Mathematical analysis of multi layers optic fiber models

Abstract

This work is devoted to the development of qualitative methods for the study of nonlinear heterogeneous structures, models of which are elliptic equations, which describe complex nonlinear processes in heterogeneous media. They may also include the structures, consisting of several parts (phases or layers): multiphase solid and liquid materials; optic
fiber and optic cable layers, anisotropic medium, etc. Relevance of the chosen direction is due to the fact that many processes in heterogeneous environments under conditions of high temperatures, heavy loads and significant deformations are described using nonlinear differential equations with discontinuous (singular) data (coefficients, right-hand side,
boundary and initial conditions, etc.). At the same time, the concept of weak solutions that meet the modern needs of mathematical physics arose. Nonlinear differential equations have a complex structure, which actually makes them impossible to study by finding solutions in an explicit form. Therefore, the development of qualitative methods for their investigations
becomes an extremely important tool. This paper considers mathematical models of multilayer optic fiber and cable, which consist of 3 and 5 different materials respectively with different properties. Using potential theory, the behavior of a weak solution of this equation at a fixed point is estimated and analyzed by the value of the nonlinear Wolff potential from the right hand side. We study pointwise properties that play a key role in the further
study: expansion of positivity Harnack’s inequalities, regularities and others. The paper discusses also the application of the obtained theoretical results for the problem of modeling and analyzing of optic fiber and optic cable modern technologies.