We discover a new class of topological solitons. These solitons can exist in a space of infinite volume like, e.g., R n , but they cannot be placed in any finite volume because the resulting formal solutions have infinite energy. Therefore, these objects are interpreted as totally incompressible solitons. As a first particular example, we consider (1 þ 1)-dimensional kinks in theories with a nonstandard kinetic term or, equivalently, in models with the so-called runaway (or vacuumless)

more »

... s. But incompressible solitons also exist in higher dimensions. As specific examples, in (3 þ 1) dimensions we study Skyrmions in the dielectric extensions of both the minimal and BPS Skyrme models. In the latter case, the Skyrmionic matter describes a completely incompressible topological perfect fluid.