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September  2005, 4(3): 665-682. doi: 10.3934/cpaa.2005.4.665

A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions

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

1. 

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

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

Received  August 2004 Revised  May 2005 Published  June 2005

We consider the Ginzburg-Landau equation with a positive parameter, say lambda, and solve all equilibrium solutions with periodic boundary conditions. In particular we reveal a complete bifurcation diagram of the equilibrium solutions as lambda increases. Although it is known that this equation allows bifurcations from not only a trivial solution but also secondary bifurcations as lambda varies, the global structure of the secondary branches was open. We first classify all the equilibrium solutions by considering some configuration of the solutions. Then we formulate the problem to find a solution which bifurcates from a nontrivial solution and drive a reduced equation for the solution in terms of complete elliptic integrals involving useful parametrizations. Using some relations between the integrals, we investigate the reduced equation. In the sequel we obtain a global branch of the bifurcating solution.
Keywords: Ginzburg-Landau equation, secondary bifurcation., periodic boundary conditions, bifurcation diagram.
Mathematics Subject Classification: 35B32, 35J6.
Citation: Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
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