Integrable Semi-Discretization for a Modified Camassa-Holm Equation with Cubic Nonlinearity

TL;DR

This paper develops an integrable semi-discretization of the modified Camassa-Holm equation with cubic nonlinearity, based on discrete KP equations and reciprocal transformations, providing explicit soliton solutions and showing convergence to the continuous equation.

Contribution

It introduces a novel integrable semi-discrete version of the mCH equation derived from discrete KP equations, including explicit solutions and a continuum limit analysis.

Findings

Derived semi-discrete bilinear equations from discrete KP

Constructed general soliton solutions in Gram determinant form

Proved convergence of semi-discrete to continuous mCH equation

Abstract

In the present paper, an integrable semi-discretization of the modified Camassa-Holm (mCH) equation with cubic nonlinearity is presented. The key points of the construction are based on the discrete Kadomtsev-Petviashvili (KP) equation and appropriate definition of discrete reciprocal transformations. First, we demonstrate that these bilinear equations and their determinant solutions can be derived from the discrete KP equation through Miwa transformation and some reductions. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solutions in the Gram-type determinant form. Finally, we obtain an integrable semi-discrete analog of the mCH equation by introducing dependent variables and discrete reciprocal transformation. It is also shown that the semi-discrete mCH equation converges to the continuous one in the continuum limit.