Global well-posedness and multi-tone solutions of a class of nonlinear nonlocal cochlear models in hearing

DOI10.1088/0951-7715/17/2/020zbMATH Open1051.35013arXivnlin/0303048OpenAlexW2134268091WikidataQ115415022 ScholiaQ115415022MaRDI QIDQ4810819

Yingyong Qi, Jack Xin

Publication date: 16 August 2004

Published in: Nonlinearity (Search for Journal in Brave)

Abstract: 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

2 f_{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.

Full work available at URL: https://arxiv.org/abs/nlin/0303048

zbMATH Keywords

boundary forcing damped dispersive partial differential equations smooth quasi-periodic solutions

Mathematics Subject Classification ID

Stability in context of PDEs (35B35) Vibrations in dynamical problems in solid mechanics (74H45) Wave equation (35L05) Almost and pseudo-almost periodic solutions to PDEs (35B15) Methods involving semicontinuity and convergence; relaxation (49J45) Biomechanical solid mechanics (74L15) Initial-boundary value problems for first-order hyperbolic systems (35L50) Resonance in context of PDEs (35B34) General biology and biomathematics (92B05)