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

Pages: 665 - 682, Volume 4, Issue 3, September 2005

doi:10.3934/cpaa.2005.4.665       Abstract        Full Text (282.4K)       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 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, periodic boundary conditions, bifurcation diagram, secondary bifurcation.
Mathematics Subject Classification:  35B32, 35J60.

Received: August 2004;      Revised: May 2005;      Published: June 2005.