Linear stability analysis of travelling waves for a pseudo-parabolic Burgers’ equation

We investigate the linear stability of non-monotone travelling wave solutions of the pseudo-parabolic Burgers’ type equation ∂u ∂t = ∂u ∂x2 + 2u ∂u ∂x + τ ∂u ∂x2∂t with τ > 0 . The monotonicity of the waves depending on the strength of the parameter τ and the far-filed values. The most part of the paper is devoted to prove that the linear stability is determined by the spectrum of the linearised operator. This step is necessary since the linearised operator does not fall into the class of sectorial operators, for which the result is well-known. An Evans function is defined and used to search for instabilities numerically. The numerical results yield the conclusion that no eigenvalues with positive real part appear and hence that stability of the waves holds.