Solutions of nonlinear Schrödinger equation for interfacial waves propagating between two ideal fluids

Abstract

We present solutions to the problem of waves propagating at an interface between two inviscid fluids of infinite extent and differing densities. The method of multiple scale is employed to obtain a dispersion relation and nonlinear Schrödinger (NLS) equation, which describes the behavior of the system for the fluid interface. The dispersion relation of the model NLS equation is studied. The solutions of the NLS equation are derived analytically by using the complex tanh-function method and the function transformation method into a sine-Gordon equation. Also, diagrams are drawn to illustrate the elevation of the interface, the slip velocity, and the conservation of power. We observe that the elevation of the interface is in the form of traveling quasi-solitary waves that decrease as the wave number increases. We see that the slip velocities also bring a nonlinear and periodic characters. Finally, we observe that the conservation of power is in the form of traveling waves. Also, as the wave number increases, the conservation of power is more accurate in fluctuating around zero.