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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.
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