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

Abstract

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