Lump Solutions to the ( 3 + 1 )-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

were 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 fewdecades, lumphas 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) equation u푦푡 + 3u푥푧 − 3u푥u푥푦 − 3u푥푥u푦 − u푥푥푥푦 = 0, (2)


Introduction
Lump is a type of localized rational solutions.Lump solutions for the (2+1)-dimensional Kadomtsev-Petviashvili I equation were 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) equation which 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][13][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.

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.
From the point of view of the (, , )-space, at each time , the lump (23) attains its extreme values at the lines which move with the velocity In general, for arbitrary real constants , , , , let  = −2.Then as  → 0, the limit of (11) yields the following result.where
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  5 and  10 can be set zero by the translation (, , , ) → ( 耠 ,  耠 ,  耠 ,  耠 ) defined by where  0 ,  0 ,  0 ,  0 are some constants; secondly, since the parameters  1 and  6 can not be zero simultaneously, without loss of generality, we may assume  1 ̸ = 0, and then from the form of (35), we can set  1 = 1; thirdly, since (34) is invariant under the Galilean transformation (, , , ) → ( 耠 ,  耠 ,  耠 ,  耠 ) given by  and ℎ 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.

Figure 1 :
Figure 1: The 3D plot of lump (13) and the corresponding contour plot.

Figure 2 :
Figure 2: The 3D plot of lump (18) and the corresponding contour plot.

Figure 3 :
Figure 3: The 3D plot of lump (23) and the corresponding contour plot.