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The Hirota-Satsuma-Ito equation in (2+1)-dimensions passes the three-soliton test. This paper aims to generalize this equation to a new one which still has abundant interesting solution structures. Based on the Hirota bilinear formulation, a symbolic computation with a new class of Hirota-Satsuma-Ito type equations involving general second-order derivative terms is conducted to require having lump solutions. Explicit expressions for lump solutions are successfully presented in terms of coefficients in a generalized Hirota-Satsuma-Ito equation. Three-dimensional plots and contour plots of a special presented lump solution are made to shed light on the characteristic of the resulting lump solutions.

In the classical theory of differential equations, the main question is to study the existence of solutions to given equations, including many nonlinear equations describing real-world problems. Cauchy problems are to deal with the existence, uniqueness, and stability of solutions satisfying initial data. Laplace’s method is developed for solving Cauchy problems for linear ordinary differential equations and the Fourier transform method for linear partial differential equations. In modern soliton theory, the isomonodromic transform method and the inverse scattering transform method have been designed to solve Cauchy problems for nonlinear ordinary and partial differential equations [

However, the Hirota bilinear method provides us with a working approach to soliton solutions, historically found for nonlinear integrable equations [

Lump solutions are a class of analytic rational solutions which are localized in all directions in space, originated from solving integrable equations in (2+1)-dimensions (see, e.g., [

In this paper, we would like to generalize the Hirota-Satsuma-Ito (HSI) equation in (2+1)-dimensions to a new one which still has abundant interesting solution structures. Hirota bilinear forms are the starting point for our discussion (see, e.g., [

It is known that the Hirota-Satsuma shallow water wave equation [

We start to search for positive quadratic solutions to the gHSI bilinear equation (

From (

When one takes

Particularly taking

Profiles of

All the exact solutions generated above add valuable insights into the existing theories on soliton solutions and dromion-type solutions, developed through various powerful solution techniques including the Hirota perturbation approach, the Riemann-Hilbert approach, the Wronskian technique, symmetry reductions, and symmetry constraints (see, e.g., [

We have studied a generalized (2+1)-dimensional Hirota-Satsuma-Ito (HSI) equation to explore different equations which possess lump solutions, through symbolic computations with Maple. The results enrich the theory of lumps and solitons, providing a new example of (2+1)-dimensional nonlinear equations, which possess beautiful lump structures. Three-dimensional plots and contour plots of a specially chosen lump solution were made by using the plot tool in Maple.

Many nonlinear equations possess lump solutions, which include (2+1)-dimensional generalized KP, BKP, KP-Boussinesq, Sawada-Kotera, and Bogoyavlensky-Konopelchenko equations [

We finally remark that we could add one more term to the gHSI equation (

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The work was supported in part by NSFC under Grants nos. 11301454, 11301331, and 11371086, NSF under Grant no. DMS-1664561, the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17KJB110020), Emphasis Foundation of Special Science Research on Subject Frontiers of CUMT under Grant no. 2017XKZD11, and the Distinguished Professorships by Shanghai University of Electric Power, China, and North-West University, South Africa.