Multiscale expansion of the lattice potential KdV equation on functions of an infinite slow-varyness order

We present a discrete multiscale expansion of the lattice potential Korteweg-de Vries (lpKdV) equation on functions of an infinite order of slow varyness. To do so, we introduce a formal expansion of the shift operator on many lattices holding at all orders. The lowest secularity condition from the expansion of the lpKdV equation gives a nonlinear lattice equation, depending on shifts of all orders, of the form of the nonlinear Schrodinger equation.