On the elliptical solutions of models connected to the short pulse equation

Abstract

In the present paper, we consider a special hierarchy of equations comprising the short pulse equation, the sine-Gordon integrable hierarchy and the elastic beam equation. These equations are highly non-linear and rely on transformations to arrive at solutions. Previously, recursion operators and hodograph mappings were successful in reducing these equations. However, we show that via the conservation laws or the one-parameter Lie group, the special hierarchy may be integrated and will admit the exact solutions that feature elliptical functions.