An integrable semi-discretization of the modified Camassa-Holm equation with linear dispersion term

TL;DR

This paper develops an integrable semi-discrete version of the modified Camassa-Holm equation with linear dispersion, derived from bilinear equations related to the discrete KP equation, and demonstrates its convergence to the continuous form.

Contribution

The paper introduces a novel semi-discrete integrable model of the modified Camassa-Holm equation derived via bilinear equations and Miwa transformation, expanding discretization methods for integrable systems.

Findings

Derived semi-discrete bilinear equations from the discrete KP equation.

Constructed general soliton solutions in determinant form.

Showed convergence of the semi-discrete equation to the continuous one.

Abstract

In the present paper, we are concerned with integrable discretization of a modified Camassa-Holm equation with linear dispersion term. The key of the construction is the semi-discrete analogue for a set of bilinear equations of the modified Camassa-Holm equation. Firstly, we show that these bilinear equations and their determinant solutions either in Gram-type or Casorati-type can be reduced from the discrete KP equation through Miwa transformation. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete 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 semi-discrete analogue of the modified Camassa-Holm equation. It is also shown that the semi-discrete modified Camassa-Holm equation converges to the continuous one in the continuum limit.