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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.

N -soliton solutions and the Hirota conditions in (1 + 1)-dimensions