Dynamics of anharmonic lattices: Solitons and the central-peak problem in one dimension

The properties of a model classical Hamiltonian describing one-dimensional anharmonic lattices is studied by a new approach. The Hamiltonian represents particles sitting on one or the other side of a double-well potential and interacting with each other through harmonic forces. The model can describe order-disorder or displacive phase transitions. We look for solutions of the equations of motion of the form $\Psi (x,t)=\Sigma i^{}{\Psi }_l^i(x,t){\Psi }_{\mathrm{nl}}^i(x,t)$, where ${\Psi }_l^i(x,t)$ is the $i\mathrm{th}$ harmonic solution of the linear problem and obtain differential equations for the functions ${\Psi }_{\mathrm{nl}}^i(x,t)$ with the assumption that ${\Psi }_{\mathrm{nl}}^i$ vary slowly compared with ${\Psi }_l^i$. The solutions for ${\Psi }_{\mathrm{nl}}^i(x,t)$ are shown to be solitons, which are stationary traveling pulses whose properties have been extensively studied recently. The physical interpretation of solitons seems to be in terms of moving domains and dislocations, which transfer particles from one side of the well to the other during their passage. The energy of the solitons is calculated in terms of their amplitude and their velocity. The effect on the dynamic structure factor of the new solutions is considered. Solitons lead to a frequency width of the phonon and also give a quasielastic peak—the "central peak." The height and width of the central peak and of the phonons depend on the density of the solitons thermally excited and their velocity distribution. The spatial and temporal correlations due to the solitons increase exponentially with decreasing temperature and start becoming important about a temperature which is just the mean-field transition temperature of the conventional theory. In an appendix we study another anharmonic problem by similar methods.