An inverse solitary wave problem related to microstructured materials

Abstract

An inverse problem for determining coefficients of a one-dimensional wave equation of nonlinear microstructured material is considered. The solution of the problem is based on measurements gathered from two independent solitary waves. Uniqueness for the inverse problem is proved and a stability estimate is derived.

Internal Variables and Scale Separation in Dynamics of Microstructured Solids

“…In linear case, the dependence of phase velocity on microstructure gives a solid ground for solving such an inverse problem [33]. In nonlinear case, the distortion of solitary waves due to microstructure can be used [34].…”

Nonlinear wave motion and complexity

“…Besides general analysis of complexity of waves in solids, the applications are important. For example, the knowledge of the influence of the microstructure on phase velocities or asymmetry of solitary pulses opens new ways for solving the inverse problems (Janno and Engelbrecht, 2005b, 2005c. Certainly there many challenges for further studies.…”

On the propagation of solitary waves in Mindlin-type microstructured solids

“…For numerical simulation of wave propagation in nonlinear dispersive media with the microstructure a model derived by Engelbrecht and Pastrone [2,11,12] is employed in the present paper. The model is based on Mindlin's and Eringen's earlier works [4,17].…”

“…Here A, B,C, D are material parameters responsible for the linear part of the model and N, M are responsible for the nonlinearity in macro-and microscale, respectively [11,12]. Making use of the free energy function (2) and Euler-Lagrange equations, we obtain the equations of motion…”