  We reveal intrinsic topological vector potentials underlying the nonlinear
waves governed by one-dimensional nonlinear Schr\"{o}dinger equations by
investigating the Berry connection of the linearized Bogoliubov-de-Gennes (BdG)
equations in an extended complex coordinate space. Surprisingly, we find that
the density zeros of these nonlinear waves exactly correspond to the degenerate
points of the BdG energy spectra and can constitute monopole fields with a
quantized magnetic flux of elementary $\pi$. Such a vector potential consisting
of paired monopoles with opposite charges can completely capture the essential
characteristics of nonlinear wave evolution. As an application, we investigate
rogue waves and explain their exotic property of ``appearing from nowhere and
disappearing without a trace'' by means of a monopole collision mechanism. The
maximum amplification ratio and multiple phase steps of a high-order rogue wave
are found to be closely related to the number of monopoles. Important
implications of the intrinsic topological vector potentials are discussed.
