Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals
Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals
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DOI：
10.3934/dcds.2007.19.609
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发表时间：
2007-09
影响因子：
1.1
通讯作者：
S. Kosugi;Y. Morita;Shoji Yotsutani
中科院分区：
文献类型：
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作者：
S. Kosugi;Y. Morita;Shoji Yotsutani
研究了以扩散系数和密度总质量为参数的一维Cahn-Hilliard方程的定态解.利用Jacobi椭圆函数和完全椭圆积分，在整个参数空间内完全求解方程。除了计算Grinfeld和Novick-Cohen研究的定态解外，我们还提供了解的精确表达式。我们还说明了整体分歧图连同渐近行为的解决方案的扩散系数为零。
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.