Determine explicit parameter constraints in the n=1 double-root case for two-component vector Fokas–Lenells solutions

Determine explicit, convenient, and simple conditions on the parameters α1, α2 ∈ ℝ with α1 ≠ α2, and complex γ, r2 for the n=1 Darboux data in the generic two-component vector Fokas–Lenells equation, in the double-root scenario where the cubic equation for r has a simple root r1 and a double root r2 with r1 = i(α1+α2) + (2γ)^{-1} − 2 r2, such that the background amplitudes |A1|^2 and |A2|^2 given by |A1|^2 = ((2 r2 − i α2)(2 γ r2 − 1 − 2 i α1 γ)^2)/(4 α1^2 γ (α2 − α1)) and |A2|^2 = −((2 r2 − i α1)(2 γ r2 − 1 − 2 i α2 γ)^2)/(4 α2^2 γ (α2 − α1)) are real and positive.

Background

In Section 6 the authors develop exact solutions of the two-component vector Fokas–Lenells equation using a vectorial Darboux transformation with a plane-wave seed. The generic case α1 ≠ α2 leads to a cubic equation for an auxiliary parameter r that governs the linear system underlying the transformation.

In the double-root configuration (n=1), the cubic admits a simple root r1 and a double root r2. Under this condition the seed background amplitudes |A1|^2 and |A2|^2 are expressed explicitly in terms of α1, α2, γ, and r2. Physical relevance requires these amplitudes to be real and positive, which imposes constraints on the parameters.

The authors note that, although such constraints must exist, they were not able to present them in a convenient and simple closed form, leaving the task of deriving explicit conditions as an unresolved point.