Canonical reduction of self-dual Yang–Mills equations to Fitzhugh–Nagumo equation and exact solutions

The (constrained) canonical reduction of four-dimensional self-dual Yang–Mills theory to two-dimensional Fitzhugh–Nagumo and the real Newell–Whitehead equations are considered. On the other hand, other methods and transformations are developed to obtain exact solutions for the original two-dimensional Fitzhugh–Nagumo and Newell–Whitehead equations. The corresponding gauge potential Aμ and the gauge field strengths Fμν are also obtained. New explicit and exact traveling wave and solitary solutions (for Fitzhugh–Nagumo and Newell–Whitehead equations) are obtained by using an improved sine-cosine method and the Wu’s elimination method with the aid of Mathematica.