FLAT BLOW-UP SOLUTIONS FOR THE COMPLEX GINZBURG LANDAU EQUATION - Université Paris Dauphine
Journal Articles Archive for Rational Mechanics and Analysis Year : 2024

FLAT BLOW-UP SOLUTIONS FOR THE COMPLEX GINZBURG LANDAU EQUATION

Solution explosive plate pour l'équation complexe de Ginzburg Landau

Giao Ky Duong
Hatem Zaag

Abstract

In this paper, we consider the complex Ginzburg Landau equation $u_t = (1+i \beta) \Delta u +(1+i \delta)|u|^{p-1}u - \alpha u$ where β, δ, α are in $R$. The study aims to investigate the finite time blowup phenomenon. In particular, for fixed β P R, the existence of finite time blowup solutions for an arbitrary large |δ| is still unknown. Especially, Popp et al [24] formally conjectured that there is no blowup (collapse) in such case. In this work, considered as a breakthrough, we give a counter example to this conjecture. We show the existence of blowup solutions in one dimension with δ arbitrarily given and β " 0. The novelty is based on two main contributions: an investigation of a new blowup scaling (flat blowup regime) and a suitable modulation.
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Dates and versions

hal-04197208 , version 1 (05-09-2023)

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Giao Ky Duong, Nejla Nouaili, Hatem Zaag. FLAT BLOW-UP SOLUTIONS FOR THE COMPLEX GINZBURG LANDAU EQUATION. Archive for Rational Mechanics and Analysis, 2024, 248 (6), pp.117. ⟨10.1007/s00205-024-02052-1⟩. ⟨hal-04197208⟩
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