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Charge migration is the electronic response that immediately follows localized ionization or excitation in a molecule, before the nuclei have time to move. It typically unfolds on sub-femtosecond time scales and most often corresponds to dynamics far from equilibrium, involving multi-electron interactions in a complex chemical environment. While charge migration has been documented experimentally and theoretically in multiple organic and inorganic compounds, the general mechanism that regulates it remains unsettled. In this work we use tools from nonlinear dynamics to analyze charge migration that takes place along the backbone of conjugated hydrocarbons, which we simulate using time-dependent density functional theory. In this electron-density framework we show that charge migration modes emerge as attosecond solitons and demonstrate the same type of solitary-wave dynamics in both simplified model systems and full three-dimensional molecular simulations. We show that these attosecond-soliton modes result from a balance between dispersion and nonlinear effects tied to time-dependent multi-electron interactions.%Our soliton-mode mechanism, and the nonlinear tools we use to analyze it, pave the way for understanding migration dynamics in a broad range of organic molecules.%For instance, we demonstrate the opportunities for chemically steering charge migration via molecular functionalization, which can alter both the initially localized electron perturbation and its subsequent time evolution.