Local Bifurcations in the Cahn–Hilliard and Kuramoto–Sivashinsky Equations and in Their Generalizations

• Published in: Computational mathematics and mathematical physics Vol. 59; no. 4; pp. 630 - 643
• Main Authors: Kulikov, A. N., Kulikov, D. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.04.2019; Springer Nature B.V
• Subjects: Asymptotic methods; Bifurcations; Boundary value problems; Canonical forms; Computational Mathematics and Numerical Analysis; Initial conditions; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear evolution equations; Stability analysis; local bifurcations; nonlinear boundary value problem; asymptotic formulas; normal form; stability
• ISSN: 0965-5425; 1555-6662
• DOI: 10.1134/S0965542519040080

A periodic boundary value problem for a nonlinear evolution equation that takes the form of such well-known equations of mathematical physics as the Cahn–Hilliard, Kuramoto–Sivashinsky, and Kawahara equations for specific values of its coefficients is studied. Three bifurcation problems arising when the stability of the spatially homogeneous equilibrium states changes are studied. The analysis of these problems is based on the method of invariant manifolds, the normal form techniques for dynamic systems with an infinite-dimensional space of initial conditions, and asymptotic methods of analysis. Asymptotic formulas for the bifurcation solutions are found, and stability of these solutions is analyzed. For the Kuramoto–Sivashinsky and Kawahara equations, it is proved that a two-dimensional local attractor exists such that all solutions on it are unstable in Lyapunov’s sense.