Wavelet and Linear Algebra

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.