Solutions for the Mikhailov-Shabat-Yamilov difference-differential equations and generalized solutions for the Volterra and the Toda lattice equations

We present two types of mixed 1-rational N-soliton solutions and two types of special solutions for four types of Volterra-related difference-differential equations arising in Mikhailov, Shabat and Yamilov's lists. We also find new expressions of mixed 1-rational N-soliton solutions for the Volterra and the Toda lattice equations based on the invariance of Gibbon and Tabor's equation (J. Math. Phys. 26 (1985), 1956) under the fractional linear transfermation. By taking appropriate limits of wave numbers, we find some new rational solutions for the Volterra and the Toda lattice equations. We also present elliptic function solutions for the Volterra and the Toda lattice equations different from known ones based on the same formulation.