  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 (KP) equation, the nonlinear
Schrodinger (NLS) equation, the $\lambda \phi^4$ model, the sine-Gordon
equation and the Boussinesq equation by making appropriate linear
superpositions of known periodic solutions. This unusual procedure for
generating solutions is successful as a consequence of some powerful, recently
discovered, cyclic identities satisfied by the Jacobi elliptic functions.
