Abstract
The extended fifth-order Korteweg-de Vries (KdV) equation ut + 6uux + uxxx + ε(α1u2ux + γ1uuxxx + γ2uxuxx + β1u5x) = 0 models long internal waves in a two-layer stratified fluid with next-order dispersive and nonlinear corrections. Whether these four correction terms preserve integrability under a purely local change of the dependent variable, without a non-local term, had not been settled as an exact statement. We apply the local near-identity transformation u = v + ε(av2 + bvxx) and compute its effect on the four correction coefficients to O(ε). We evaluate the time derivative of the third KdV conserved density under the same transformation, examine the compatibility of the 2 × 2 AKNS Lax scheme with the quadratic term 6uux, and carry out a linear dispersion analysis of the corrected equation. The local transformation removes the coefficients α1 and γ2 completely, for any values, but cannot remove γ1 or β1 for any choice of the transformation parameters. The equation reduces exactly to KdV if and only if γ1 = β1 = 0. The third KdV invariant is conserved along solutions under the same condition. The symmetric AKNS reduction q = u, r = −u is structurally incompatible with the 6uux term, the reduction q = u, r ≡ −1 is the correct one. The correction to the group-phase velocity gap from fifth-order dispersion is obtained in closed form, ΔV(k, ε) − ΔV(k, 0) = 4εβ1k4. The results give an exact, checkable criterion for which correction terms in the fifth-order internal-wave equation are removable by local means and which reflect genuine higher-order physics, relevant to estimating wave-packet asymmetry in stratified oceanic flows.