Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 091, 14 pages      arXiv:2404.18372      https://doi.org/10.3842/SIGMA.2024.091

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

Bao-Feng Feng a, Heng-Chun Hu b, Han-Han Sheng cd, Wei Yin ae and Guo-Fu Yu d
a) School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, Texas 78541, USA
b) College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P.R. China
c) Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, P.R. China
d) School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, P.R. China
e) Department of Mathematics, South Texas College, McAllen, Texas 78501, USA

Received April 30, 2024, in final form October 07, 2024; Published online October 12, 2024

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.

Key words: modified Camassa-Holm equation; discrete KP equation; Miwa transformation.

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