Spectral transverse instability of solitary waves in Korteweg fluids

The motion of Korteweg fluids is governed by the Euler–Korteweg model, which admits planar solitary waves for nonmonotone pressure laws such as the van der Waals law below critical temperature. In an earlier work with Danchin, Descombes and Jamet, it was shown by variational arguments and numerical computations that some of these solitary waves are stable in one space dimension. The purpose here is to study their stability with respect to transverse perturbations in several space dimensions. By Evans functions techniques and Rouchéʹs theorem, it is shown that transverse perturbations of large wave length always destabilize solitary waves in the Euler–Korteweg model, whereas energy estimates show that perturbations of short wave length tend to stabilize them.