A note on Bäcklund transformations for the Harry Dym equation

The Bäcklund transformations are revisited for the Harry Dym equation. The associated nonlinear superposition formula, which consists of two equations — one for the field variable and the other for the spatial variable, is also presented. Interpreted as discrete systems, nonlinear superposition formula is identified as the lattice potential modified Korteweg–de Vries equation and a special case of Q1 equation in the ABS list. A particular limit of the nonlinear superposition formula is considered and the resulting Bäcklund transformation is shown to be related to the finite transformation of the nonlocal symmetries.