A simple inverse problem in solitonic chain deformations

We present a method that permits to obtain the nearest neighbor interaction potential starting from the shape of a spatial deformation of the chain of atoms. The deformations are considered to be traveling pulses. This potential does not necessarily have an analytical expression. Most of this work is focused on the usual formalism for solving Hamilton equations of motion. The concept of dual systems provides a highly efficient method to relate displacements from the equilibrium with generalized momenta and interaction potentials. We introduce generating functions from which sinusoidal waves and solitons are generated; then the corresponding interaction potentials are found. We arrive at the Hamiltonian and the energy of the system. A new kind of solitonic deformations is found for nonlinear chains which corresponds to an interaction potential similar to the Toda exponential potential but with a different dispersion relation to the speed of long-wavelength sinusoidal waves. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)