N-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation

Within the Hirota bilinear formulation, we construct N-soliton solutions and analyze the Hirota N-soliton conditions in (2+1)-dimensions. A generalized algorithm to prove the Hirota conditions is presented by comparing degrees of the multivariate polynomials derived from the Hirota function in N wave vectors, and two weight numbers are introduced for transforming the Hirota function to achieve homogeneity of the related polynomials. An application is developed for a general combined nonlinear equation, which provides a proof of existence of its N-soliton solutions. The considered model equation includes three integrable equations in (2+1)-dimensions: the (2+1)-dimensional KdV equation, the Kadomtsev-Petviashvili equation, and the (2+1)-dimensional Hirota-Satsuma-Ito equation, as specific examples. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

This study focuses on constructing N-soliton solutions within the Hirota bilinear formulation and analyzing the Hirota N-soliton conditions in (2+1)-dimensions. A generalized algorithm for proving the Hirota conditions is presented, along with the introduction of two weight numbers to achieve homogeneity of the related polynomials. An application is developed for a general combined nonlinear equation to provide proof of existence of its N-soliton solutions.