Diffusive Mixing of Stable States in the Ginzburg-Landau Equation

Details Diffusive Mixing of Stable States in the Ginzburg-Landau Equation Diffusive Mixing of Stable States in the Ginzburg-Landau Equation

1998-01-27

arxiv.org/abs/patt-sol/9801003v1

Nonlinear Sciences

Pattern Formation and Solitons

28 pages, LaTeX

Scientific paper

10.1007/s002200050495

For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as $x \to \pm\infty$, to periodic stationary states with different wave-numbers $\eta_\pm$. These solutions are stable with respect to small perturbations, and approach as $t \to +\infty$ a universal diffusive profile depending only on the values of $\eta_\pm$. This extends a previous result of Bricmont and Kupiainen by removing the assumption that $\eta_\pm$ should be close to zero. The existence of the diffusive profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.

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