Abstract
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.
1. Introduction
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 [1–3]. Explicitly solvable differential equations include various constant-coefficient and linear differential equations, but it is extremely difficult to compute exact solutions to variable-coefficient or nonlinear equations.
However, the Hirota bilinear method provides us with a working approach to soliton solutions, historically found for nonlinear integrable equations [4, 5]. Soliton solutions are analytic ones exponentially localized in all directions in space and time. Let a polynomial determine a Hirota bilinear differential equation:in (2+1)-dimensions, where , and are Hirota’s bilinear derivatives. The corresponding partial differential equation with a dependent variable is determined usually by one of the logarithmic transformations: and . Within the Hirota bilinear formulation, soliton solutions are expressed throughwhere means the sum over all possibilities for taking either 0 or 1, and the wave variables and the phase shifts are defined byandin which , and , , satisfy the corresponding dispersion relation and , , are arbitrary phase shifts.
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., [6–8]). Taking long wave limits of -soliton solutions can generate special lumps [9]. Many integrable equations in (2+1)-dimensions exhibit the remarkable richness of lump solutions (see, e.g., [6, 7]). Such equations contain the KPI equation [10], whose special lump solutions have been derived from -soliton solutions [11], the three-dimensional three-wave resonant interaction [12], the BKP equation [13, 14], the Davey-Stewartson equation II [9], the Ishimori-I equation [15], and the KP equation with a self-consistent source [16]. An important step in the process of getting lumps is to determine positive quadratic function solutions to bilinear equations [6]. Then, through the mentioned logarithmic transformations, we present lump solutions to nonlinear equations (see, e.g., [6] for the case of Hirota bilinear equations and [7] for the case of generalized bilinear equations).
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., [6, 7, 17, 18] for other equations). We will consider a general class of HSI type equations while keeping the existence of lump solutions. A general such generalized HSI equation in (2+1)-dimensions and its lump solutions will be determined through symbolic computations with Maple. For a special presented lump solution, three-dimensional plots and contour plots will be made via the Maple plot tool, to shed light on the characteristic of the presented lump solutions. A few concluding remarks will be given in the last section.
2. Lump Solutions
It is known that the Hirota-Satsuma shallow water wave equation [4], has a bilinear form, under the logarithmic transformations and . An integrable -dimensional extension of the Hirota-Satsuma equation reads which passes the Hirota three-soliton test [19], and has a bilinear form under the logarithmic transformation :Equation (7) is called the Hirota-Satsuma-Ito (HSI) equation in (2+1)-dimensions [19]. We would like to add three terms to generalize the abovementioned HSI equation to a new one which still possesses abundant interesting solution structures: This generalized HSI equation has a bilinear form under the logarithmic transformation :Precisely, under , we have the relation In what follows, we would like to determine lump solutions to the gHSI equation in (2+1)-dimensions (9), through symbolic computations with Maple.
We start to search for positive quadratic solutions to the gHSI bilinear equation (10) to generate lump solutions to the gHSI equation (9): Plugging this function into the gHSI bilinear equation (10) generates a system of nonlinear algebraic equations on the parameters , . Conducting direct symbolic computation to solve this system gives a set of solutions for the parameters whereand all other ’s are arbitrary. The involved three constants are defined as follows:Those formulas in (12) and (13) were obtained under a simplification process with Maple.
From (12), we can easily see that it is sufficient to guarantee if we require and, thus, the function defined by (12) and (13) under the abovementioned condition and leads to lump solutions to the gHSI equation in (2+1)-dimensions (9).
When one takes one obtains the original HSI equation in (2+1)-dimensions (7), and the function by (12) and (13) presents a class of lump solutions to the HSI equation (7):whereand all other ’s are arbitrary. Solving the abovementioned parameter solutions on and for and and substituting the resulting expressions for and into the formula for in (19), we getIt is easy to see thatand, thus, the conditions of guarantee that (16) with (11) and (20) will present lump solutions to the HSI equation (7) [20].
Particularly takingwe obtain a special gHSI equation as follows: which has a Hirota bilinear form under the logarithmic transformation (16). Associated with (16) with (11) and (22) present the lump solution to the special gHSI equation (24): Three three-dimensional plots and contour plots of this lump solution are made via Maple plot tools, to shed light on the characteristic of the presented lump solutions, in Figure 1.
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., [21–31]).
3. Concluding Remarks
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 [32–36]. Some recent studies also demonstrate the strikingly high richness of lump solutions to linear partial differential equations [37] and nonlinear partial differential equations in (2+1)-dimensions (see, e.g., [38–41]) and (3+1)-dimensions (see, e.g., [42–48]). Diversity of lump solutions supplements exact solutions generated from different kinds of combinations (see, e.g., [49–52]) and can yield various Lie-Bäcklund symmetries, which can be used to determine conservation laws by symmetries and adjoint symmetries [53–55]. Moreover, diverse interaction solutions [35] have been exhibited for many integrable equations in (2+1)-dimensions, including lump-soliton interaction solutions (see, e.g., [56–58]) and lump-kink interaction solutions (see, e.g., [59–62]).
We finally remark that we could add one more term to the gHSI equation (9) to formulate a more generalized HSI bilinear equation,where , , are all constants, but we failed to drive any lump solution to the corresponding nonlinear equation on . The first term in the abovementioned bilinear equation is crucial in determining lump solutions but the last term brings the difficulty to work out lump solutions. There is no hint on how to solve any big system of resulting nonlinear algebraic equations. Nevertheless, some general considerations on the existence of lumps have been made for the Hirota bilinear case [6] and the generalized bilinear cases [7].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
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.