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

• Published in: Mathematics and computers in simulation Vol. 190; pp. 270 - 279
• Main Author: Ma, Wen-Xiu
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2021
• Subjects: (2+1)-dimensional integrable equations; [formula omitted]-soliton solution; Hirota [formula omitted]-soliton condition; Hirota N-soliton condition; N-soliton solution; (2+1)-dimensional integrable equations
• ISSN: 0378-4754
• DOI: 10.1016/j.matcom.2021.05.020

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.