Stability analysis of finite amplitude interfacial waves in a two-layer fluid in the presence of depth uniform current

A fourth-order nonlinear evolution equation of interfacial progressive waves in two-layer fluids of finite depths is derived in the case when there is a depth uniform current in the lower fluid. Based on this equation, stability analysis is then determined of a plane progressive wave. Discourses are provided for both air–water interface and a Boussinesq approximation. Graphs are plotted for maximum growth rate of instability as a function of wave steepness. Two-dimensional instability regions in the perturbed wavenumber plane and three-dimensional contour plots of growth rate of instability are also drawn. Starting from third-order nonlinear Schrödinger equation in one spatial dimension, we have additionally found the effect of depth uniform current on Peregrine breather. The present fourth-order analysis shows significant deviation from the third-order analysis and produces results consistent with the exact numerical results.