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In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All single traveling wave solutions to the equation can be obtained. As an example, we give the solutions to (3 + 1)-dimensional breaking soliton equation.

The Nonlinear Partial Differential Equation; Complete Discrimination System for Polynomial; Direct Integral Method; Traveling Wave Transform; (3 + 1)-Dimensional Breaking Soliton Equation

For the past decades, to deal with nonlinear partial differential equations (PDEs), many methods have been developed. These methods have been widely applied to many PDEs to obtain the exact solutions. Recently, a method named the complete discrimination system for polynomial method has been proposed by Liu [

where

In this paper, we take into account (3 + 1)-dimensional breaking soliton equation, and it reads as

where a, b, c, d and e are arbitrary constants.

Equation (2) was originally proposed by Lin [

For Equation (2), we take the traveling wave transformation

Integrating Equation (3) with respect to

where

Let

Then we have

Or equivalently

Integrating the Equation (7) once with respect to

where

Let

Then Equation (8) becomes

where

Denote

According to the complete discrimination system, we give the corresponding single traveling wave solutions to Equation (2).

Case 1.

When

The corresponding solutions to Equation (2) are

Case 2.

The corresponding solution to Equation (2) is

Case 3.

When

When

where

Case 4.

When

where

In Equations (24) (25) and (27), we give the expression of some signals as follow

The solutions

From the descriptions above, we use the complete discrimination system for polynomial and direct integral method to obtain all possible traveling wave solutions to (3 + 1)-dimensional breaking soliton equation. This method is direct and effective. With the same method, some of other equations can be dealt with.

I would like to thank the referees for their valuable suggestions.