Abstract
This paper is devoted to the study of lump solutions to the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. First we use a direct method to construct a class of exact solutions which contain six arbitrary real constants. Then we use these solutions to generate lump solutions with four real parameters. We also determine the amplitude and velocity of these lumps.
1. Introduction
Lump is a type of localized rational solutions. Lump solutions for the (2+1)-dimensional Kadomtsev-Petviashvili I equationwere first obtained by Manakov et al. [1]. Then Satsuma and Ablowitz [2] constructed more lump solutions by taking a long wave limit of the corresponding soliton solutions. More recently, by use of the bilinear form, Ma [3] generalized the results of [1, 2] and got a larger class of lump solutions. Over the past few decades, lump has been an active area in the study of nonlinear evolution equations. They can be used to describe nonlinear patterns in plasma [4], in optical media [5], in the Bose-Einstein condensate [6], and so forth. Lump solutions have been obtained for some other equations, such as the Ishimori equation [7], the Jimbo-Miwa equation [8], and the Sawada-Kotera equation [9].
Recently much literature is devoted to the study of (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equationwhich can be used to model fluid dynamics, plasma physics, and weakly dispersive media [10, 11]. Various methods have been applied to (2) to construct its soliton and multiple wave solutions [12–14]. In this paper, we develop another method to construct its lump solutions. We first use a direct method to obtain a class of exact solutions which contain six arbitrary real constants (see Theorem 1). Then we show that the limits of these solutions can generate lump solutions (see Theorem 4). We also determine the amplitude and velocity of these lumps.
2. Exact Solutions
In this section, we seek exact solutions to the (3+1)-dimensional BKP equation (2) in the following form:where, , and are real constants to be determined. Substituting (3) into the left hand side of (2) and then factoring we get the following three determining equations:Solving for from (5) and (6) givesandSubstituting (8) and (9) into (7) yieldsTherefore, we have the following.
Theorem 1. The (3+1)-dimensional BKP equation (2) admits the following exact solutions:where , , are arbitrary real constants, and are given by (8), (9), and (10), respectively.
3. Lump Solutions
In this section, we use (11) to construct lump solutions to the (3+1)-dimensional BKP equation (2). We first give some examples and then give a general result.
Example 1. Let , , , , . Then (11) becomeswhere , . As , the limit of (12) gives a lump solution to (2):where
The lump (13) attains its maximum value at and minimum value at . So its amplitude is . The profiles of the lump (13) in the -space are given in Figure 1.
From the point of view of the -space, at each time , the lump (13) attains its extreme values at the linesNote that the planes are fixed (do not depend on ), while the plane moves along its normal vector with the speed . Therefore the velocity of the moving lines (15) is
Example 2. Let , , , , . Then (11) becomeswhere , . As , the limit of (17) gives a lump solution to (2):where
The lump (18) attains its maximum value at and minimum value at . So the amplitude of the lump (18) is . The profiles of the lump (18) in the -space are given in Figure 2.
From the point of view of the -space, at each time , the lump (18) attains its extreme values at the lineswhich move with the velocity
Example 3. Let , , , , . Then (11) becomeswhere , . As , the limit of (22) gives a lump solution to (2):where
The lump (23) attains its maximum value at and minimum value at . So the amplitude of the lump (23) is . The profiles of the lump (23) in the -space are given in Figure 3.
From the point of view of the -space, at each time , the lump (23) attains its extreme values at the lineswhich move with the velocity
In general, for arbitrary real constants , let . Then as , the limit of (11) yields the following result.
Theorem 4. The (3+1)-dimensional BKP equation (2) admits the following rational solutions:whereand are arbitrary real constants.
Remark 5. If the parameters satisfy the following condition,then (27) are lump solutions.
For given parameters satisfying condition (31), the lump (27) attains its extreme values at the linesin the -space. Therefore the amplitude of the lump (27) isThe velocity of the lump (27) in the -space can also be determined. Here we omit the details.
4. Discussion
The (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation is the Jimbo-Miwa equation. If we replace with , then (2) becomesBased on the bilinear form of (34), Yang and Ma [8] construct ten classes of its lump solutions in the formwhere ,, and are some constants.
In this section, we point out that up to the invariance after a translation and a Galilean transformation, all the solutions obtained in [8] can be simplified and be rewritten in a unified form of (27). This is because, firstly, the parameters and can be set zero by the translation defined bywhere are some constants; secondly, since the parameters and can not be zero simultaneously, without loss of generality, we may assume , and then from the form of (35), we can set ; thirdly, since (34) is invariant under the Galilean transformation given bywhere is an arbitrary real number, we can set . Then under the condition and are of the forms of (29) and (30), and a direct calculation shows that each class of solutions in [8] can be rewritten in the unified form of (27).
From the mathematical point of view, the lump solutions (27) are limits of the exact solutions (11) as the parameter approaches zero. It is worth studying the physical meaning of this limit.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work is supported by the Natural Science Foundation of Jiangsu Province through BK20151304.