Short wave limit of the Novikov equation and its integrable semi-discretizations

In this paper, we are concerned with one of the generalized short wave equations proposed by Hone et al (2018 Lett. Math. Phys 108 927). We show that the derivative form of this equation can be viewed as a short wave limit of the Novikov (sw-Novikov) equation. Furthermore, this generalized short wave equation and its derivative form are found to be connected to period 3 reduction of two-dimensional CKP(BKP)-Toda hierarchy, same as the short wave limit of the Depasperis-Procesi (sw-DP) equation. We propose a two-component short wave equation which contain the sw-Novikov equation and sw-DP equation as two special cases. As a main result, we construct two types of integrable semi-discretizations via Hirota's bilinear method and provide multi-soliton solution to the semi-discrete sw-Novikov equation.