Degenerate Riemann theta functions, Fredholm and wronskian representations of the solutions to the KdV equation and the degenerate rational case

We degenerate the finite gap solutions of KdV equation from general formulation given in terms abelian functions when gaps tend to points, get Fredholm determinants and wronskians. For this we establish a link between Riemann theta functions, This gives bridge algebro-geometric approach Darboux dressing method. construct also multi-parametric rational equation.