Damping of periodic waves in the classically important nonlinear wave systems-nonlinear Schrödinger, Korteweg-deVries (KdV), and modified KdV-is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m = m(t). The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m = 0 to soliton waves when m = 1.
Fracture healing is a complex process involving numerous cell types, whose actions are regulated by many factors in their local environment. Mechanical factors are known to exert a strong influence on the actions of these cells and the progression of the repair process. While prior studies have investigated the effect of physical forces on cell differentiation, biofactor expression, and mechanical competence of repair, the mechanosensory and response mechanisms are poorly understood. This study was designed to explore the influence of a controlled mechanical environment on temporal aspects of the bone repair process. Specifically, this study examines how the timing of an applied strain influences local cell behavior during fracture repair, and how this load affects the migration of systemically introduced mesenchymal stem cells (MSCs) to the fracture site.
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