Asymptotic solutions for localized vibrational modes in strongly anharmonic periodic systems

Sievers and Takeno (ST) have recently argued that a periodic system of particles interacting through harmonic and quartic anharmonic potentials can exhibit odd-parity localized vibrational modes for sufficiently strong anharmonicity. In the present paper, we demonstrate that this behavior is a fundamental property of the underlying pure anharmonic system. For a monatomic one-dimensional periodic chain of particles interacting via nearest-neighbor purely anharmonic potentials of any even order, it is shown that within the rotating-wave approximation of ST, the odd-parity vibrational mode pattern of a simple linear triatomic molecule, and the even-parity mode pattern of a simple diatomic molecule, yield exact solutions of the classical equations of motion in the asymptotic limit of increasing order of anharmonicity. These localized vibrational modes may be centered on any lattice site. The odd-parity solution remains a very good approximation even for the lowest-order case of pure quartic anharmonicity, whereas the new even-parity solution requires a relatively minor correction for this case. Our results are obtained by directly studying the equations of motion. For a system with just harmonic plus sufficiently strong quartic anharmonic interactions, our odd-parity solution corresponds to that found by ST using lattice Green's-function techniques.