The modified Camassa–Holm equation with self-consistent sources

Abstract The modified Camassa–Holm equation serves as the dual counterpart to the modified KdV equation. Notably, the modified KdV equation has been shown to possesses the corresponding equation with the self-consistent sources (SCSs). In this paper, we derive the modified Camassa–Holm equation with SCSs based on its bi-Hamiltonian structure. Subsequently, we obtain its Lax pair. Utilizing the Lax pair of the modified Camassa–Holm equation and its variant form with the SCSs, we also derive the Darboux transformations for both the modified Camassa–Holm equation and its counterpart with SCSs. Additionally, we propose a novel approach for constructing the Darboux transformation for these equations.