Analytic solution of the cubic parameter equation for Solution XXII

Solve analytically the cubic equation governing the ratio of parameters in Solution XXII of the asymmetric phi^4 equation arising from the n = 1/2 generalized Fokas–Lennels equation, and thereby obtain the ratio y in terms of D and m.

Background

For Solution XXII, the authors reduce the parameter constraints to a cubic equation for the ratio y = A/B as a function of D and m. They further simplify it through the substitution y = z − D, but do not obtain an analytic expression for the required ratio.

This leaves the explicit parameterization of the corresponding periodic pulse solution unresolved for general D and m.