Virtual Dirac Monopoles underlying the Nontrivial Phases of Rogue Waves

We uncover the virtual monopoles underlying the nontrivial phases of the one-dimensional nonlinear excitations of rogue waves with extending the Dirac magnetic monopole theory to a complex plane. We find that the density zeros of the nonlinear waves on the extended complex plane constitute the virtual monopole fields with a quantized flux of elementary π. We then explain the exotic property of rogue waves by means of a virtual monopole collision mechanism, and find that the maximum amplitude amplification ratio and multiple phase steps of the high-order rogue waves are closely related to the number of their contained monopoles. These results open a new avenue to study topological properties of nonlinear waves and provide an alternative way to understand their dynamics.