Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization

In Part I soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between the Q3 equation and the NQC equations, and then by reductions. In that work, the central role was played by a Cauchy matrix. In this work we use a different approach, and we derive the N -soliton solutions following Hirota's direct and constructive method. This leads to Casoratians and bilinear difference equations. We give here details for the H-series of equations and for Q1; the results for Q3 have been given earlier.