Title: Existence and spectral stability analysis of a family of standing periodic wave solutions for a modified Eckhaus equation with an additional term
Abstract: We study the existence and spectral stability properties of a family of periodic standing wave solutions to a modified version of Eckhaus equation that incorporates an additional term. The family consists of small amplitude waves with finite fundamental period which emerge from a Hopf bifurcation around a critical value of a specific parameter. It is shown that the Floquet (continuous) spectrum of the linearization around the periodic standing waves intersects the unstable half plane of complex values with positive real part. To conclude this, we decompose the linearized operator as the sum of a constant coefficient operator, followed by a first order perturbation and then a second order perturbation. We prove that the small-amplitude stationary periodic wave solutions are spectrally unstable by applying techniques from perturbation theory of linear operators.