Superposed periodic kink and pulse solutions of coupled nonlinear equations

TL;DR

This paper introduces new periodic solutions for coupled nonlinear equations using Jacobi elliptic functions, showing that superpositions of kinks and pulses are possible, extending the understanding of solutions in such systems.

Contribution

It presents novel superposed periodic solutions for coupled nonlinear equations, expanding the class of known solutions with explicit elliptic function expressions.

Findings

Solutions expressed in terms of Jacobi elliptic functions.

Superpositions of kinks and pulses are possible in coupled systems.

Superposition does not hold for certain parameter values.

Abstract

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