Global well-posedness and multi-tone solutions of a class of nonlinear nonlocal cochlear models in hearing
We study a class of nonlinear nonlocal cochlear models of the transmission line type, describing the motion of basilar membrane (BM) in the cochlea. They are damped dispersive partial differential equations (PDEs) driven by time dependent boundary forcing due to the input sounds. The global well-posedness in time follows from energy estimates. Uniform bounds of solutions hold in case of bounded nonlinear damping. When the input sounds are multi-frequency tones, and the nonlinearity in the PDEs is cubic, we construct smooth quasi-periodic solutions (multi-tone solutions) in the weakly nonlinear regime, where new frequencies are generated due to nonlinear interaction. When the input is two tones at frequencies $f_1$, $f_2$ ($f_1 < f_2$), and high enough intensities, numerical results illustrate the formation of combination tones at $2 f_1 -f_2$ and $2f_2 -f_1$, in agreement with hearing experiments. We visualize the frequency content of solutions through the FFT power spectral density of displacement at selected spatial locations on BM.