We revise soliton and lump solutions described by the cylindrical Kadomtsev-Petviashvili (cKP) equation and construct new exact solutions relevant to physical observation. In the first part of this study, we consider basically axisymmetric waves described by the cylindrical Kortweg-de Vries equation and analyze approximate and exact solutions to this equation. Then, we consider the stability of the axisymmetric solitons with respect to the azimuthal perturbations and suggest a criterion of soliton instability. The results of our numerical modeling confirm the suggested criterion and reveal lump emergence in the course of the development of the modulation instability of ring solitons in the unstable case. In the next part of this study, which will follow shortly, we will present exact solutions to the cKP equation describing weakly nonlinear waves in media with positive dispersion subject to the modulation instability of solitons with respect to small azimuthal perturbations.
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