Integration of the modified Korteweg-de Vries hierarchy with an integral type of source

Abstract

The modified Korteweg-de Vries hierarchy with an integral type of source (mKdVHWS), which consists of the reduced AKNS eigenvalue problem with r = q and the mKdV hierarchy with an extra term of the integration of a square eigenfunction, is investigated. We propose a method to find the explicit evolution equation for the eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations of scattering data corresponding to the mKdVHWS, and solve the equation in the mKdVHWS by inverse scattering transformation.