The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

The evolution equation for the second-order internal solitary waves in stratified fluids, that are incomepressible inviscid ones of mixed stratification in which the density varies with depth in lower layer but keeps uniform in upper layer with a finite depth, is derived directly from the Euler equation by using the balance between the nonlinearity and the dispersion with the scaling μ=O(ɛ), and by introducing the Gardner-Morikawa transformation, asymptotic expansions and the matching of the solutions for the upper and lower layers of fluid via the computer algebraic operation. The evolution equation and its solution derived for the first-order wave amplitude are consistent with the classical ones, and the desired equation governing the second-order amplitude is reduced as follows ∂f2∂τ-c1∂f2∂ξ+α∂(f1f2)∂ξ+β∂∂ξ∫-∞∞f2(ξ')g(ξ-ξ')dξ'=G2(f1,c1,κ1;c0,ϕ) Where g(ξ)=-12π∫-∞∞kcoth(kl)ejkξdk and G2(f1,c1,κ1;c0,ϕ) is the inhomogeneous term, f1 first-order wave amplitude.