2026/06/13 by Tuncay Aktosun, Ramazan Ercan, Ivan Toledo +1 · 1 voice
Physics and Astronomy · #nlin.SI
Associated with the third-order linear differential operator, we present the Marchenko integral equation using as input the bound-state poles of a transmission coefficient and the time-evolved bound-state dependency constants. We derive the \mathbf N-soliton solution to the Sawada--Kotera equation, for an arbitrary positive integer \mathbf N, by recovering that soliton solution from the solution to our Marchenko integral equation. Our method explains the origin of the 2\mathbf N real parameters appearing in the \mathbf N-soliton solution formula obtained by the ad-hoc method of Hirota. We show that \mathbf N of those parameters are related to the \mathbf N bound-state poles of the left transmission coefficient and the remaining \mathbf N parameters are related to the bound-state dependency constants. Our Marchenko integral equation corresponds to the ``GLM (Gel'fand--Levitan--Marchenko) integral equation'' Kaup relentlessly but unsuccessfully tried to obtain.