Нелокальные солитоны в нелинейной цепочке атомов

A nonlinear 1D chain with nonlocal interaction is considered. Using the multiscale expansion method, a nonlocal equation is obtained that describes the propagation of envelope waves in a medium. The properties of the obtained equation were studied and exact soliton-like solutions were constructed by the Darboux method of transformations.

Equivalent Robin boundary conditions for acoustic and elastic media

We present equivalent conditions and asymptotic models for a diffraction problem of acoustic and elastic waves. The mathematical problem is set with a Robin boundary condition. Elastic and acoustic waves propagate in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. This approach leads to solve only elastic equations. We derive and validate equivalent conditions up to the third order for the elastic displacement. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.

Multiscale homogenization for fluid and drug transport in vascularized malignant tissues

A system of differential equations for coupled fluid and drug transport in vascularized (malignant) tissues is derived by a multiscale expansion. We start from mass and momentum balance equations, stated in the physical domain, geometrically characterized by the intercapillary distance (the microscale). The Kedem–Katchalsky equations are used to account for blood and drug exchange across the capillary walls. The multiscale technique (homogenization) is used to formulate continuum equations describing the coupling of fluid and drug transport on the tumor length scale (the macroscale), under the assumption of local periodicity; macroscale variations of the microstructure account for spatial heterogeneities of the angiogenic capillary network. A double porous medium model for the fluid dynamics in the tumor is obtained, where the drug dynamics is represented by a double advection–diffusion–reaction model. The homogenized equations are straightforward to approximate, as the role of the vascular geometry is recovered at an average level by solving standard cell differential problems. Fluid and drug fluxes now read as effective mass sources in the macroscale model, which upscale the interplay between blood and drug dynamics on the tissue scale. We aim to provide a theoretical setting for a better understanding of the design of effective anti-cancer therapies.

Thin Layer Models for Electromagnetism

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer. These models appear as first order approximations of the electromagnetic field. They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer, that makes possible to replace the thin layer by approximate conditions. We present the advantages and the drawbacks of several approximations together with numerical validations and simulations. The main motivation of this work concerns the computation of electromagnetic field in biological cells. The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane, which provides a specific behavior of the electromagnetic field at low frequencies.