Construction of Exact Traveling Wave for KP-Type Equation Based on Symbolic Computation - Advanced Research on Computer Education, Simulation and Modeling - page 283

α

2 2 3

a = 0, a = C , ω = (3 β − 12 α a + a B ) / 4 α . (10)

1 2 0

4

and

0 aABC are arbitrary constants.
The traveling wave solution of the KP-type equation are given by

, , ,

α C

2

Ua () ξ =+ F (), ξ

0

4

where ξ =++ α x β y ω t , ABC ,,,, α β are arbitrary constants, and a

2 ,

ω are given

in (10).
From the proposition, if we choose

2 2 2

2 (0)

B = ϑ −

CA =− =− ϑϑ (0) (0) and

2 4

2
4 (0),

− ϑ

we can get the solution to the KP-type equation in terms of theta functions:

2 ()

ϑ ξ

α C

1

uxyt (, ,) = a + , (11)

0

2

4

ϑ () ξ

3

a

2 ,

ω

where, are given in (10).

To grasp the characteristc of solutions of (1), we dipict the figure of the solution
(, ,)

ABC ,,,, α β are arbitrary constants, and

uxyt by using the mathematica, their properties and profiles are displayed in fig-
ures (1)-(4) under chosen parameters:
3,

ac d === 1 , α === 1, βγ 8, 0 , C =− 1 and t = 2 for 2D figure (3), x = 2

for 2D figure (4).
From figures (1)-(4), it is easy to see that the solution uxyt is periodic wave ( , , )

solution.