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December  2007, 19(4): 609-629. doi: 10.3934/dcds.2007.19.609

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

Satoshi Kosugi 1, , Yoshihisa Morita 2, and  Shoji Yotsutani 3,

1. 

Department of Applied Mathematics and Informations, Ryukoku University, Seta, Otsu, 520-2194
2. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194
3. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194

Received  December 2006 Revised  June 2007 Published  September 2007

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: exact solution., global bifurcation diagram, Cahn Hilliard equation, Jacobi elliptic function, complete elliptic integral.
Mathematics Subject Classification: 34B15, 35C05, 35B32, 33E0.
Citation: Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 609-629. doi: 10.3934/dcds.2007.19.609
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