Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals

Pages: 609 - 629, Volume 19, Issue 4, December 2007

doi:10.3934/dcds.2007.19.609       Abstract        Full Text (268.8K)       Related Articles

Satoshi Kosugi - Department of Applied Mathematics and Informations, Ryukoku University, Seta, Otsu, 520-2194, Japan (email)
Yoshihisa Morita - Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194, Japan (email)
Shoji Yotsutani - Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194, Japan (email)

Abstract: We investigate stationary solutions of the one-dimensional Cahn-Hilliard equation with the diffusion coefficient and the total mass of the density as two given parameters. We solve the equation completely in the whole parameter space by using the Jacobi elliptic functions and complete elliptic integrals. In addition to counting the stationary solutions, which was studied by Grinfeld and Novick-Cohen, we provide an exact expression of the solutions. We also illustrate global bifurcation diagrams together with the asymptotic behavior of the solutions as the diffusion coefficient vanishes.

Keywords:  Cahn Hilliard equation, global bifurcation diagram, Jacobi elliptic function, complete elliptic integral, exact solution.
Mathematics Subject Classification:  34B15, 35C05, 35B32, 33E05.

Received: December 2006;      Revised: June 2007;      Published: September 2007.