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We analyze the low-energy dynamics of quasi-one-dimensional, large-S quantum antiferromagnets with easy-axis anisotropy, using a semiclassical nonlinear sigma model. The saddle point approximation leads to a sine-Gordon equation which supports soliton solutions. These correspond to the movement of spatially extended domain walls. Long-range magnetic order is a consequence of a weak interchain coupling. Below the ordering temperature, the coupling to nearby chains leads to an energy cost associated with the separation of two domain walls. From the kink-antikink two-soliton solution, we compute the effective confinement potential. At distances large compared to the size of the solitons the potential is linear, as expected for pointlike domain walls. At small distances the gradual annihilation of the solitons weakens the effective attraction and renders the potential quadratic. From numerically solving the effective one-dimensional Schrödinger equation with this nonlinear confinement potential we compute the soliton bound state spectrum. We apply the theory to CaFe2O4, an anisotropic S=5/2 magnet based upon antiferromagnetic zigzag chains. Using inelastic neutron scattering, we are able to resolve seven discrete energy levels for spectra recorded slightly below the Néel temperature TN≈200 K. These modes are well described by our nonlinear confinement model in the regime of large spatially extended solitons.