A remarkable exact mapping, valid for low-enough energy scales and close to a sharp boundary distribution of hadronic matter, from the (3 + 1)-dimensional Skyrme model to the sine-Gordon theory in (1 + 1) dimensions in the attractive regime is explicitly constructed. Besides the intrinsic theoretical interest to be able to describe the prototype of nonintegrable theories (namely, quantum chromodynamics in the infrared regime) in terms of the prototype of integrable relativistic field theories (namely, sine-Gordon theory in (1 + 1) dimensions), we will show that this mapping can be extremely useful to analyze both equilibrium and out-of-equilibrium features of baryonic distributions in a cavity.