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In this paper, the Auto-B ?cklund transformation connected with the homogeneous balance method (HB) and the extended tanh-function method are used to construct new exact solutions for the time-dependent coefficients Calogero-Degasperis (VCCD) equation. New soliton and periodic solutions of many types are obtained. Furthermore, the soliton propagation is discussed under the effect of the variable coefficients.

Recently, investigation of exact solutions for nonlinear partial differential equations (NPDEs) with variable coefficients plays an important role in modern nonlinear science because NPDEs with variable coefficients reflect the real thing even more than those with constant.

One of the most important NPDEs is the time-dependent coefficients Calogero-Degasperis (VCCD) equation [

where

The objective of this paper is to apply the auto-Bäcklund transformation method and the extended tanh- function method on the VCCD equation, to find more general new solitonic and periodic exact solutions.

We can obtain Auto-Bäcklund transformation by using HB method [

Step 1: We consider the exact solution of (1) in the form

where

According to the HB method n can be determined by balancing the linear term of the highest order derivative and the highest nonlinear term of u in (1).

Therefore,

Substituting (3) into (1), we get

Step 2: To make (4) as a linear equation in f we assume that,

So that, we have the following relations

Substitute from relations (6) into (4) and equating the linear coefficients

Step 3: To solve the previous system, assume that

where

By substitution from (8-10) into (3) using (5), we obtain the following one-soliton solution for the VCCD equation under condition (9)

By using the following two useful formulas [

We obtain the following kink-type soliton and periodic solutions respectively

Analogously, we assume that

In this section, we are going to find more new exact solutions for the VCCD equation using direct integration and extended tanh-function method [

where

By substitution in (1), we have

To make the previous Equation (19) be an ordinary differential equation, we have found

Therefore, (19) becomes

By Integrating (22) twice, we get

where

Now, we apply the extended tanh function method used in [

where

This Riccati equation has the following solutions

Substitute from (25) into Equation (24) and balance the term

Therefore,

where

By substitution from (30) and (27) in (29), we have got the following exact solutions for Equation (24)

and

the relation

where

where

The following part of this section is devoted to analyzing the influences of the variable coefficients on the solitonic propagation. From the expression of

from the previous equation, we have found that there are three arbitrary constants

Therefore, the propagation direction of the soliton is decided by the sign of v and the solitonic velocity depend on the variables

The previous figures indicate that how the variable coefficients

trajectory is not a straight line anymore. It exhibits as a parabolic and periodic-type propagation respectively.

By using the HB method, we have obtained Auto-Bäcklund transformation and new exact solitary and periodic solutions for the VCCD equation. Also by using a travelling wave transformation, we have reduced the VCCD equation to an ordinary differential equation, by the extended tanh function method we have been able to obtain many other new exact solitary and periodic-type solutions. Some remarks have been found on the obtained so-n lutions

Remark 1: The obtained Bäcklund transformation is more easy and simple in calculations than that obtained in [

Remark 2: The combination between the two functions

Remark 3: All solutions obtained in this paper have been satisfied by Mathematica program.