The third-order asymptotic solutions in the Lagrangian description for interfacial internal waves in a three layer fluid system

In this paper, we discuss the interfacial internal waves with a rigid boundary in a three-layer fluid system, where the density of the upper layer fluid is smaller than that of the lower layer. With the Lagrangian matching conditions at the interfaces, the first-order solutions, the second-order solutions and the third-order asymptotic solutions for the interfacial internal waves are obtained in the Lagrangian description using the perturbation method, and the mass transport velocity, the wave frequency, the mean level and the particle trajectory are also given. The results show that the discontinuities across the interfaces appear for the mass transport velocity, wave frequency and mean level, but we find that these discontinuities may disappear if the water depth ratio and the density ratio of the three layer fluids satisfy certain conditions.