Moving boundary problems for a canonical member of the WKI inverse scattering scheme: conjugation of a reciprocal and Möbius transformation

Abstract

Reciprocal links between certain solitonic systems and their hierarchies are well-established. Moreover, the AKNS and WKI inverse scattering schemes are known to be connected by a composition of gauge and reciprocal transformations. Here, a reciprocal transformation allied with a Möbius-type mapping is applied to a class of Stefan-type problems for the solitonic Dym equation to generate a novel exact parametric solution to a class of moving boundary problems for a canonical member of the WKI system.

Miura-reciprocal transformations and localizable Poisson pencils

“…A distinguished subgroup of R I is the group of projective reciprocal transformations P. The group was introduced in [FPV14] as the invariance group of a class of a Hamiltonian operators. A recent instance of such a transformation in the literature can be found here [Rog22].…”

“…Indeed, an important class of such transformations is that of reciprocal transformations, which play an important role in Mathematical Physics (see e.g. [Rog69, Rog68, Fer89, Fer91, FP03, XZ06, AG07, Abe09, BS09, LZ11, AL13] and the recent paper [Rog22] with historical notes in the Introduction).…”