<i>N</i>-soliton solutions and the Hirota conditions in (1 + 1)-dimensions

Abstract

Abstract We analyze N -soliton solutions and explore the Hirota N -soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N -soliton solutions to all equations in two classes.