Periodic solutions of nonlinear equations obtained by linear superposition

We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach we find periodic solutions of the Kadomtsev-Petviashvili equation, the nonlinear Schrodinger equation, the lambdaphi(4) model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known-periodic solutions. This unusual procedure for generating solutions of nonlinear differential equations is successful as a consequence of. some powerful, recently discovered, cyclic identities satisfied by the Jacobi elliptic functions.