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The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’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.

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 [

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 [

In this paper, we will consider the following (3+1)-dimensional generalized BKP equation:

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 (

The paper is organized as follows. In Section

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 [

Assuming that a group of functions

Obviously, we have

Under (

Theorem

From the compatibility conditions

If the coefficient matrix

In principle, we can construct general Wronskian solutions of (

It is well known that the corresponding Jordan form of a real matrix

(I)

(II)

Suppose

In this case, if the eigenvalue

From (

If the eigenvalue

When

Similarly, we have a 2-soliton to (

The shape of the 1-soliton to (

The shape of the 2-soliton to (

When

The shape of the 1-positon of zero-order to (

The shape of the 1-positon of first-order to (

When

The shape of the 1-negaton of first-order to (

The shape of the 1-negaton of second-order to (

We are now presenting examples of Wronskian interaction solutions among different kinds of Wronskian solutions to the (3+1)-dimensional generalized BKP equation (

Let us assume that there are two sets of eigenfunctions

In what follows, we would like to show a few special Wronskian interaction solutions. Let us first choose different sets of eigenfunctions:

Through three Wronskian interaction solutions between any two of a rational solution, a single soliton and a single positon read as

One Wronskian interaction solution involving the three eigenfunctions is given by

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.

The authors declare that they have no competing interests.

This work is supported by the National Natural Science Foundation of China (Grant nos. 11402194, 11501442, and 71271171), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 2013JK0584), and the Science and Technology Research and Development Program in Shaanxi Province of China (Grant no. 2014KW03-01).