Bell polynomials approach for two higher-order KdV-type equations in fluids

The present paper investigates the higher-order Sawada-Kotera-type equation and the higher-order Lax-type equation in fluids. The Bell polynomials approach is employed to directly bilinearize the two equations. For the Lax-type equation, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair and infinitely many conservation laws are obtained by means of binary Bell polynomials. Moreover, based on its bilinear form, N-soliton solutions are also obtained. For the Sawada-Kotera-type equation, with the help of the Riemann theta function and Hirota bilinear method, its one periodic wave solution is obtained. A limiting procedure is presented to analyze in detail the relations between the one periodic wave solution and one soliton solution.