Project

Theoretical and Numerical Investigation of Nonlinear Mathematical Models

The real processes in the nature and the society are essentially nonlinear. Due to the lack of general theory of the nonlinear mathematical models, their investigation is still a serious challenge to the researchers. The goal of the project is theoretical and numerical study of the mathematical models of important for technology and society nonlinear processes.

The research is organized into seven work packages:
WP1. Investigation of the dynamics of solitary waves as solutions of nonlinear wave and amplitude equations.
WP2. Investigation of nonlinear dispersive equations that model the distribution of surface waves with/without surface tension and the dynamics of nonlinear lattices.
WP3. Investigation of the dynamics of drops in viscous flows.
WP4. Investigation of the dynamics of the processes in multilayer superconductors of Josephson type.
WP5. Theoretical and numerical analysis of financial and ecological mathematical models.
WP6. Theoretical and numerical study of non-Newtonian flows in heterogeneous porous media using fractional calculus.
WP7. Discrete methods preserving basic properties of solutions of Hamiltonian systems of differential equations.

The methods of investigation will be tailored to the peculiarities of the nonlinear mathematical models – nonuniqueness of their solutions, occurrence of singularities from smooth initial data, localization of the solutions in space and time (self-dispersing and blow up in finite time). Important characteristics of the real processes will be obtained through a rigorous study of the mathematical models and further recommendations for their optimization will be made. At the same time advanced efficient and reliable numerical methods and algorithms for solving the discrete problems will be developed and analyzed by using multiprocessor clusters and grid computing.
The research is in harmony with the National strategies 2014-2020.

Keywords: nonlinear nonintegrable equations, localized solutions, dispersive equations, equations of sine-Gordon type, solitons, finite time blow up, multiphase viscous flow, Van der Waals forces, superconductors of Josephson type, coherent fluksons, non-Newtonian flows, viscoelastic flows, Darcy's law, Riemann-Liouville fractional derivative, finite element method, finite difference methods, parallel algorithms, multiprocessor clusters, grid computing.