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Trial equation method is a powerful tool for obtaining exact solutions of nonlinear differential equations. In this paper, the improved Boussinesq is reduced to an ordinary differential equation under the travelling wave transformation. Trial equation method and the theory of complete discrimination system for polynomial are used to establish exact solutions of the improved Boussinesq equation.

In every field of engineering technology, science research, natural world and human society activities, nonlinear phenomena occupy an important position. The investigation of exact solutions of nonlinear evolution equations helps us understand these phenomena better. With the development of soliton theory and the application of computer symbolic system such as Matlab and Mathematica, many powerful methods for obtaining exact solutions of nonlinear evolution equations are presented, such as the inverse scattering method [

In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours. The objective of this paper is to apply Liu’s method and the theory of complete discrimination system for polynomial [12-16] to find the exact solutions of the nonlinear differential equation.

The objective of this section is to outline the use of trial equation method for solving a nonlinear partial differential equation (PDE). Suppose we have a nonlinear PDE for in the form

where is a polynomial, which includes nonlinear terms and the highest order derivatives and so on.

Step 1 Taking the wave transformation, reduces Equation (1) to the ordinary differential equation (ODE).

Step 2 Take trial equation method

Integrating the Equation (3) with respect to once, we get

where m, and integration constant are to be determined. Substituting Equations (3), (4) and other derivative terms into Equation (2) yields a polynomial of μ. According to the balance principle we can determine the value of m. Setting the coefficients of to zero, we get a system of algebraic equations. Solving this system, we can determine values of and integration constant.

Step 3 Rewrite Equation (4) by the integral form

According to the complete discrimination system of the polynomial, we classify the roots of and solve the integral equation (5). Thus we obtain the exact solutions to Equation (1).

The improved Boussinesq equation [17,18] reads as

Taking the traveling wave transformation and, we can obtain the corresponding reduced ODE.

we take the trial equation as follows

According to the trial equation method of rank homogeneous equation, balancing with (or) gets, so Equation (8) has the following specific form

Integrating Equation (9) with respect to once, we yield

where values of and the integration constant d are to be determined latter. By Equation (9) and Equation (10), we derive the following formula

Substituting Equations (9), (10) and (11) into Equation (7), we have

where

Let the coefficient be zero, we will yield nonlinear algebraic equations. Solving the equations, we will determine the values of. We get and d are two arbitrary constants. When the above conditions are satisfied, we use the complete discrimination system for the third order polynomial and have the following solving process.

Let

Then Equation (10) becomes

where is a function of. The integral form of Equation (18) is

Denote

According to the complete discrimination system, we give the corresponding single traveling wave solutions to Equation (6).

Case1. has a double real root and a simple real root. Then we have

When, the corresponding solutions are

Case2. has a triple root. Then we have

The corresponding solution is

Case 3. has three different real roots. Then we have

When, we take the transformation as follows

According to the Equation (19), we have

where. On the basis of Equation (30) and the definition of the Jacobi elliptic sine function, we have

The corresponding solutions is

when, we take the transformation as follows

The corresponding solutions is

where.

Case 4. has only a real root. Then we have

when, we take the transformation as follows

According to the Equation (19), we have

where

On the basis of Equation (37) and the definition of the Jacobi elliptic cosine function, we have

The corresponding solutions is

In Equations (23), (24), (25), (27), (32), (34), and (39), the integration constant has been rewritten, but we still use it. The solutions are all possible exact traveling wave solutions to Equation (6). We can see it is easy to write the corresponding solutions to the improved Boussinesq equation.

Trial equation method is a systematic method to solve nonlinear differential equations. The advantage of this method is that we can deal with nonlinear equations with linear methods. This method has the characteristics of simple steps and clear effectivity. Based on the idea of the trial equation method and the aid of the computerized symbolic computation, some exact traveling wave solutions to the improved Boussinesq equation have been obtained. With the same method, some of other equations can be dealt with.

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

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