Matrix Methods of Searching for Lax Pairs and a Paper by Estevez

Lax pairs are useful in studying nonlinear partial differential equations, although finding them is often difficult. A standard approach for finding them was developed by Wahlquist and Estabrook [1]. It was designed to apply to for equations with two independent variables and generally produces incomplete Lie algebras (called “prolongation structures”), which can be written as relations among certain matrices and their commutators. Extending the method to three variable problems is more difficult. One still gets matrix equations, but now with a more complicated structure. Exploration of a Lax pair in a paper by Estevez [3] suggested a variation of the method. This paper will discuss how that can be used to obtain her Lax pair.