Integral-Type Darboux Transformations for the mKdV Hierarchy with Self-Consistent Sources

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.

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