Initial–boundary value problems for Fuss-Winkler-Zimmermann and Swift–Hohenberg nonlinear equations of 4th order

Abstract

This paper presents results of the investigationof bifurcations of stationary solutions of the Swift–Hohenbergequation and dynamic descent to the points of minimal values of thefunctional of energy for this equation, obtained with the use of themodification of the Lyapunov–Schmidt variation method and somemethods from the theory of singularities of smooth functions.Nonstationary case is investigated by the construction of paths ofdescent along the trajectories of the infinite-dimensional SHdynamical system from arbitrary initial states to points ofthe minimum energy.