Traveling waves near shear flows for the inhomogeneous Euler equations with non-constant density

We investigate the existence and nonexistence of traveling wave solutions near monotonic shear flows with non-constant background density for the two-dimensional inhomogeneous Euler equations in a finite channel. For any small $\tau>0$, first, we construct nontrivial traveling waves with velocity and density in $H^{5/2-\tau}$ and $H^{3/2-\tau}$, respectively, showing that inviscid damping fails at these regularities. Second, when the distorted Rayleigh operator has no eigenvalues, we prove that such traveling wave solutions cannot exist in higher regularity spaces ($H^{5/2+\tau}$ for velocity and $H^{3/2+\tau}$ for density).