Integral-Type Darboux Transformations for the mKdV Hierarchy with Self-Consistent Sources**The project supported in part by the Research Grants Council of Hong Kong Special Administrative Region, China (Project No. 9040466), and the City University of Hong Kong (Project No. 7001041), as well as by the Special Funds for Chinese Major Basic Research Project “Nonlinear Science”

Abstract

We present integral-type Darboux transformation for the mKdV hierarchy and for the mKdV hierarchy with self-consistent sources. In contrast with the normal Darboux transformation, the integral-type Darboux transformations can offer non-auto-Bácklund transformation between two -th mKdV equations with self-consistent sources with different degrees. This kind of Darboux transformation enables us to construct the -soliton solution for the mKdV hierarchy with self-consistent sources. We also propose the formulas for the times repeated integral-type Darboux transformations for both mKdV hierarchy and mKdV hierarchy with self-consistent sources.