INTEGRABILITY OF 7TH-ORDER KDV-LIKE EVOLUTION-EQUATIONS USING FORMAL SYMMETRIES AND PERTURBATIONS OF CONSERVED-DENSITIES

Seventh-order evolution equations of the form u(t) = d/dx where G is a differential polynomial with certain scaling properties are classified using the integrability test of Mikhailov-Shabat-Sokolov based on the existence of formal symmetries. The integrability of these equations is checked using Kodama's method based on the perturbation of the conserved densities of the KdV equation. According to these tests, the only integrable KdV-like seventh-order equations are the KdV, Sawada-Kotera, and Kaup equations. It is also shown that these equations are characterized by the existence of two conserved densities.