From the Self‐Dual Yang–Mills Equation to the Fokas–Lenells Equation

A reduction from the Miura transformation of the self‐dual Yang–Mills (SDYM) equations to the unreduced Fokas–Lenells (FL) system is described in this paper. It has been known that the SDYM equation can be formulated from the Cauchy matrix schemes of the matrix Kadomtsev–Petviashvili (KP) hierarchy and of the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. We show that the reduction can be realized in these two Cauchy matrix schemes, respectively. Each scheme allows us to construct solutions for the unreduced FL system. We prove that these solutions obtained from different schemes are equivalent under a certain reflection transformation of coordinates. Using conjugate reduction, we obtain solutions of the FL equation. The paper adds an important example to Ward's conjecture on the reductions of the SDYM equation. It also reveals the Cauchy matrix structures of the Kaup–Newell hierarchy.