On a family of KP multi-line solitons associated to rational degenerations of real hyperelliptic curves and to the finite non-periodic Toda hierarchy

We continue the program started in Abenda and Grinevich (2015) of associating rational degenerations of M-curves to points in GrTNN(k,n) using KP theory for real finite gap solutions. More precisely, we focus on the inverse problem of characterizing the soliton data which produce Krichever divisors compatible with the KP reality condition when Γ is a certain rational degeneration of a hyperelliptic M-curve. Such choice is motivated by the fact that Γ is related to the curves associated to points in GrTP(1,n) and in GrTP(n−1,n) in Abenda and Grinevich (2015). We prove that the reality condition on the Krichever divisor on Γ singles out a special family of KP multi-line solitons (T-hyperelliptic solitons) in GrTP(k,n), k∈[n−1], naturally connected to the finite non-periodic Toda hierarchy. We discuss the relations between the algebraic-geometric description of KP T-hyperelliptic solitons and of the open Toda system. Finally, we also explain the effect of the space-time transformation which conjugates soliton data in GrTP(k,n) to soliton data in GrTP(n−k,n) on the Krichever divisor for such KP solitons.