Linear Superposition in Nonlinear Equations

TL;DR

This paper demonstrates that linear superpositions of known solutions to nonlinear equations like KdV can generate new solutions, thanks to novel identities of elliptic functions.

Contribution

It introduces a method to construct new solutions for nonlinear equations using linear combinations and elliptic function identities.

Findings

Linear combinations produce new solutions to KdV and mKdV equations.

New identities of elliptic functions enable this superposition approach.

The method broadens the solution space for nonlinear integrable equations.

Abstract

Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by virtue of some remarkable new identities satisfied by the elliptic functions.