Reduction of the self-dual Yang-Mills equations by Bessel and Legendre functions

Two integrable nonlinear equations are derived by dimensional reductions of the SU(2) self-dual Yang-Mills equations. To derive the equations, we use Hirota's method and some properties of special functions, namely those of the Bessel and Legendre functions. Solutions of these equations are represented in terms of Toeplitz determinants whose elements are a superposition of the special functions.