Spectral stability of wave trains in the Kawahara equation
The authors consider the Kawahara equation \[ \partial_t u= \partial^5_x u- \varepsilon\partial^3_x u+ u\partial_x u,\tag{1} \] where \(\varepsilon\) is a real parameter. The goal of the authors is to study the spectral stability of the periodic wave trains at the tail of the generalized solitary waves of (1), with respect to localized bounded perturbations. They prove that periodic travelling waves with speed \(c\) are spectrally stable, that is, the spectrum of the linearization about these waves is contained in the imaginary axis, provided their amplitude \(A\) satisfies \(A= o(|c|^{5/4})\).
scientific article; zbMATH DE number 5126600