Eur. Phys. J. Plus (2020) 135: 119
https://doi.org/10.1140/epjp/s13360-019-00054-7

Regular Article

Bell polynomials and lump-type solutions to the Hirota–Satsuma–Ito equation under general and positive quadratic polynomial functions

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

^* e-mail: stslyj@mail.sysu.edu.cn

Received: 14 July 2019
Accepted: 12 October 2019
Published online: 23 January 2020

Abstract

In this article, we explore the Hirota–Satsuma–Ito (HSI) equation in (2+1)-dimensions which possess lump solutions. We first used the concept of Bell’s polynomials to derive the bilinear form of the equation. Then, we proceed to derive a quadratic function solution of the bilinear form and then expand it as the sums of squares of linear functions satisfying some conditions. Most importantly, we acquire a lump-type solution containing 11 parameters along with some non-zero conditions necessary for the existence of the solutions. Then, lump solutions are derived from the from the lump-type solutions by choosing a set of the constant. The solutions obtained in this paper further enrich the literature the ones reported in previous time using different Hirota bilinear approaches and the category of nonlinear partial differential equations (NPDEs) which possess lump solutions, particularly the HSI equation. The physical interpretation of the results is discussed and represented graphically.