Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

Abstract

We give a dispersionless Toda-like extension to the dispersionless Harry Dym (dDym) hierarchy. The extended dDym (EdDym) hierarchy has a dressing formulation, and its underlying solution structure can be investigated through the twistor construction. We show that additional symmetries of the solution space generate Bäcklund transformations of the EdDym hierarchy. We present some examples of constructing new solutions of the (2+1)-dimensional dDym and EdDym equations via Bäcklund transformations.

New extension of dispersionless Harry Dym hierarchy

“…In 1999, the dHD hierarchy was defined by a given classical r-matrix on a Poisson algebra [16]. Some progresses have been made for the dHD hierarchy such as hodograph solutions [6], Miura map and bi-Hamiltonian formulation [7], additional symmetries and Bäcklund transformation [5] and Hydrodynamic reduction [4]. We notice that the Miura map between the dmKP hierarchy and the dHD hierarchy is triggered by the "eigenfunctions "of the dmKP hierarchy and depends on a transformation of independent variables.…”