**6. Conclusion**

18 Will-be-set-by-IN-TECH

where *ξ*<sup>1</sup> = *a*1*x* + *b*1*y* + *c*1*t*, *ξ*<sup>2</sup> = *a*2*x* + *b*2*y* + *c*2*t*, *ξ*<sup>3</sup> = *a*3*x* + *b*3*y* + *c*3*t* and *aj*, *bj*, *cj*, *j* = 1, 2, 3,

<sup>3</sup> + 3 *a*<sup>1</sup>

<sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>b</sup>*<sup>1</sup>

<sup>3</sup>*b*1*a*<sup>1</sup>

<sup>5</sup>*b*<sup>1</sup> + 3 *a*<sup>1</sup>

<sup>7</sup>*b*1)( *a*1 <sup>2</sup> + *a*<sup>2</sup> <sup>2</sup><sup>3</sup> )−<sup>1</sup>

<sup>4</sup>*a*<sup>2</sup> <sup>−</sup> <sup>6</sup> *<sup>a</sup>*1*b*1*b*<sup>2</sup>

<sup>2</sup>*a*<sup>1</sup> + 3 *b*<sup>2</sup>

<sup>4</sup> + 9 *a*<sup>1</sup>

2*b*1

<sup>5</sup> <sup>−</sup> *<sup>a</sup>*1*b*<sup>2</sup>

<sup>2</sup> − *<sup>a</sup>*2*b*1)

 =

S\_12

–5

0

5

y

**Figure 9.** Contourplot of the breather-type multi-solitary waves solutions with the different parameters.

10

15

2*a*1

3*a*2

<sup>3</sup>*a*<sup>2</sup> + 3 *a*1*a*<sup>2</sup>

<sup>4</sup>*b*<sup>2</sup> + *a*<sup>1</sup>

<sup>2</sup>*a*<sup>2</sup> + *b*1*b*2*a*<sup>2</sup>

*δ*1 2

S\_1

–15 –10 –5 0 5 10 15 x

 1.5 <sup>−</sup>1 0.05 −1.1 1.2 2

S\_2

3*b*2 3

2

<sup>6</sup>*b*<sup>2</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*1*a*<sup>2</sup>

2*b*2 3

2*a*2

3*a*2

<sup>5</sup> + 9 *a*<sup>2</sup>

2*a*2

<sup>3</sup> + 3 *a*<sup>2</sup>

<sup>2</sup>*a*<sup>2</sup> + *a*1*a*<sup>2</sup>

<sup>3</sup> + *b*2*a*<sup>1</sup> + *a*1*a*<sup>2</sup>

 *a*<sup>1</sup> *b*<sup>1</sup> *δ*<sup>1</sup> *a*<sup>2</sup> *b*<sup>2</sup> *δ*<sup>2</sup>

Fig.9 shows the plot of two kinds of interaction behavior between two single solitons with

respectively. From the first picture of Fig.9, we can see that two single solitons interact strongly to make a resonance breather-wave solution from a point at which two incident solitons meet

 and

and *δ*1, *δ*<sup>2</sup> with the following relations:

<sup>5</sup> <sup>−</sup> <sup>3</sup> *<sup>a</sup>*<sup>2</sup>

3*b*1

2*b*1

<sup>5</sup> + 2 *a*<sup>2</sup>

<sup>7</sup> + 3 *a*2*b*1*a*<sup>1</sup>

<sup>2</sup>*b*<sup>2</sup> <sup>−</sup> <sup>9</sup> *<sup>a</sup>*<sup>1</sup>

<sup>2</sup>*b*<sup>2</sup> + 3 *a*<sup>2</sup>

3*a*1 <sup>3</sup> <sup>−</sup> *<sup>a</sup>*<sup>1</sup>

<sup>3</sup> + *a*2*b*<sup>1</sup> + *a*1*a*<sup>2</sup>

different parameters and a breather wave, where

 =

S\_1

–15 –10 –5 0 5 10 15 x

S\_12

<sup>2</sup>*a*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*<sup>2</sup>

<sup>5</sup> <sup>−</sup> <sup>3</sup> *<sup>b</sup>*<sup>2</sup>

<sup>2</sup>*a*<sup>2</sup> + 2 *a*<sup>1</sup>

*a*1 <sup>2</sup> + *a*<sup>2</sup> 2

*a*1 <sup>2</sup> + *a*<sup>2</sup> 2

<sup>2</sup>*b*2*a*<sup>1</sup>

<sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>1</sup> 3*a*2

<sup>2</sup>*b*1*b*<sup>2</sup> + *a*1*b*<sup>1</sup>

 *a*1 <sup>2</sup> + *a*<sup>2</sup> <sup>2</sup><sup>3</sup>

 <sup>−</sup>1.5 1 0.05 −1.1 1.2 2

<sup>2</sup> − *<sup>b</sup>*2*a*1) (*a*<sup>1</sup>

(*a*<sup>1</sup> <sup>2</sup> + *a*<sup>2</sup> 2) 2

<sup>2</sup> + 9 *a*<sup>1</sup>

<sup>4</sup>*a*<sup>1</sup> <sup>−</sup> <sup>6</sup> *<sup>a</sup>*2*b*1*b*<sup>2</sup> <sup>−</sup> <sup>2</sup> *<sup>a</sup>*<sup>1</sup>

<sup>6</sup> + 9 *a*<sup>2</sup>

<sup>2</sup>*a*2*b*1*b*<sup>2</sup>

<sup>3</sup>*b*1*b*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>3</sup> *<sup>b</sup>*<sup>1</sup>

*<sup>c</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>a</sup>*<sup>1</sup>

*c*<sup>3</sup> = (3 *b*2*a*<sup>1</sup>

] − 3 *a*<sup>1</sup>

*<sup>b</sup>*<sup>3</sup> <sup>=</sup> *<sup>a</sup>*2*a*<sup>1</sup>

*δ*2 <sup>2</sup> =

+9 *a*1*a*<sup>2</sup>

*<sup>a</sup>*<sup>3</sup> <sup>=</sup> *<sup>a</sup>*2*b*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*2*a*<sup>1</sup> *a*1 <sup>2</sup> + *a*<sup>2</sup> 2

(*a*<sup>1</sup>

 *a*<sup>1</sup> *b*<sup>1</sup> *δ*<sup>1</sup> *a*<sup>2</sup> *b*<sup>2</sup> *δ*<sup>2</sup>

S\_2

–5

0

5

y

10

15

Using Homoclinic test approach, Extend Homoclinic test approach, Three-wave method and Introducing parameters and small perturbation method, we obtain novel solutions of Potential Kadomtsev-Petviashvili equation and Kadomtsev-Petviashvili equation such as periodic solitary wave,breather solitary wave, breather homoclinic wave, breather heteroclinic wave, cross kink wave,kinky kink wave, periodic kink wave, two-solitary wave, doubly periodic wave, doubly breather solitary wave. Moreover, we observed that there were differently spatiotemporal features in two-wave and multi-wave propagations including the degeneracy of soliton, periodic bifurcation and soliton deflexion of two-wave, fission and fusion of breather two-wave and so on. In future, we intend to study the stability and the interactions patterns of *N*-wave solutions in KP equation. What's more, can we obtain similar results to another integrable or non-integrable system? How can one use the soliton fission and fusion of models to study the practically observed soliton fission and fusion in the experiments?
