Study of weakly nonlinear differential operators on graphs with small edges
GOAL
Project Goal
To consider weakly nonlinear elliptic equations on graphs with small edges and to study in as much detail as possible the behavior of solutions as the lengths of these edges decrease.
ABOUT THE PROJECT
Relevance
This project is aimed at studying topical problems for differential equations on graphs. In particular, such problems are motivated by the theory of quantum and optical waveguides. The project will study weakly nonlinear elliptic differential equations on graphs with small edges. The aim of the research is to describe the behavior of solutions as the lengths of the small edges decrease and, where possible, to characterize the dependence of the solutions on these lengths.
The study of elliptic operators on stratified sets will be extended to weakly nonlinear equations, where the nonlinearity is contained in the free term. Nonlinearities of this kind reflect weakly nonlinear interactions that often arise in the theory of optical waveguides. The nonlinearity itself is assumed to be sufficiently weak to allow the application of standard solvability and uniqueness results from the theory of monotone operators. The main questions concern the convergence of the solution of the perturbed problem and its dependence on the small parameter characterizing the small edges. The scientific significance and novelty of this project lie primarily in the fact that it is the first work to consider nonlinear equations on graphs with small edges.
The key question is whether the main effects identified in the linear case are preserved or whether the nonlinearity can destroy them.
PLANS
Expected Results
- The basic results of the theory of monotone operators will be transferred and adapted to equations on graphs. Model examples of graphs with small edges and weakly nonlinear operators on them will be identified and rigorously formulated, for which the solutions of the corresponding equations can be constructed most explicitly. Solutions of model problems for weakly nonlinear equations will be studied and their dependence on the edge lengths determined.
- General graphs with small edges will be described and problems for weakly nonlinear equations on such graphs will be formulated. The solvability of such equations will be studied for the case when the nonlinearity is present only on the finite edges. The dependence of the solution of the problem for a weakly nonlinear equation on a general graph with small edges on the length of the small edges will be studied for the case when the nonlinearity is present only on the finite edges.
- The solvability of the problem for a weakly nonlinear equation on a general graph with small edges will be studied for the case when the nonlinearity is present on the small edges. The dependence of the solution on the length of the small edges will be studied for the case when the nonlinearity is present only on the small edges.
- 2 articles will be published in journals ranked in the top three quartiles by impact factor in Web of Science or having a CiteScore percentile of at least 50 in Scopus, as well as 1 article in a journal recommended by the CQASHE.
TEAM
Research Team
Konyrkulzhaeva M.N.
IITU JSC
Scopus ID: 57199508869
ORCID: 0000-0002-7174-4774
Kanguzhin B.E.
Al-Farabi Kazakh National University, Professor, Scientific Consultant
Scopus ID: 35776348600
ORCID: 0000-0001-5504-6362
Borisov D.I.
Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of RAS (Ufa); M. Akmullah Bashkir State Pedagogical University (Ufa), Scientific Consultant
Scopus ID: 6603696942
ORCID: 0000-0001-9631-1791
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