Construction of Exact Traveling Wave for KP-Type Equation Based on Symbolic Computation - Advanced Research on Computer Education, Simulation and Modeling - page 280

Construction of Exact Traveling Wave for KP-Type
Equation Based on Symbolic Computation

Qingfu Li 1,* and Ruihua Cheng 2

1 School of mathematics and information science, Pingdingshan University,
Pingdingshan 467000, China
2 Department of mathematics and information science,
Henan University of Finance and Law, Zhengzhou, 450002, China
[e-mail]

Abstract. The (2+1)-dimensional KP-type equation arising from the soliton hi-
erarchy associated with new spectral problem is studied, it is the compatible
condition of Lax triad. With the aid of symbolic computation system Mathe-
matica, theta function periodic solutions for the (2+1)-dimensional KP-type
equation are constructed by the auxiliary equation method.

Keywords: Riemann-Theta function, the (2+1)-dimensional KP-type equation,
auxiliary method.

1 Introduction

There are many mathematical models described by nonlinear partial differential Equa-
tion (NLPDE), especially some basic equations in physics and mechanics, To investi-
gate the exact solutions for these NLPDE plays an important role in the study of
nonlinear physical phenomena. In recent years, direct search for exact solutions to
NLPDE has become more and more attractive partly due to the availability of com-
puter symbolic systems like Maple or Mathematica which allows us to perform some
complicated and tedious algebraic calculation on computer, and helps us to find new
exact solutions to NLPDE, such as Homogeneous balance method[1], tanh-function
method[2], sine-cosine method[3], Jacobi elliptic functions method[4], F-expansion
method[5,6] and so on. In this paper, we apply auxiliary equation method [7] to seek
exact t h e t a function periodic solutions for (2+1)-dimensional KP-type equation:

3 1 3

⎛ ⎞ − 1 2

ω =∂ ω + ω − ω

⎠ . (1)

⎜ ⎟

t x yy xx

4 16 2

⎝ x

By taking full advantages of elliptic equation:

2 4

Fx ABFx CFx ′ () =+ () + () . (2)

* This work has been supported by the Basic and advanced technology projects of Henan Prov-
ince (No.102300410262) and the National Science Foundation of Henan Province
(No.2010A110001).