Commutativity of Pfaffianization and Bäcklund transformation: The semi-discrete Toda equation

We present a modified version of the semi-discrete Toda lattice, which also serves as a Bäcklund transformation for the semi-discrete Toda lattice, and we give N-soliton solution of this modified semi-discrete Toda equation in terms of Casorati determinants. Commutativity of Pfaffianization and Bäcklund transformations is generalized to the semi-discrete Toda lattice, which has enabled us to construct Bäcklund transforation formulae for a coupled semi-discrete Toda lattice.