English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, we consider the complex Ginzburg Landau equation  $$ \partial_t u = (1 + i \beta ) \Delta u + (1 + i \delta) |u|^{p-1}u -\alpha u \text{ where } \beta, \delta, \alpha \in \mathbb R. $$  The study aims to investigate the finite time blowup phenomenon. Inparticular, for fixed $ \beta\in \mathbb R$, the existence of finite timeblowup solutions for an arbitrary large $|\delta|$ is still unknown.Especially, Popp, Stiller, Kuznetsov, and Kramer formally conjectured in 1998that there is no blowup (collapse) in such a case. In this work, considered asa breakthrough, we give a counter example to this conjecture. We show theexistence of blowup solutions in one dimension with $\delta $ arbitrarily givenand $\beta =0$. The novelty is based on two main contributions: aninvestigation of a new blowup scaling (flat blowup regime) and a suitablemodulation.

Flat blow-up solutions for the complex Ginzburg Landau equation