New Exact Solutions for the ( 3 + 1 )-Dimensional Generalized BKP Equation

TheWronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based onHirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.


Introduction
In recent years, the problem of finding exact solutions of nonlinear evolution equations (NLEEs) is very popular for both mathematicians and physicists.Because seeking exact solutions of NLEEs is of great significance in nonlinear dynamics, many methods such as the inverse scattering transformation [1], Hirota's bilinear method [2], the Darboux transformation [3], the sine-cosine method [4],   /-expansion method [5,6], and the transformed rational function method [7] have been proposed.The Wronskian method which is based on the bilinear form of the NLEEs was proposed by Freeman and Nimmo in [8,9].It is a fairly powerful tool to construct exact solutions of NLEEs in terms of the Wronskian determinant.By means of the method, the exact solutions of some NLEEs are obtained [10][11][12][13][14][15][16].
The study of the BKP equation has attracted a considerable size of research work.These equations were studied using the Hirota method, the multiple exp-function algorithm, the Pfaffian technique, Riemann theta functions, the extended homoclinic test approach, and Bäcklund transformation by many authors [17][18][19][20][21][22][23][24][25][26].In this paper, based on the Wronskian method, the new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions of the (3+1)-dimensional generalized BKP equations are investigated.
In this paper, we will consider the following (3+1)dimensional generalized BKP equation: When  = , this (3+1)-dimensional generalized BKP equation reduces to the BKP equation [27,28]: By the dependent variable transformation the (3+1)-dimensional generalized BKP equation (1) becomes a bilinear form where   ,   ,   , and   are the Hirota operators [2]: We will show this (3+1)-dimensional generalized BKP equation has a class of Wronskian solutions with all generating functions for matrix entries satisfying a linear system of partial differential equations involving a free parameter.Rational solutions, solitons, positons, negatons, and interaction solutions to (1) among Wronskian determinant solutions are constructed and a few plots of particular solutions are made.
The paper is organized as follows.In Section 2, we derive a Wronskian formulation for the (3+1)-dimensional generalized BKP equation.In Section 3, Wronskian solutions to the (3+1)-dimensional generalized BKP equation are obtained.Section 4 presents the conclusion.

A Wronskian Formulation
The Wronskian technique is a powerful tool to construct exact solutions to bilinear differential or difference equations.To use the Wronskian technique, we adopt the compact notation introduced by Freeman and Nimmo [8,9]: where where  is an arbitrary nonzero constant, then the Wronskian determinant  = | N − 1| defined by (6) solves the bilinear equation (5).
Based on Remark 1, we only need to consider case of ( 8)- (11) under Λ/ = 0, that is, the following conditions: where Λ = (  ) is an arbitrary real constant matrix.Moreover, Remark 2 tells us that an invertible constant linear transformation on Φ in the Wronskian determinant does not change the corresponding Wronskian solution, and thus, we only have to solve (21) under the Jordan form of Λ.

Wronskian Solutions
In principle, we can construct general Wronskian solutions of (1) associated with two types of Jordan blocks of the coefficient matrix Λ.But it is not easy.In this section we will present a few special Wronskian solutions to the generalized BKP equation, together with examples of exact solutions.
It is well known that the corresponding Jordan form of a real matrix has the following two types of blocks: (I) (II) where   ,   ,   are all real constants.The first type of blocks has the real eigenvalue   with algebraic multiplicity   (Σ  =1   = ), and the second type of blocks has the complex eigenvalue  ±  =   ±   √ −1 with algebraic multiplicity   .
3.1.Rational Solutions.Suppose Λ has the first type of Jordan blocks.Without loss of generality, let In this case, if the eigenvalue  1 = 0, ( 1 ) becomes of the following form: From condition (21), we get Such functions   ( ≥ 1) are all polynomials in , , , and , and a general Wronskian solution to the (3+1)-dimensional generalized BKP equation ( 1) is rational and is called a rational Wronskian solution of order  1 − 1.
, and the rational Wronskian solution of secondorder is given by (32)

Solitons, Positons, and Negatons.
If the eigenvalue   ̸ = 0, (  ) becomes of the following form: We start from the eigenfunction   (  ) determined by General solutions to this system in two cases of   > 0 and   < 0 read as respectively, where  1 ,  2 ,  3 , and  4 are arbitrary real constants.By an inspection, we find that Therefore, through this set of eigenfunctions, we obtain a Wronskian solution to (1): which corresponds to the first type of Jordan blocks with a nonzero real eigenvalue.
If  =  or  = 0, we simply say that it is an -positon of order  or an -negaton of order .
(1) Solitons.An -soliton solution is a special -negaton: with   being given by   = cosh (√−  ( +  + √ 2 − 4  ) +   ) ,  odd, where  1 <  2 < ⋅ ⋅ ⋅ <   < 0 and   (1 ≤  ≤ ) are arbitrary real constants.For example, a 1-soliton to (1) is given by where Similarly, we have a 2-soliton to (1): where where  < 0 and  is an arbitrary function of √ −.Similarly, these two kinds of negatons are equivalent to each other.(57) Of course, we have more general Wronskian interaction solutions among three or more kinds of solutions such as rational solutions, positons, solitons, breathers, and negatons.Roughly speaking, it increases the complexities of rational solutions, positons, solitons, and negatons, respectively, to add zero, positive, negative eigenvalues to the spectrum of the coefficient matrix.

Conclusion
In summary we have extended the Wronskian method to a (3+1)-dimensional generalized BKP equation by its bilinear form.Moreover, we obtained some rational solutions, solitons, positons, negatons, and their interaction solutions to this equation by solving the systems of linear partial differential equations.All these show the richness of the solution space of the (3+1)-dimensional generalized BKP equation and the resulting solutions are expected to help understand wave dynamics in weakly nonlinear and dispersive media.

Figures 1 and 2
of three-dimensional plots show the -soliton to (1) defined by (40) on the indicated specific regions, with specific values being chosen for the parameters.