Abstract We present analytic integral solutions for the second-order induced gravitational waves (GWs). After presenting all possible source terms, we calculate explicitly GWs by linear scalar and tensor perturbations during matter- radiation-dominated epochs.

In this paper, we study analytic and loop solutions of the K(2,2) equation(focusing branch), which is first proposed by Rosenau. The implicit analytic and loop solutions are obtained by using the dynamical system approach. Moreover, we investigate how the famous Rosenau-Hyman compactons can be recovered as limits of classical solitary wave solutions forming analytic homoclinic orbits for the re...