Finite-gap solutions to the mKdV equation: classical and alternative formulae

The Baker–Akhiezer function for the real modified Korteweg–de Vries equation is constructed. A classical theta functional formula for the finite-gap solution to the real mKdV equation and other equations from the corresponding hierarchy is obtained. Alternative formulae for finite-gap solutions to the real mKdV equation are found using the reduction theorem for the multidimensional Riemann theta function.