction–diffusion system and reduce the first component to a travelling-wave ordinary differential equation labeled (39). They provide explicit non-constant solutions for the case A < 0 and proceed to construct exact solutions for the coupled second equation accordingly.

However, for the parameter regime A > 0, the authors explicitly note that non-constant exact solutions of the reduced ODE are unknown. Resolving this would extend the catalogue of exact solutions obtainable via nonclassical reductions for this reaction–diffusion setting and potentially lead to new non-Lie solutions for the full system.