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

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.

Automated summary

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