On the global existence for the modified Camassa–Holm equation

Abstract The modified Camassa–Holm (mCH) equation is a model to describe the unidirectional propagation of shallow water waves. Without knowledge of any conservation quantities in the higher order Sobolev spaces for this type of the quasi‐linear evaluation equation, obtaining global well‐posedness for the mCH equation appears to be quite challenging. In this paper, we address the existence of global solutions in a weighted Sobolev space to the Cauchy problem for the mCH equation. The key to prove this result is establish bijectivity between potential and reflection coefficient by using inverse scattering transform method based on the Riemann–Hilbert problem associated with the Cauchy problem for the mCH equation.