A theorized object called a geon, "an electromagnetic or gravitational wave which is held together in a confined region by the gravitational attraction of its own field energy", would seem like a match for what you're talking about. The wiki article mentions that exact solutions involving geons have been found (one is discussed in this paper), though it's not clear if they would be stable. For numerical simulations of various scenarios involving strong gravitational waves whose behavior is highly nonlinear, see this pdf. And here is a pdf of John Wheeler's original 1955 paper proposing them, which contains an image of a toroidal electromagnetic geon (he later notes that gravitational wave geons would be spherical rather than toroidal) consisting of two electromagnetic waves traveling in opposite directions, held to their distorted paths by their own gravity:
