Time-independent harmonics dispersion relation for time-evolving nonlinear waves

"We present a theory for the dispersion of generated harmonics in a traveling nonlinear wave. The harmonics dispersion relation (HDR), derived by the theory, provides direct and exact prediction of the collective harmonics spectrum in the frequency–wave number domain and does so without prior knowledge of the u = u(x, t) solution. It is valid throughout the evolution of a distorting unbalanced wave or the steady-steady propagation of a balanced wave with waveform invariance. The new relation is shown to be a special case of the general nonlinear dispersion relation (NDR), which is also derived. The theory is examined on a diverse range of cases of one-dimensional elastic waves and shown to hold irrespective of the spatial form of the initial wave profile, type and strength of the nonlinearity, and the level of dispersion in the linear limit. Another direct outcome of the general NDR is an analytical condition for soliton synthesis."

R. Khajehtourian, M. I. Hussein

https://www.science.org/toc/sciadv/7/50