Solutions to the ABS lattice equations via generalized Cauchy matrix approach

The usual Cauchy matrix approach starts from a known plain wave factor vector $r$ and known dressed Cauchy matrix $M$. In this paper we start from a matrix equation set with undetermined $r$ and $M$. From the starting equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The starting matrix equation set admits more choices for $r$ and $M$ and in the paper we give explicit formulae for all possible $r$ and $M$. As applications, we get more solutions than usual multi-soliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation and some lattice equations in ABS list.