Linear Superposition for a Large Number of Nonlinear Equations

TL;DR

This paper reveals that many nonlinear equations, both continuous and discrete, admit superposed solutions constructed from Jacobi elliptic functions, expanding the known solution space for such equations.

Contribution

It introduces a novel superposition principle for nonlinear equations using Jacobi elliptic functions, applicable to both single and coupled systems.

Findings

Solutions in terms of $ ext{cn}(x,m)$ and $ ext{dn}(x,m)$ imply superposed solutions.

Existence of superpositions involving $ ext{dn}^2(x,m)$ and combinations with $ ext{cn}(x,m)$ and $ ext{dn}(x,m)$.

Superposed solutions are valid for both continuum and discrete nonlinear equations.

Abstract

We demonstrate a kind of linear superposition for a large number of nonlinear equations, both continuum and discrete. In particular, we show that whenever a nonlinear equation admits solutions in terms of Jacobi elliptic functions $\cn(x,m)$ and $\dn(x,m)$, then it also admits solutions in terms of their sum as well as difference, i.e. $\dn(x,m) \pm \sqrt{m}\, \cn(x,m)$. Further, we also show that whenever a nonlinear equation admits a solution in terms of $\dn^2(x,m)$, it also has solutions in terms of $\dn^2(x,m) \pm \sqrt{m}\, \cn(x,m)\, \dn(x,m)$ even though $\cn(x,m)\, \dn(x,m)$ is not a solution of that nonlinear equation. Finally, we obtain similar superposed solutions in coupled theories.