Reply to the Comment by Zhang and Li [J. Math. Phys. 56, 084101, 2015] on the Paper [J. Math. Phys. {\bf 56}, 032104, 2015]}

TL;DR

This paper defends a previous claim that nonlinear equations with solutions in Jacobi elliptic functions also have solutions in related forms, clarifying and correcting a misunderstanding in prior critique.

Contribution

The authors clarify and validate their original assertion about solutions in Jacobi elliptic functions, addressing and correcting the critique by Zhang and Li.

Findings

The original claim about solutions in Jacobi elliptic functions is correct.

The flaw in Zhang and Li's argument is identified and explained.

The assertion about related solutions in nonlinear equations is reaffirmed.

Abstract

In a recent paper Zhang and Li have doubted our claim that whenever a nonlinear equation has solutions in terms of the Jacobi elliptic functions $\cn(x,m)$ and $\dn(x,m)$, then the same nonlinear equation will necessarily also have solutions in terms of $\dn(x,m) \pm \sqrt{m} \cn(x,m)$. We point out the flaw in their argument and show why our assertion is indeed valid.