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

TL;DR

The paper addresses whether nonlinear equations can admit linear superposition of quadratic functions. By applying a traveling-wave reduction to a local, fifth-order KdV-type equation $u_t + \alpha u^2 u_x + \beta u_{xxx} + \gamma u_{xxxxx} = 0$, it constructs exact solutions built from quadratic Jacobi elliptic functions and their complex/PT-invariant combinations. It reports 12 explicit quadratically superposed solutions: 6 single-quadratic, 2 real quadratically superposed periodic, and 4 complex PT-invariant periodic plus 2 hyperbolic PT-invariant cases, all with fixed amplitude ratios and parameter constraints linking $\alpha,\beta,$ $\gamma$, $\eta$, and $v$, including hyperbolic limits $m=1$ yielding $\sech^2(y)$ and $\sech(y)\tanh(y)$ forms. The work demonstrates a rich linear-superposition structure in a nonlinear, local equation and opens questions about stability, broader model classes, and potential physical applications of these quadratically superposed states, including their PT-symmetric properties.

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), $φ^3$ and a few other nonlinear equations.