From optics to hydrodynamics, shock and rogue waves are widespread. Although they appear as distinct phenomena, transitions between extreme waves are allowed. However, these have never been experimentally observed because control strategies are still missing. We introduce the new concept of topological control based on the one-to-one correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the nonlinear Schrödinger equation solutions through Riemann theta functions. We demonstrate the concept experimentally by reporting observations of supervised transitions between waves with different genera. Considering the box problem in a focusing photorefractive medium, we tailor the time-dependent nonlinearity and dispersion to explore each region in the state diagram of the nonlinear wave propagation. Our result is the first realization of topological control of nonlinear waves. This new technique casts light on shock and rogue waves generation and can be extended to other nonlinear phenomena.
Bibliographical noteFunding Information:
We acknowledge M. Conforti, S. Gentilini, P.G. Grinevich, A. Mussot, P.M. Santini, S. Trillo, and V.E. Zakharov for fruitful conversations on related topics. We thank MD Deen Islam for technical support in the laboratory. The present research was supported by PRIN 2015 NEMO project (grant number 2015KEZNYM), H2020 QuantERA QUOMPLEX (grant number 731473), H2020 PhoQus (grant number 820392), PRIN 2017 PELM (grant number 20177PSCKT), Sapienza Ateneo (2016 and 2017 programs), and Ministry of Science and Technology of Taiwan (105-2628-M-007-003-MY4).
© 2019, The Author(s).