Soliton injection and local evaporation of a KdV soliton condensate

Determine whether injecting a localized soliton component into a KdV soliton condensate produces local evaporation of the condensate and rigorously characterize the resulting spatiotemporal dynamics within the continuous binary Darboux transformation framework by modeling the condensate background via a nonnegative spectral measure ρ in the scattering data and the injected soliton via a narrowly supported measure σ added to the negative-spectrum measure.

Background

The paper develops a continuous binary Darboux transformation that acts directly on scattering data for step-type KdV solutions, enabling controlled modification of the negative spectrum while preserving the right reflection coefficient. In this framework, deterministic soliton gases and condensates can be superimposed on general step-like backgrounds by adjusting the spectral measure.

Numerical experiments suggest that adding a soliton to a dense soliton condensate may cause a localized depletion or "evaporation" of the condensate. The authors highlight this phenomenon as an explicit open problem, proposing to model the condensate by a background measure dρ and the injected soliton by a narrowly supported measure dσ within their operator-theoretic setting.