Analytic solution of the cubic parameter equation for Solution XXIII

Solve analytically the cubic equation governing the ratio of parameters in Solution XXIII of the asymmetric phi^4 equation arising from the n = 1/2 generalized Fokas–Lennels equation, and thereby obtain z, or equivalently A/B, in terms of D and m beyond the special case m = 1/2.

Background

For Solution XXIII, the parameter conditions lead to a cubic equation for y = A/B, which is transformed into a cubic equation for z. The authors explicitly report that they cannot solve this equation analytically in general, although they do solve it in the special case m = 1/2.

The unresolved problem is therefore to derive the parameter ratio for general values of D and m, completing the explicit description of this periodic pulse solution.