Lump Solutions to a (2+1)-Dimensional Fifth-Order KdV-Like Equation.
The (2+1)-dimensional fifth-order KdV equation  is
36[u.sub.t] + [u.sub.5x] + 15[u.sub.x][u.sub.xx] + 15[uu.sub.3x] + 45[u.sup.2][u.sub.x] - 5[u.sub.xxy]
- 15[uu.sub.y] - 15[u.sub.x] [integral] [u.sub.y]dx - 5 [integral] [u.sub.yy]dx = 0, (1)
which is the (2+1)-dimensional analogue of the Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation . When [u.sub.y] = 0, (1) reduces to the Sawada-Kotera equation
[u.sub.t] + [u.sub.5x] + 15[u.sub.x][u.sub.xx] + 15[uu.sub.3x] + 45[u.sup.2][u.sub.x] = 0. (2)
Konopelchenko and Dubovsky  were the first to come up with (1). Lv et al.  obtained the symmetry transformations for (1) by using its Lax pair. Lu  constructed four sets of bilinear Backlund transformations in order to obtain multisoliton solutions. Wazwaz  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 , and the B-KP equation . Through Hirota bilinear equations, one of the authors (Ma)  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 , the (2+1)-dimensional Boussinesq equation , the BKP equation , the (3+1)-dimensional Jambo-Miwa , and the KdV equation . In addition to Hirota bilinear forms, generalized bilinear derivatives  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 transformation
u = 2 [(ln f).sub.xx] (3)
with f = f(x, y, t), the (2+1)-dimensional fifth-order KdV equation (1) becomes the following (2+1)-dimensional Hirota bilinear equation:
[mathematical expression not reproducible] (4)
where the Hirota derivatives [D.sub.x], [D.sub.y], and [D.sub.t] are defined in .
Based on a prime number p, a kind of generalized bilinear operators is introduced as [24, 29]
[mathematical expression not reproducible] (5)
where m, n [greater than or equal to] 0, [mathematical expression not reproducible] mod p.
For example, if we assume p = 5, we have
[[alpha].sub.5] = -1
[[alpha].sup.2.sub.5] = 1,
[[alpha].sup.3.sub.5] = -1,
[[alpha].sup.4.sub.5] = [[alpha].sup.5.sub.5] = 1,
[[a.sup.6.sub.5] = -1, ... (6)
With p = 5, we can generalize (4) into
[mathematical expression not reproducible] (7)
Equation (7) is a generalized bilinear fifth-order KdV equation. Under the transformations
u = 6 [(ln f).sub.x],
v = 6 [(ln f).sub.y] (8)
which were suggested by the Bell polynomial theories [29-31], (7) is transformed into the following fifth-order KdV-like nonlinear differential equation:
[mathematical expression not reproducible] (9)
Therefore, if f solves the bilinear equation (4) or (7), then u = 6[(ln f).sub.xx] or u = 6[(ln f).sub.x] 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 assumption
f = [g.sup.2] + [h.sup.2] + [a.sub.9],
g = [a.sub.1]x + [a.sub.2]y + [a.sub.3]t + [a.sub.4],
h = [a.sub.5]x + [a.sub.6]y + [a.sub.7]t + [a.sub.8] (10)
where [a.sub.i], 1 [less than or equal to] i [less than or equal to] 9, 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 x, y, and t to zero using Maple symbolic computation, we obtain a set of algebraic equations in [a.sub.i] (1 [less than or equal to] i [less than or equal to] 9); solving the set of algebraic equations with the aid of Maple, we attain the following two classes of solutions.
[mathematical expression not reproducible] (11)
and [a.sub.i] = [a.sub.i] (i = 2, 4, 5, 8, 9) are real free parameters which need to satisfy [a.sub.5] [not equal to] 0 to make the corresponding solutions f to be well defined and [a.sub.9] > 0 to guarantee the positiveness of f.
The parameters in the sets (11) generate a class of positive quadratic function solutions to (7):
[mathematical expression not reproducible] (12)
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:
[mathematical expression not reproducible] (13)
where the function f is defined by (10), and the functions g and h are given as follows:
[mathematical expression not reproducible] (14)
[mathematical expression not reproducible] (15)
where [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.5], [a.sub.6], [a.sub.8] are arbitrary constants to be determined with the following restricted conditions:
[mathematical expression not reproducible] (16)
[[DELTA].sub.1] makes the corresponding solutions f well defined and [[DELTA].sub.2] assures that the solution f is positive, while [[DELTA].sub.3] guarantees the localization of the solutions u in all directions in the (x, y)-plane.
Since these parameters are arbitrary, the solutions of (9) are more general. The parameters [a.sub.1], [a.sub.5] indicate that the wave velocity in the x direction is arbitrary and [a.sub.2], [a.sub.6] illustrate the arbitrariness of the wave velocity in the y direction. The parameters [a.sub.4], [a.sub.8] represent the invariance of variables and [a.sub.3], [a.sub.7] show the wave frequency which are represented by other quantities.
This set of parameters, in turn, generates positive quadratic function solutions to (7):
[mathematical expression not reproducible] (17)
Consequently, a kind of lump solutions to (9) through the transformation u = 6[(ln f).sub.xx] and (10) is achieved as follows:
[mathematical expression not reproducible] (18)
where the functions g and h are given by
[mathematical expression not reproducible] (19)
Choosing a special value for the free parameters in Cases 1 and 2, we construct specific lump solutions u 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,
[a.sub.2] = 4,
[a.sub.4] = 0,
[a.sub.5] = 2,
[a.sub.8] = 0,
[a.sub.9] = 1, (20)
f = 34225/26244 [tsup.2] - 370/243 ty + 148/9 [y.sup.2] - 350/81 tx + 4[x.sup.2] - 8/3 xy + 1 (21)
and lump solution
u = 1259712 (27025[t.sup.2] - 113400tx + 115560ty + 104976[x.sup.2] - 69984xy - 408240[y.sup.2] - 26244)/ [(34225[t.sup.2] - 113400tx - 39960ty + 104976[x.sup.2] - 69984xy + 431568[y.sup.2] + 26244).sup.2] (22)
If we take a particular choice of the parameters in Case 2 as
[a.sub.1] = 1,
[a.sub.2] = -1/2,
[a.sub.4] = 0,
[a.sub.5] = 0,
[a.sub.6] = 3,
[a.sub.8] = 0, (23)
then we have
f = 34225/26244 [t.sup.2] - 370/243 ty + 148/9 [y.sup.2] - 350/81tx + 4[x.sup.2] - 8/3xy + 1 (24)
and lump solution
u = 248832 (27025[t.sup.2] - 113400tx + 115560ty + 104976[x.sup.2] - 69984xy - 408240[y.sup.2] - 26244)/ [(34225[t.sup.2] - 113400tx - 39960ty + 104976[x.sup.2] - 69984xy + 431568[y.sup.2] + 26244).sup.2] (25)
Figure 1 shows the profile of lump solutions in Case 1 with the special choice of the parameters (20) at t = 0, while Figure 2 shows the profile of lump solutions in Case 2 with the special choice of the parameters (23) at t = 0.
In this paper, we studied a new (2+1)-dimensional fifth-order KdV-like equation, obtained by using the generalized Hirota bilinear formulation with p = 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  and interaction solutions between lumps and solitons for many integrable equations in (2+1)-dimensions. We refer the reader to  for lump-kink interaction solutions and [34,35] for lump-soliton interaction solutions.
All data are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Sumayah Batwa (iD) (1,2) and Wen-Xiu Ma (iD) (1,3,4,5,6)
(1) Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
(2) Mathematics Department, King Abdulaziz University, Jeddah 21589, Saudi Arabia
(3) Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
(4) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
(5) College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China
(6) International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
Correspondence should be addressed to Sumayah Batwa; firstname.lastname@example.org
Received 22 January 2018; Accepted 31 May 2018; Published 2 July 2018
Academic Editor: Guozhen Lu
Caption: Figure 1: Plots of lump solution in Case 1 with [a.sub.2] = 4, [a.sub.4] = 0, [a.sub.5] = 2, [a.sub.8] = 0, [a.sub.9] = 1 at t = 0: (a) 3D plot, (b) density plot, and (c) x-curves.
Caption: Figure 2: Plots of lump solution in Case 2 with [a.sub.1] = 1, [a.sub.2] = -1/2, [a.sub.4] = 0, [a.sub.5] = 0, [a.sub.6] = 3, [a.sub.8] = 0 at t = 0: (a) 3D plot, (b) density plot, and (c) x-curves.
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|Title Annotation:||Research Article; Korteweg-de Vries|
|Author:||Batwa, Sumayah; Ma, Wen-Xiu|
|Publication:||Advances in Mathematical Physics|
|Date:||Jan 1, 2018|
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