Linear Superposition of Quadratic Functions in a Fifth Order KdV-Type Equation

TL;DR

This paper demonstrates that a fifth order KdV-type equation allows for multiple real and complex PT-invariant solutions through linear superpositions of quadratic Jacobi elliptic functions, unlike simpler equations like KdV.

Contribution

It introduces a novel superposition principle for quadratic Jacobi elliptic functions in a fifth order KdV-type equation, expanding the known solution space.

Findings

Multiple real and complex PT-invariant solutions identified

Linear superposition of quadratic functions established

Contrasts with partial superposition in simpler equations

Abstract

We show that a fifth order KdV-type equation admits several real as well as complex parity-time reversal or PT-invariant solutions with linear superposition of quadratic functions involving Jacobi elliptic functions of the form ${\rm dn}^2(x,m)$, ${\rm cn}(x,m){\rm dn}(x,m)$, ${\rm sn}(x,m) {\rm cn}(x,m)$ and ${\rm sn}(x,m){\rm dn}(x,m)$. These results must be contrasted with only partial superposition of such functions in Korteweg-de Vries (KdV), $\phi^3$ and a few other nonlinear equations.