The resonance interaction of vibrational modes in one-dimensional nonlinear lattices

Resonances of vibrational modes were for the first time revealed for the example of the one-dimensional random Morse lattice. The observation of resonances was possible because of lattice deformation, when, at certain relative deformation values, vibrational modes satisfied the conditions of double (m (i) omega (i) + m (j) omega (j) = 0) or triple (m (i) omega (i) + m (j) omega (j) + m (k) omega (k) = 0) resonances. Of all the resonances observed, the resonance with the frequency ratio omega(2): omega(1) = 2: 1 was studied in detail. The dependences of mode lifetimes and the degree of energy exchange between them on such parameters as resonance frequency detuning, excitation energy level, etc. were determined. A model of two nonlinearly coupled harmonic oscillators was considered in detail on the assumption of a one-to-one correspondence between oscillators and vibrational modes. A consideration of the model problem of oscillators revealed analytic dependences of the dynamic behavior of vibrational modes on control parameters. Excellent agreement between the numerical results for the Morse lattice and analytic conclusions was obtained. It was shown that, for the Fermi-Pasta-Ulam lattice, the resonance interaction of vibrational modes was controlled by the same rules as with the Morse lattice.