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

Wen-Xiu Ma

doi:10.1515/ijnsns-2020-0214

Article

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

Wen-Xiu Ma

Published/Copyright: March 1, 2021

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 23 Issue 1

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.

Keywords: (1 + 1)-dimensional integrable equations; Hirota N-soliton condition; N-soliton solution

MSC 2010: 35Q51; 35Q53; 37K40

Corresponding author: Wen-Xiu Ma, Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China; Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia; Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA; School of Mathematics, South China University of Technology, Guangzhou 510640, China; and School of Mathematical and Statistical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa, E-mail: mawx@cas.usf.edu

Funding source: NSFC

Award Identifier / Grant number: 11975145, 11972291

Acknowledgments

The work was supported in part by NSFC under the Grants 11975145 and 11972291. The author would also like to thank Yushan Bai, Yehui Huang, Xing Lü, Solomon Manukure, Morgan McAnally and Fudong Wang for their valuable discussions.

Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: NSFC under the grants 11975145 and 11972291.
Conflict of interest statement: The author declares no conflicts of interest regarding this article.

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Received: 2020-09-19

Accepted: 2021-01-16

Published Online: 2021-03-01

Published in Print: 2022-02-23

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2020-0214

Keywords for this article

(1 + 1)-dimensional integrable equations; Hirota N-soliton condition; N-soliton solution