| Dispersion and nonlinearity are two important characteristics of the interfacial waves in two-layer fluid, and the research on them not only helps to uncover the mechanisms of their formation, evolvement and attenuation, but also can guide the ocean and coastal engineering in reality.By introducing the two small parameters, nonlinearity and dispersion, we consider the interfacial waves propagating along the interface between a two-fluid system, and the fluid in each layer is homogeneous, irrotational, inviscid and incompressible. First, we derive the higher-order Boussinesq equations for the interfacial waves in terms of the velocities at the still water level, the velocities at arbitrary locations and the depth-averaged velocities, respectively. The derived equations not only include the results before for the interfacial waves in a two-fluid system, but also can be reduced to the many cases for the surface waves. Then, we analyze the characteristics of dispersion, nonlinearity and shoaling coefficients of the equations in terms of the velocities at arbitrary locations, the results demonstrate that they can fit well with the dispersion, nonlinearity and shoaling coefficients of the original equations when we choose the suitable nondimensional parameters and keep up to lower order. Subsequently, we improve the equations in terms of the depth-averaged velocities and the velocities at arbitrary locations, the dispersion relations of the improved equations are much better. Finally we consider the higher-order Boussinesq equations with background flows including uniform basic flows, uniform shear flows and Poiseuille flows, and present the second-order asymptotic solutions of the velocities and the second-order Stokes solutions of the associated elevations of the interfacial waves in the cases of uniform basic flows and uniform shear flows as well, which reveal the nonlinear interactions between waves and flows and among the interfacial waves.
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