An integrable semidiscretization of the modified Camassa–Holm equation with linear dispersion term

In the present paper, we are with integrable discretization of a modified Camassa–Holm (mCH) equation linear dispersion term. The key construction is semidiscrete analog for set bilinear equations mCH equation. First, show that these and their determinant solutions either in Gram-type or Casorati-type can be reduced from discrete Kadomtsev–Petviashvili (KP) through Miwa transformation. Then, by scrutinizing reduction process, obtain general soliton solution form. Finally, defining dependent variables hodograph transformations, able to derive an It also shown converges continuous one continuum limit.