1998/01/27 by Thierry Gallay, Alexander Mielke · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Thermodynamics and Statistical Mechanics #Exponential growth #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Mixing (physics) #Monotone polygon #Physics #Quantum mechanics #Scaling #Spectral Theory in Mathematical Physics #Statistical physics #Term (time) #Zero (linguistics) #nlin.PS #patt-sol
paper · pdf · doi:10.1007/s002200050495
28 pages, LaTeX
arxiv created 1998/01/27 · openalex publication_date 1998/12/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as x → ±∞, to periodic stationary states with different wave-numbers η_±. These solutions are stable with respect to small perturbations, and approach as t → +∞ a universal diffusive profile depending only on the values of η_±. This extends a previous result of Bricmont and Kupiainen by removing the assumption that η_± 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.