The 3+1 Dimensional Toda Molecule Equation and Its Multiple Soliton Solutions

Abstract

We consider the 3+1 dimensional Toda equation Δ log V_n−V_n+1+2V_n−V_n−1=0 where Δ≡(∂⁄∂x)²+(∂⁄∂y)²+(∂⁄∂z)². The boundary condition is molecule type which is V_n=0 at finite n corresponding to both ends of the finite chain. We take the form of the solution to be V_n(x, y, z)=V_n(r, θ, φ)=r⁻²V_n(θ, φ) where r, θ and φ are usual spherical coordinates. We have found explicit multiple soliton solutions for V_n(θ, φ), which correspond to the superpositions of arbitrary number of axially symmetric solutions with each symmetry axis directing to arbitrary different directions.