A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions
Satoshi Kosugi - Department of Applied Mathematics and Informations, 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.
Received: August 2004; Revised: May 2005; Published: June 2005.
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