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Construction of Blowup Solutions for the Complex Ginzburg-Landau Equation with Critical Parameters
About this Title
Giao Ky Duong, Nejla Nouaili and Hatem Zaag
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 285, Number 1411
ISBNs: 978-1-4704-6121-8 (print); 978-1-4704-7481-2 (online)
DOI: https://doi.org/10.1090/memo/1411
Published electronically: April 25, 2023
Keywords: Blow-up profile,
Complex Ginzburg-Landau equation
Table of Contents
Chapters
- 1. Introduction
- 2. Formal approach
- 3. Formulation of the problem
- 4. Existence
- 5. Single point blow-up and final profile
- A. Spectral properties of $\mathcal {L}_{\beta }$
- B. Details of expansions of the potential terms: $V_1$ and $V_2$
- C. Details of some expansions of $B(q)$
- D. Details of expansions of $R^*(y,s, \theta ’(s))$
- E. Formal derivation of constants $b$ and $\mu$
- F. Cancellation of some coefficient in the ODE of $\tilde q_2$
Abstract
We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its profile. In the first part, we formally construct a blow-up solution. In the second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.- Igor S. Aranson and Lorenz Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys. 74 (2002), no. 1, 99–143. MR 1895097, DOI 10.1103/RevModPhys.74.99
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