Periodic solutions of nonlinear equations obtained by linear superposition

Abstract

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 Schrö dinger equation, the λφ 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.