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In the paper, we will discuss the Kadomtsev-Petviashvili Equation which is used to model shallow-water waves with weakly non-linear restoring forces and is also used to model waves in ferromagnetic media by employing the method of variable separation. Abundant exact solutions including global smooth solutions and local blow up solutions are obtained. These solutions would contribute to studying the behavior and blow up properties of the solution of the Kadomtsev-Petviashvili Equation.

The Kadomtsev-Petviashvili (KP) equation [

where

This means that the two KP equations have different physical structures and different properties [

It is well known that searching for exact solutions of nonlinear evolution equation arising in mathematical physics plays an important role in nonlinear science fields, since they can provide much physical information and more insight into the physical aspects of the problem and thus lead to further applications [

This paper will study global smooth solution and local blow up solution of the KP equation by means of the method of variable separation [

We consider the KP-I equation

Setting

Now we suppose the additive separable solution of Equation (3) as

where

Substituting Equation (4) into Equation (3), we discover that

by simple transposition, we get

In order to obtain nontrivial solution of separation of variables, we demand that

Case 1:

In this case, Equation (6) is reduced to

by solving Equation (7), We can be easy to get

So, the global smooth solution of Equation (2) is

where C_{1}, C_{2} and C_{3} are arbitrary constants.

Case 2:

In this case, Equation (6) is transformed into

The left side of the Equation (10) is the function about variable z, and the right side is a function about variable y, so

where C_{1}, C_{2} and C_{3} are undetermined constants.

Substituting Equation (11) into

In the meantime, Equation (10) is transformed into two order homogeneous linear differential equation with constant coefficients as follows

by solving Equation (13), We obtain

where C_{4} and C_{5} are arbitrary constants.

So, in this case, the global smooth solution of Equation (2) is

where C_{4} and C_{5} are arbitrary constants.

Case 3:

In this case, It is assumed that

by assumption, we get

where C_{1}, C_{2} and C_{3} are undetermined constants.

Substituting Equation (17) into

We obtain two group of solutions by solving Equation (18) as follows

1)

2)

Accordingly, the equation

Solving Equation (19), we have_{4} and C_{5} are arbitrary constants.

Solving Equation (20), we have

So, we obtain two group of global smooth solutions of Equation (2) as follows:

where C_{3}, C_{4} and C_{5} are arbitrary constants, and

Case 4:

In this case, Equation (6) is transformed into

Solving

where C_{1}, and C_{2} are undetermined constants.

Substituting Equation (24) into the equation

Solving the equation

where C_{3} and C_{4} are arbitrary constants.

So, we obtain the global smooth solutions of Equation (2) as follows:

where C_{2}, C_{3} and C_{4} are arbitrary constants, and

We look for separable solution of the multiplicative form of Equation (3)

where

Plugging the form (27) into the nonlinear diffusion Equation (3), we obtain

Then

where

Solving Equation (30), we will discuss both cases as follows:

Case 1:

when

Substituting Equation (31) into Equation (32), we have

Solving Equation (33), the solution _{3} is a arbitrary constant.

So, in this case, the Equation (2) possesses local blow up solution as follows

where C_{1} and C_{3} are arbitrary constants with

Case 2:

when

Substituting Equation (31) into Equation (35), we have_{1} is a arbitrary constant.

Solving Equation (36), the solution _{5} and C_{6} are arbitrary constants.

In this case, we can not get the blow up solution of Equation (2).

It is well known that the method of variable separation is one of the most universal and efficient means for studying linear partial differential equations. Several methods of variable separation for nonlinear partial differential have been suggested until recently. This paper applies the method of variable separation to obtain global smooth solutions and local blow up solutions of the KP equation. These solutions can be used to qualitative or numerical analysis for properties of the KP equation. In the future, we will try to seek for the generalized variable separation solutions by the form of solution