Nonlocal symmetries and Darboux transformations of the Camassa–Holm equation and modified Camassa–Holm equation revisited

By the nonlocal symmetry approach, Hernández-Heredero and Reyes [J. Phys. A: Math. Theor. 42, 182002 (2009)] and Bies et al. [J. Math. Phys. 53, 073710 (2012)] obtained Darboux Transformations (DTs) of the Camassa–Holm equation and the modified Camassa–Holm equation. However, the wave function does not appear in their DTs explicitly. We introduce wave functions to the DTs and show how they are related to the binary DT of the first negative flow in the Korteweg–de Vries (KdV) hierarchy. Furthermore, we connect nonlocal symmetries of the Camassa–Holm equation and the modified Camassa–Holm equation with those of the negative KdV equation and the negative modified KdV equation, respectively.