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In high energy proton-nucleus collisions, the single- and double-inclusivesoft gluon productions at the leading order have been calculated andphenomenologically studied in various approaches for many years. These studiesdo not take into account the saturation and multiple rescatterings in the fieldof the proton. The first saturation correction to these leading order results(the terms that are enhanced by the combination $\alpha_s^2 \mu^2$, where$\mu^2$ is the proton's color charge squared per unit transverse area) has notbeen completely derived despite recent attempts using a diagrammatic approach.This paper is the first in a series of papers towards analytically completingthe first saturation correction to physical observables in high energyproton-nucleus collisions. Our approach is to analytically solve the classicalYang-Mills equations in the dilute-dense regime using the Color GlassCondensate effective theory and compute physical observables constructed fromclassical gluon fields. In the current paper, the Yang-Mills equations aresolved perturbatively in the field of the dilute object (the proton).Next-to-leading order and next-to-next-to-leading order analytic solutions areexplicitly constructed. A systematic way to obtain all higher order analyticsolutions is outlined

First Saturation Correction in High Energy Proton-Nucleus Collisions: I. Time evolution of classical Yang-Mills fields beyond leading order