Periodic Stationary Patterns Governed by a Convective Cahn-Hilliard Equation

We investigate bifurcations of stationary periodic solutions of a convective Cahn-Hilliard equation,$u_t + Duu_x + (u - u^3 + u_{xx})_{xx} = 0$, describing phase separation in driven systems, and study the stability of the main family of these solutions. For the driving parameter$D < D_0 = \sqrt{...

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Bibliographic Details
Published inSIAM journal on applied mathematics Vol. 66; no. 2; pp. 700 - 720
Main Authors Zaks, Michael A., Podolny, Alla, Nepomnyashchy, Alexander A., Golovin, Alexander A.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2005
Subjects
Applied mathematics
Contour lines
Dynamical systems
Eigenvalues
Equilibrium
Investigations
Mathematics
Nooses
Periodic orbits
Physics
Respect
Symmetry
Wavelengths
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Summary:We investigate bifurcations of stationary periodic solutions of a convective Cahn-Hilliard equation,$u_t + Duu_x + (u - u^3 + u_{xx})_{xx} = 0$, describing phase separation in driven systems, and study the stability of the main family of these solutions. For the driving parameter$D < D_0 = \sqrt{2}/3$, the periodic stationary solutions are unstable. For$D > D_0$, the periodic stationary solutions are stable if their wavelength belongs to a certain stability interval. It is therefore shown that in a driven phase-separating system that undergoes spinodal decomposition the coarsening can be stopped by the driving force, and formation of stable periodic structures is possible. The modes that destroy the stability at the boundaries of the stability interval are also found.
ISSN:0036-1399
1095-712X
DOI:10.1137/040615766