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

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.