Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity

In this paper we investigate spectral stability of traveling wave solutions to 1-\(D\) quantum hydrodynamics system with nonlinear viscosity in the \((\rho,u)\), that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum stability.