Three-wave resonant interactions in the diatomic chain with cubic anharmonic potential: theory and simulations

Abstract

We consider a diatomic chain characterized by a cubic anharmonic potential. After diagonalizing the harmonic case, we study in the new canonical variables, the nonlinear interactions between the acoustical and optical branches of the dispersion relation. Using the wave turbulence approach, we formally derive two coupled wave kinetic equations, each describing the evolution of the wave action spectral density associated to each branch. An H-theorem shows that there exist an irreversible transfer of energy that leads to an equilibrium solution characterized by the equipartition of energy in the new variables. While in the monoatomic cubic chain, in the large box limit, the main nonlinear transfer mechanism is based on four-wave resonant interactions, the diatomic one is ruled by a three wave resonant process (two acoustical and one optical wave): thermalization happens on shorter time scale for the diatomic chain with respect to the standard chain. Resonances are possible only if the ratio between the heavy and light masses is less than 3. Numerical simulations of the deterministic equations support our theoretical findings.

Nonlinear topological edge states: From dynamic delocalization to thermalization

“…The above observations are in line with the recent works on nonlinear lattice thermalization [14,17,45,46,52,53,75,76]. In those studies, the broadening of the width of the distribution of frequency shifts is attributed to the presence of the nonlinearity in the system, which introduces a degree of stochasticity (chaoticity) in the modes' interactions [71,73].…”

Nonlinear Topological Edge States: from Dynamic Delocalization to Thermalization

“…The above observations are in line with the recent works on nonlinear lattice thermalization [14,17,45,46,51,52,74,75]. In those studies, the broadening of the width of the distribution of frequency shifts is attributed to the presence of the nonlinearity in the system, which introduces a degree of stochasticity (chaoticity) in the modes' interactions [70,72].…”

“…More specifically, how the Floquet stability of the topological edge mode [32] connects to the mode's resonance theory (e.g., discrete mode resonances [45], Chirikov resonance-overlap [72]) of thermalization of discrete lattice model? As such, expressing the mode resonance conditions of the present system along the directions of [45,46,51,75] is relevant for this problem. Additionally, understanding how the above mechanisms change with the type of initial conditions and/or the nonlinearity of the system is also an interesting line in order to construct a more general framework of thermalization on topological lattice systems.…”

“…The later type of mode resonance is also relevant for the quartic KG system [17,49]. Furthermore, in the case of the diatomic α-FPUT chain, three-wave resonances are responsible for the renormalization of frequencies [75]. As such, it is pertinent to give a closer look toward which resonant processes are involved in the present model of Fig 1, something we plan to tackle in future endeavours.…”

Feynman rules for forced wave turbulence

“…A 3. This includes: elastic plate wave turbulence[8][9][10][11][12][13][14], turbulence in surface gravity waves[15][16][17][18][19], in gravitational waves[20][21][22][23], acoustic turbulence[24], planetary Rossby waves[25], waves on bubbles[26], emergent hydrodynamics[27], in a diatomic chain[28], in plasmas[29,30], optical waves[31][32][33], and rotating waves[34,35], and in FPUT chains[36].…”