Bulletin of the
Korean Mathematical Society BKMS

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 [13]. As compared to [18], 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.

Keywords: Spectral instability, rolls, generalized Swift-Hohenberg equa\-tions

MSC numbers: Primary 35B35

Supported by: The first author was supported by a research grant from Hankyong National University for an academic exchange program in 2022. The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. 2022R1F1A1074414).