Loop Groups and Discrete KdV Equations

A study is presented of fully discretized lattice equations associated with the KdV hierarchy. Loop group methods give a systematic way of constructing discretizations of the equations in the hierarchy. The lattice KdV system of Nijhoff et al. arises from the lowest order discretization of the trivial, lowest order equation in the hierarchy, bt = bx. Two new discretizations are also given, the lowest order discretization of the first nontrivial equation in the hierarchy, and a “second order” discretization of bt = bx. The former, which is given the name full lattice KdV has the (potential) KdV equation as a standard continuum limit. For each discretization a Bäcklund transformation is given and soliton content analyzed. The full lattice KdV system has, like KdV itself, solitons of all speeds, whereas both other discretizations studied have a limited range of speeds (being discretizations of an equation with solutions only of a fixed speed).