Reciprocal transformations of Harry Dym hierarchy

Abstract In this paper, we work out the reciprocal transformations generated by the adjoint eigenfunctions, and write down the corresponding changes of the Lax operators, eigenfunctions and adjoint eigenfunctions under this kind of new reciprocal transformations at first. Then we discuss the successive actions of the reciprocal transformations generated by the eigenfunctions and the adjoint eigenfunctions. Also we investigate the commutativity of different types of the reciprocal transformations. The above results are very useful to construct the new type solutions.