Diffusive Mixing of Stable States in the Ginzburg-Landau Equation

Abstract

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.

28 pages, LaTeX

References in corpus (1)

• Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

Cited by in corpus (4)

• Universal wavenumber selection laws in apical growth
• Nonlinear stability and asymptotic behavior of periodic wave trains in reaction-diffusion systems against $C_{\mathrm{ub}}$-perturbations
• Nonlinear stability of periodic roll solutions in the real Ginzburg-Landau equation against $C_{\mathrm{ub}}^m$-perturbations
• Dynamics on unbounded domains; co-solutions and inheritance of stability