  Based on our previous work to the Degasperis-Procesi equation (J. Phys. A 46
045205) and the integrable semi-discrete analogue of its short wave limit (J.
Phys. A 48 135203), we derive an integrable semi-discrete Degasperis-Procesi
equation by Hirota's bilinear method. Meanwhile, $N$-soliton solution to the
semi-discrete Degasperis-Procesi equation is provided and proved. It is shown
that the proposed semi-discrete Degasperis-Procesi equation, along with its
$N$-soliton solution converge to ones of the original Degasperis-Procesi
equation in the continuous limit.
