Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

We study the spectral (in)stability of one-dimensional solitary and cnoidal waves of various Boussinesq systems. These systems model three-dimensional water waves (i.e., the surface is two-dimensional) with or without surface tension. We present the results of numerous computations examining the spectra related to the linear stability problem for both stationary solitary and cnoidal waves with various amplitudes, as well as multipulse solutions. The one-dimensional nature of the wave forms allows us to separate the dependence of the perturbations on the spatial variables by transverse wave number. The compilation of these results gives a full view of the two-dimensional stability problem of these one-dimensional solutions. We demonstrate that line solitary waves with elevated profiles are spectrally stable with respect to one-dimensional perturbations and long transverse perturbations. We show that depression solitary waves are spectrally stable with respect to one-dimensional perturbations, but unstable with respect to transverse perturbations. We also discuss the instability of multipulse solitary waves and cnoidal-wave solutions of the Boussinesq system.