El Naschie’s ε(∞) space–time and new results in scale relativity theories

New results in fractal space–time theory are established: the fractal operator for the fractal dimension D = 2 implies a generalized Schrödinger equation in the Nottale’s scale relativity theory, while the fractal operator for the fractal dimension D = 3 implies the Korteweg–de Vries equation in the one-dimensional case. The connection with El Naschie’s ε(∞) theory result by means of the Cooper-type pairs or through wave-particle duality.