Toda Molecule Equations

Abstract

This chapter focuses on the Toda molecule equations. It is known that the hierarchy of the Kadomtsev-Petviashvili (KP) equation reduces by “reductions” to a variety of soliton equations, such as the KdV equation, the modified KdV or Gardner equation, the Boussinesq equation, the coupled KdV equation, the classical Boussinesq—Kaups higher order water wave—equation, the nonlinear Schrödinger equation exhibiting dark-soliton solution, and the Drinfel'd-Sokolov-Wilson equation. In 1967, Toda introduced a nonlinear equation describing a one-dimensional nonlinear lattice, for which he found an exact two-soliton solution. Soliton phenomena have been observed using nonlinear electrical networks simulating the Toda lattice. The Jacobi formula for the Casorati determinant is discussed, and a discrete analogue of the two-dimensional Toda molecule equation is reviewed. The chapter also transforms the two-dimensional and one-dimensional Toda molecule equations into the bilinear forms, and describes their exact solutions.