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 (Formula presented.) -soliton solution to the Sawada–Kotera equation, for an arbitrary positive integer (Formula presented.), by recovering that soliton solution from the solution to our Marchenko integral equation. Our method explains the origin of the (Formula presented.) real parameters appearing in the (Formula presented.) -soliton solution formula obtained by the ad-hoc method of Hirota. We show that (Formula presented.) of those parameters are related to the (Formula presented.) bound-state poles of the left transmission coefficient and the remaining (Formula presented.) 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.