Mutual extraction of Bäcklund transformations and Lax representations for the Korteweg–de Vries equation

Hoseini, Sayed Mohammad; Reza, Abedini Mohajeri

doi:10.22072/wala.2026.2082722.1487

Mutual extraction of Bäcklund transformations and Lax representations for the Korteweg–de Vries equation

Document Type : Research Paper

Authors

Sayed Mohammad Hoseini ¹

Abedini Mohajeri Reza ²

¹ Mathematics Dep. Vali-E-Asr University of Rafsanjan

² ‎Vali-e-Asr Rafsanjan University

10.22072/wala.2026.2082722.1487

Abstract

‎In this paper‎, ‎we investigate the intrinsic relationship between Bäcklund transformations and Lax representations for the Korteweg–de Vries (KdV) equation‎. ‎Viewing the KdV equation within the framework of integrable hierarchies‎, ‎we analyze how its Bäcklund transformations encode the underlying spectral structure‎. ‎We demonstrate that the Lax pair of the KdV equation can be systematically derived from its Bäcklund transformations‎, ‎and conversely‎, ‎that the Bäcklund transformations can be reconstructed directly from the associated Lax representation‎. ‎This bidirectional correspondence clarifies the geometric and algebraic role of Bäcklund transformations as discrete symmetries of the KdV equation and highlights their interpretation as Darboux-type transformations acting on the spectral problem‎.

Keywords

Integrable equations, Lax pair, Bä

cklund transformations, Korteweg-de Vries equation, Spectral Problem