Abstract
A (2+1)-dimensional fifth-order KdV-like equation is introduced through a generalized bilinear equation with the prime number . The new equation possesses the same bilinear form as the standard (2+1)-dimensional fifth-order KdV equation. By Maple symbolic computation, classes of lump solutions are constructed from a search for quadratic function solutions to the corresponding generalized bilinear equation. We get a set of free parameters in the resulting lump solutions, of which we can get a nonzero determinant condition ensuring analyticity and rational localization of the solutions. Particular classes of lump solutions with special choices of the free parameters are generated and plotted as illustrative examples.
1. Introduction
The (2+1)-dimensional fifth-order KdV equation [1] iswhich is the (2+1)-dimensional analogue of the Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation [2]. When , (1) reduces to the Sawada-Kotera equationKonopelchenko and Dubovsky [3] were the first to come up with (1). Lv et al. [4] obtained the symmetry transformations for (1) by using its Lax pair. Lü [5] constructed four sets of bilinear Bäcklund transformations in order to obtain multisoliton solutions. Wazwaz [6] derived multiple soliton solutions and multiple singular soliton solutions for (1). Equation (1) has a widespread adoption in many physical branches, such as conserved current of Liouville equation, two-dimensional quantum gravity gauge field, and conformal field theory [7–13].
In recent years, there has been a growing interest in finding exact solutions of nonlinear evolution equations, such as the rational solutions and the rogue wave solutions, which are exponentially localized in certain directions. Lump solutions are a type of rational function solutions, localized in all directions in the space. Lump solutions have been studied for many nonlinear partial differential equations such as the KPI equation [14, 15], the three-dimensional three-wave resonant interaction equation [16], and the B-KP equation [17]. Through Hirota bilinear equations, one of the authors (Ma) [18] introduced a new method to construct lump solutions to the KP equation. Following Ma’s method, the lump solutions for more nonlinear evolution equations have been found, for instance, the dimensionally reduced p-gKP and p-gBKP [19], the (2+1)-dimensional Boussinesq equation [20], the BKP equation [21], the (3+1)-dimensional Jambo-Miwa [22], and the KdV equation [23]. In addition to Hirota bilinear forms, generalized bilinear derivatives [24] are used to find rational function solutions to the generalized KdV, KP, and Boussinesq equations [25–27].
In this paper, we investigate lump solutions for a fifth-order KdV-like equation. The organization of the paper is as follows: In Section 2, we formulate a new fifth-order KdV-like equation from generalized bilinear differential equations of KdV type. In Section 3 with the help of Maple, we obtain lump solutions for the constructed equation and analyze their dynamics. Then we draw some figures for a particular classes of lump solutions to show some properties. In the last section, conclusions and some remarks are given.
2. A New (2+1)-Dimensional Fifth-Order KdV-Like Equation
Under the dependent variable transformationwith , the (2+1)-dimensional fifth-order KdV equation (1) becomes the following (2+1)-dimensional Hirota bilinear equation:where the Hirota derivatives , , and are defined in [28].
Based on a prime number , a kind of generalized bilinear operators is introduced as [24, 29]where , , .
For example, if we assume , we haveWith , we can generalize (4) intoEquation (7) is a generalized bilinear fifth-order KdV equation. Under the transformationswhich were suggested by the Bell polynomial theories [29–31], (7) is transformed into the following fifth-order KdV-like nonlinear differential equation:
Therefore, if solves the bilinear equation (4) or (7), then or will solve the nonlinear equation (1) or (9).
3. Lump Solutions to the Fifth-Order KdV-Like Equation
In this section, we are going to generate lump solutions to (9) by searching for quadratic function solutions to (7) with the assumptionwhere , are real constants to be determined later. Note that using a sum involving one square, in the two-dimensional space, will not generate exact solutions which are rationally localized in all directions in the space.
Substituting (10) into (7) and equating all the coefficients of different polynomials of , , and to zero using Maple symbolic computation, we obtain a set of algebraic equations in ; solving the set of algebraic equations with the aid of Maple, we attain the following two classes of solutions.
Case 1. and are real free parameters which need to satisfy to make the corresponding solutions to be well defined and to guarantee the positiveness of .
The parameters in the sets (11) generate a class of positive quadratic function solutions to (7):and the resulting class of quadratic function solutions, in turn, yields a few classes of lump solutions to the (2+1)-dimensional fifth-order KdV-like equation (9) through the dependent variable transformation:where the function is defined by (10), and the functions and are given as follows:
Case 2. where , , , , , are arbitrary constants to be determined with the following restricted conditions: makes the corresponding solutions well defined and assures that the solution is positive, while guarantees the localization of the solutions in all directions in the -plane.
Since these parameters are arbitrary, the solutions of (9) are more general. The parameters , indicate that the wave velocity in the direction is arbitrary and , illustrate the arbitrariness of the wave velocity in the direction. The parameters , represent the invariance of variables and , show the wave frequency which are represented by other quantities.
This set of parameters, in turn, generates positive quadratic function solutions to (7):Consequently, a kind of lump solutions to (9) through the transformation and (10) is achieved as follows:where the functions and are given by
Choosing a special value for the free parameters in Cases 1 and 2, we construct specific lump solutions of (9). One special pair of positive quadratic function solutions and lump solutions with specific values of the parameters in Case 1 is given as follows. First, the selection of the parameters,leads toand lump solutionIf we take a particular choice of the parameters in Case 2 asthen we haveand lump solutionFigure 1 shows the profile of lump solutions in Case 1 with the special choice of the parameters (20) at , while Figure 2 shows the profile of lump solutions in Case 2 with the special choice of the parameters (23) at .
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4. Conclusions
In this paper, we studied a new (2+1)-dimensional fifth-order KdV-like equation, obtained by using the generalized Hirota bilinear formulation with . Through symbolic computation with Maple we constructed a few classes of lump solutions. The analyticity and localization of the resulting lump solutions are guaranteed by a nonzero determinant condition and a positivity condition. A subclass of lump solutions under special choices of the parameters involved covers the lump solutions. Contour plots with small determinant values are sequentially made to exhibit that the corresponding lump solution tends to zero when the determinant tends to zero. Recently, there have been some systematical studies on lump solutions [32] and interaction solutions between lumps and solitons for many integrable equations in (2+1)-dimensions. We refer the reader to [33] for lump-kink interaction solutions and [34, 35] for lump-soliton interaction solutions.
Data Availability
All data are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.