摘要： Controlling nonlinear optical processes is a signi?cant challenge in photonics. Shock waves, rogue waves, and solitons are widespread, from optics to hydrodynamics. Intense research is dedicated to advanced techniques for tailoring extreme waves and ?nding the conditions to induce transitions from one kind of wave to another. We develop a new strategy to supervise, modify or tune a laser beam in third-order nonlinear materials, when light propagation is ruled by the nonlinear Schr¨odinger equation (NLSE). We denote our approach topological control (TC). TC is based on the one-to-one correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the NLSE solutions by the Riemann theta function [1]. For a box-shaped wave in a focusing Kerr medium, TC allows to control transitions from dispersive shock waves (DSWs), to rogue waves (RWs) and soliton gases (SGs). We prove that our method is experimentally realizable in a photorefractive crystal [2]. Speci?cally, we use the parametric time-dependence of photorefractive nonlinearity to shape the asymptotic wave pro?le. We tailor time-dependent propagation coef?cients, as nonlinearity and dispersion, to explore each region in the state-diagram [Fig. 1(a)] and observe all the phases in the nonlinear wave evolution [Fig. 1(b)]. This new technique casts light on focusing DSWs and RW generation, and can be extended to many other nonlinear phenomena, from classical to quantum ones.