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In this work, taking advantage of the Generalized Hedgehog Ansatz, weconstruct new self-gravitating solitons in a cavity with mirror-like boundaryconditions for the SU(2) Non-linear Sigma Model and Skyrme model. Forspherically symmetric spacetimes, we are able to reduce the system to threeindependent equations that are numerically integrated. There are two branchesof well-behaved solutions. The first branch is defined for arbitrary values ofthe Skyrme coupling and therefore also leads to a gravitating soliton in theNon-linear Sigma Model, while the second branch exists only for non-vanishingSkyrme coupling. The solutions are quasi-static and in the first branch arecharacterized by two integration constants that correspond to the frequency ofthe phase of the Skyrme field and the value of the Skyrme profile at theorigin, while in the second branch the latter is the unique parametercharacterizing the solutions. These parameters determine the size of thecavity, the redshift at the boundary of the cavity, the energy of the scalarfield and the charge associated to a U(1) global symmetry. We also show thatwithin this ansatz, assuming analyticity of the matter fields, there are nospherically symmetric black hole solutions.

Solitons in a cavity for the Einstein-SU(2) Non-linear Sigma Model and Skyrme model