Improved Bell-polynomial procedure for the higher-order Korteweg–de Vries equations in fluid dynamics

Korteweg–de Vries (KdV)-typed equations are seen in fluid dynamics, plasma physics and other fields. By means of the Bell-polynomial procedure, we take two higher-order members of the KdV hierarchy, the seventh- and ninth-order Lax’s KdV equations in fluid dynamics, as the examples for studying the integrable properties of the higher-order equations. Different from lower-order equations, two new partial differential operators in the Bell-polynomial procedure are introduced to construct the “multi-dimensional” bilinear forms of such equations with several auxiliary independent variables. Through the procedure simplified via the algebraic operation of the polynomials, the Bäcklund transformations, Lax pairs and infinite conservation laws of such equations are deduced.