The Marchenko Method for Soliton Solutions to the Sawada–Kotera Equation
Studies in Applied Mathematics, vol.157, no.2, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 157 Issue: 2
- Publication Date: 2026
- Doi Number: 10.1111/sapm.70280
- Journal Name: Studies in Applied Mathematics
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, ABI/INFORM, Aerospace Database, Applied Science & Technology Source, Compendex, INSPEC, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO), Business Source Ultimate (EBSCO), Engineering Source (EBSCO), Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Keywords: inverse scattering for the third-order operator, Marchenko integral equation, Marchenko method, Sawada–Kotera equation, soliton solutions
- Open Archive Collection: AVESIS Open Access Collection
- Recep Tayyip Erdoğan University Affiliated: Yes
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.