Waves In One-dimensional Crystals Research Articles

Resonant ensembles of stationary quasi-harmonic waves in one-dimensional crystals

Using a simple mathematical model, built on geometric representations of central and noncentral interactions between material particles in a one-dimensional anharmonic chain, nonlinear resonant interactions between quasi-harmonic waves are investigated in the so-called harmonic approximation. The investigation is carried out with standard asymptotic nonlinear dynamics methods. In the first-order approximation, resonant wave triads are established that are formed at a characteristic quadratic nonlinearity of the system provided that the phase-matching conditions are satisfied. It is demonstrated that the resonant triads can be of only three different types and each resonant triad can consist of only one longitudinal and two transverse oscillation modes. In the general case, a nontrivial interaction between different resonant triplets of three different types and spectral scales is implemented in a chain. Cascade processes of the energy exchange between oscillation modes are characterized by both the complicated dynamics typical of Hamiltonian nonintegrable dynamic systems and the presence of Lyapunov-stable multiwave steady-state motions. In ideal crystalline structures, such steady-state coherent wave ensembles can significantly affect the specific heat and other phenomenological parameters of a system, especially at low temperatures. Therefore, the theoretical and experimental study of these ensembles is of great importance.

Anharmonic interactions of elastic and orientational waves in one-dimensional crystals

Planar oscillations of a chain of dumbbell-shaped particles possessing three degrees of freedom are studied. This system models the dynamics of quasi-one-dimensional crystals consisting of elongated anisotropic molecules. A system of nonlinear differential equations describing the anharmonic interaction of the elastic and orientational waves in the lattice, corresponding to different degrees of freedom of the particles, is constructed assuming a cubic interparticle interaction potential. It is shown that in the low-frequency approximation the system obtained is equivalent to the equations of the moment theory of elasticity, widely employed for describing nonlinear and dispersion properties of layered crystals and phase transformations in alloys. Some types of three-wave collinear interactions are investigated, suggesting the possibility of exciting orientational waves in organic crystals because of their nonlinear interaction with acoustic waves.