Lax-Pair Operators for Squared Eigenfunctions in the Inverse Scattering Transformations

Abstract

The algorithm of Chen, Lee and Liu enables us to construct alternative Lax-pair operators for the Korteweg-de Vries equation and the modified Korteweg-de Vries equation. These Lax-pair operators stand as the Lax-pair operators for the squared eigenfunction and the sum of squared eigenfunctions of the Ablowitz-Kaup-Newell-Segur inverse scattering transformation for these celebrated non-linear evolution equations. Present analysis completes series of our analysis showing that the integro-differential type of Lax-pair operators derived by Chen, Lee and Liu are closely related to the Lax-pair operators for the known schemes of the inverse scattering transformation.