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 income-pressible 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 mu=O(epsilon), 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 partial derivative f(2)/partial derivative tau - c(1) partial derivative f(2)/partial derivative xi + alpha partial derivative(f(1)f(2))/partial derivative xi +beta partial derivative/partial derivative xi integral(infinity)(-infinity) f(2)(xi')g(xi-xi')d xi' = (f(1),c(1),k(1),; c(0),phi) where g(xi) = 1/2 pi integral(infinity)(-infinity) k coth(kl)e(jk xi)dk and G(2) (f(1),c(1),k(1),C-0,phi) is the inhomogeneous term, f(1) is the first-order wave amplitude.