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An integrable semi-discretization of the Camassa-Holm equation is presented.The keys of its construction are bilinear forms and determinant structure ofsolutions of the CH equation. Determinant formulas of $N$-soliton solutions ofthe continuous and semi-discrete Camassa-Holm equations are presented. Based ondeterminant formulas, we can generate multi-soliton, multi-cuspon andmulti-soliton-cuspon solutions. Numerical computations using the integrablesemi-discrete Camassa-Holm equation are performed. It is shown that theintegrable semi-discrete Camassa-Holm equation gives very accurate numericalresults even in the cases of cuspon-cuspon and soliton-cuspon interactions. Thenumerical computation for an initial value condition, which is not an exactsolution, is also presented.

An integrable semi-discretization of the Camassa-Holm equation and its determinant solution