Linear Superposition in Nonlinear Equations

Several nonlinear systems such as the Korteweg–de Vries (KdV) and modified KdV equations and $\lambda {\phi }^4$ theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions.