  The link between the short wave model of the Camassa-Holm equation (SCHE) and
bilinear equations of the two-dimensional Toda lattice (2DTL) is clarified. The
parametric form of N-cuspon solution of the SCHE in Casorati determinant is
then given. Based on the above finding, integrable semi-discrete and
full-discrete analogues of the SCHE are constructed. The determinant solutions
of both semi-discrete and fully discrete analogues of the SCHE are also
presented.
