Isospectral prosperty of double Darboux transformation

Abstract

The isospectral property of the algebro-geometric double Darboux transformation is studied by an algebraic method. This is established by verifying that the algebro-geometric double Darboux transformation of the $n$-th algebro-geometric potential solves the evolution equation of the higher order KdV type. Applying these results, the second Darboux-Lamé potential, which can be regarded as the 3-elliptic solitons, is explicitly constructed by an elementary method.