Lump Solutions to a (2+1)-Dimensional Fifth-Order KdV-Like Equation

A (2+1)-dimensional fifth-order KdV-like equation is introduced through a generalized bilinear equation with the prime number p=5. 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.

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 2 Advances in Mathematical Physics 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][26][27].
In this paper, we investigate lump solutions for a fifthorder KdV-like equation. The organization of the paper is as follows: In Section 2, we formulate a new fifth-order KdVlike 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.

A New (2+1)-Dimensional Fifth-Order KdV-Like Equation
Under the dependent variable transformation where the Hirota derivatives , , and are defined in [28].

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 assumption where , 1 ≤ ≤ 9, are real constants to be determined later. Note that using a sum involving one square, in the twodimensional 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 (1 ≤ ≤ 9); solving the set of algebraic equations with the aid of Maple, we attain the following two classes of solutions.
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 ℎ are given as follows: where 1 , 2 , 4 , 5 , 6 , 8 are arbitrary constants to be determined with the following restricted conditions:  Since these parameters are arbitrary, the solutions of (9) are more general. The parameters 1 , 5 indicate that the wave velocity in the direction is arbitrary and 2 , 6 illustrate the arbitrariness of the wave velocity in the direction. The Consequently, a kind of lump solutions to (9) through the transformation = 6(ln ) and (10) is achieved as follows: where the functions and ℎ 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, 2 = 4, If we take a particular choice of the parameters in Case 2 as

Conclusions
In this paper, we studied a new (2+1)-dimensional fifthorder KdV-like equation, obtained by using the generalized Hirota bilinear formulation with = 5. 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.