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We present novel previously unexplored periodic solutions, expressed in termsof Jacobi elliptic functions, for both a coupled $\phi^4$ model and a couplednonlinear Schr\"odinger equation (NLS) model. Remarkably, these solutions canbe elegantly reformulated as a linear combination of periodic kinks andantikinks, or as a combination of two periodic kinks or two periodic pulsesolutions. However, we also find that for $m=0$ and a specific value of theperiodicity (or at a nonzero value of the elliptic modulus $m$) thissuperposition does not hold. These results demonstrate that the notion ofsuperposed solutions extends to the coupled nonlinear equations as well.

Superposed periodic kink and pulse solutions of coupled nonlinear equations