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SPECTRAL INSTABILITY OF ROLLS IN THE 2-DIMENSIONAL GENERALIZED SWIFT-HOHENBERG EQUATION

정소연|채명주
대한수학회 (The Korean Mathematical Society)|2023.09|대한수학회보 (Bulletin of the KMS)|제60권 제5호|pp. 1335-1364|DOI 10.4134/BKMS.b220695

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키워드

Spectral instabilityrollsgeneralized Swift-Hohenberg equa\-tions
주제분류수학

초록

The aim of this paper is to investigate the spectral instability of roll waves bifurcating from an equilibrium in the $2$-dimensional generalized Swift-Hohenberg equation. We characterize unstable Bloch wave vectors to prove that the rolls are spectrally unstable in the whole parameter region where the rolls exist, while they are Eckhaus stable in $1$ dimension \cite{J}. As compared to \cite{M2}, showing that the stability of the rolls in the $2$-dimensional Swift-Hohenberg equation without a quadratic nonlinearity is determined by Eckhaus and zigzag curves, our result says that the quadratic nonlinearity of the equation is the cause of such instability of the rolls.

Abstract

The aim of this paper is to investigate the spectral instability of roll waves bifurcating from an equilibrium in the $2$-dimensional generalized Swift-Hohenberg equation. We characterize unstable Bloch wave vectors to prove that the rolls are spectrally unstable in the whole parameter region where the rolls exist, while they are Eckhaus stable in $1$ dimension \cite{J}. As compared to \cite{M2}, showing that the stability of the rolls in the $2$-dimensional Swift-Hohenberg equation without a quadratic nonlinearity is determined by Eckhaus and zigzag curves, our result says that the quadratic nonlinearity of the equation is the cause of such instability of the rolls.

참고문헌 (25)

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