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In this paper, we apply the tanh-coth method and traveling wave transformation method for solving Gardner equations, including (1 + 1)-Gardner and (2 + 1)- Gardner equations. The tanh-coth method proved to be reliable and effective in handling a large number of nonlinear dispersive and disperse equations. Through tanh-coth method, we get analytical expressions of soliton solutions of Gardner equations. The one-soliton solution is characterized by an infinite wing or infinite tail.

In the study of nonlinear science, finding the exact solution of nonlinear evolution equations is an important subject. Different methods have their different types of specific applications for nonlinear evolution equations. In recent years many scholars put forward and developed several new methods for solving PDEs which based on the original method, such as Hirota’s bilinear method [

In the plasma physics, solid physics, fluid mechanics, etc., the Gardner equation is written as

which is also called the KdV-mKdV equation. The model can be well described the wave propagation in a one-dimensional nonlinear lattice with a non harmonic bound particle. Gardner equations have very important application in mathematics, physics, engineering and other fields. Different types of equations can be obtained by changing the value of

With

where the parameter

With

It is completely integrable [

The Gardner equations are used to describe many physical models, which are closely related to the study of physics. So it is very important to study it deeply.

With

Further, the (2 + 1)-dimensional Gardner Equation [

which reduces to the (1 + 1)-Gardner equation with

For

while it is the modified KP equation with

With

We had found soliton solutions, travelling wave solutions and plane periodic solutions of KdV and mKdV equations through tanh-coth method. In order to prove superiority of the tanh-coth method, we apply it on Gardner equations which are more complex and have higher dimensions.

This paper is organized as follows. In Section 2, we introduce the tanh-coth method. In Section 3, we first substitute the wave variable

A wave variable

to an ODE

Equation (9) is then integrated as long as all terms contain derivatives where integration constants are considered zeros.

Introducing an independent variable

where

where

Substituting (11) into the reduced ODE results. We then collect all coefficients of each power of

We first substitute the wave variable

that gives

Integrating once to obtain

We then balance the nonlinear term

that gives

The tanh-coth method allows us to use the substitution

Substituting (18) into (15), collecting the coefficients of each power of

We find the following sets of solutions:

Consequently, we obtain the following solutions:

Following immediately.

that

We first substitute the wave variable

that gives

Based on

Integrating once to obtain

We then balance the nonlinear term

that gives

The tanh-coth method allows us to use the substitution

Substituting (30) into (27), that gives

Collecting the coefficients of each power of

We find the following sets of solutions:

Consequently, we obtain the solutions as

where

Following immediately.

In this paper, we obtain the soliton and kink solutions of the (1 + 1)-Gardner equation and (2 + 1)-Gardner equation through the tanh-coth method. The biggest advantage is that by traveling wave transformation, the problem of solving nonlinear partial differential equations is transformed into the problem of solving nonlinear ordinary differential equations or nonlinear algebraic equations. The tanh-coth method is convenient to use, and can be further extended to solve other nonlinear partial differential equations.

The authors would like to express their sincere thanks to the referees for their enthusiastic guidance and help. This work is supported by the National Natural Science Foundation of China (No.11371326)

Lin, L., Zhu, S.Y., Xu, Y.K. and Shi, Y.B. (2016) Exact Solutions of Gardner Equations through tanh- coth Method. Applied Mathematics, 7, 2374- 2381. http://dx.doi.org/10.4236/am.2016.718186