Calculation of Families of Solitary Waves on Discrete Lattices

While the majority of investigations of solitary waves (solitons) deal with continuous solutions, it is natural to seek discrete solutions which correspond to pure, unperturbed lattices of atoms. In rare cases such as the Toda Lattice, analytic solutions exist for discrete systems. To study more complex systems for which analytic solutions are not currently available, numerical attacks have proven useful. In this paper we present a numerical method based on spectral methods which couples quadratically convergent Newton-Raphson iteration with continuation methods to generate a whole family of lattice solitons solutions as a function of velocity.