The Marchenko method for soliton solutions to the Sawada--Kotera equation

Abstract: 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.