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.