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Vibration of a chain of particles interacting by nonlinear force is investigated. Using a transformation exact solutions to the equation of motion are aimed at. For a special type of interaction potential of the form aligned (r) =abe^-br+ar+const. , \ (a, b>0) aligned exact solutions are actually obtained in terms of the Jacobian elliptic functions. It is shown that the system has N “normal modes”. Expansion due to vibration or “thermal expansion” of the chain is also discussed.
Morikazu Toda (1967) studied this question.