The proton motion is studied in hydrogen-bonded chains where the potential double-well relief is asymmetric. An exact solution of the nonlinear differential equation of motion is obtained for the protonic sublattice in which protons move in asymmetric double-minimum potentials. It is shown that the propagating protonic displacement in hydrogen-bonded solids of this type is a soliton of a bell-shaped form. In the limiting case of the symmetric double-well potential the usual kink solution is obtained.