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Construction of Exact Traveling Wave for KP-Type Equation

Based on Symbolic Computation

Junmin Wang a Ruihua Cheng b

b Department of mathematics and information science,Henan University of Finance and Law, Zhengzhou,450002,China Abstract

The (2+1)-dimensional KP-type equation arising from the soliton hierarchy associated with new spectral problem is studied, it is the compatible condition of Lax triad. With the aid of symbolic computation system Mathematica, 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 Equation (NLPDE), especially some basic equations in physics and mechanics, To investigate 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 computer 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 the t a function periodic solutions for (2+1)-dimensional KP-type equation:

123

13416

2t x yy xx x ωωωω-

??=?+- ???. (1) By taking full advantages of elliptic equation:

24()()()F x A BF x CF x '=++. (2)

and get some traveling wave solutions in terms of theta functions with the aid of symbolic computation for the first time. The (2+1)-dimensional KP-type eqation (1) arises from the soliton hierarchy associated with spectral problem [8]:

2,2(ln )/1

u v U U x v r v u λφφλ-+==+-?? ???, (3) where (,)T

q p φ=,1r is constant.

This paper is arranged as follows. In section 2, we shall illustrate the auxiliary equation method; In section 3, we apply auxiliary method [12] to seek exact solutions of (2+1)-dimensional KP-type equation; Some conclusions are given in section 4.

2. Method of Solution

For the elliptic equation (2), the following fact is needed to realize the aim of this paper.

Proposition: If we take 2224(0)(0)C A ??=-=- and 2224(0)(0),B ??=- then

()13()/()

F z z z ??= satisfies the elliptic equation (2), where theta functions are defined as following

2[]()exp{()2()()}222n z i n n z εεεε?τπτε∞=-∞'=++++'?? ???∑ ()()[](),i i i z z z ??τ?ετ=

12341100,,,1001εεεε====????????

????????????????

1,2,3,4i =. In the following, we seek traveling wave solution in the formal solution for the equation (1)

()()(0)01N N n u a a F a i i ξξ=+≠∑= (4)

by taking

(;;;;)(),

12.122u x x x t u l x k x k x t l l ξξλ==++++ (5)

where ,,,,12k k k l

λ are constants to be determined, and ,,,a A B C i are constants to be determined, ()F ξ satisfies elliptic equation (2) .

Step 1: Inserting (4) into (1) yields an ODE for ()u ξ:

(;;;)0P u u u '''= . (6)

Step 2: Determine N by considering the homogeneous balance between the governing nonlinear term(s) and highest order derivatives of ()u ξ in (6).

Step 3: Substituting (4) into (6), and using (2), and then the left-hand side of (6) can be

converted into a finite series in ()()0,1,,k F k M ξ= .

Step 4: Equating each coefficient of ()k F ξ to zero yields a system of algebraic equations for

()0,1,,a i N i = .

Step 5: Solving the system of algebraic equations, with the aid of Mathematica or Maple, ,,a k i i λ can be expressed by A, B, C (or the coefficients of ODE (6)).

Step 6: Substituting these results into (4), we can obtain the general form of travelling wave solutions for equation (1).

Step 7: From propositon, we can give theta function periodic solutions for equation (1).

3. Exact Solutions

In this section, we will make use of the auxiliary equation method and symbolic computation to find the exact solutions to the mKdV-ZK equation.

We assume that (1) has travelling wave solution in the form

(;;)(),.u x y t u x y t ξξαβω==++ (7)

Substituting (7) into (1), then (1) is transformed into the following form:

4223()()3()()0416U U U U αβαωξξαξξ??'''''-+-= ???

(8) According to step 2 in section 2, by balancing ()U ξ''' and ()()U U ξξ' in (8), we obtain 2n =, and suppose that Eq. (8) has the following solutions:

2()()(),012U a a F a F ξξξ=++ (9)

Substituting (9) along with Eq. (2) into (8) yields a polynomial equation in ()F ξ. Setting their coefficients to zero yields a set of algebraic equations for unknown parameters ,,,012a a a ω. Solving these equations, we can get the following solutions:

2230,,(312)/4.1204

a a C a a B αωβαα===-+ (10) and ,,,0a A B C are arbitrary constants. The traveling wave solution of the KP-type equation are given by

02()(),4C U a F αξξ=+

where x y t ξαβω

=++, ,,,,A B C αβ are arbitrary constants, and 2,a ω are given in (10). From the proposition ，if we choose 2224(0)(0)C A ??=-=- and 2224(0)(0),B ??=- we can get the solution to the KP-type equation in terms of theta functions:

(,,)u x y t =2()1,024()3

C a ?ξα?ξ+ (11) where x y t ξαβω=++, ,,,,A B C αβ are arbitrary constants, and 2,a ω are given in (10). To grasp the characteristc of solutions of (1) , we dipict the figure of the solution (,,)u x y t by using the mathematica, their properties and profiles are displayed in figures (1)-(4) under chosen parameters: 3,1a c d ===, 1,8,0αβγ===, 1C =- and 2t = for 2

D figure (3), 2x = for 2D figure

(4).

From figures (1)-(4), it is easy to see that the solution (,,)u x y t is periodic wave solution.

Fig. 1. Perspective view of the wave (,,)

u x y t.

Fig. 2. Overhead view of the wave(,,)

u x y t.

Fig. 3. The propogation of the wave along x-axis.

Fig. 4. The propogation of the wave along y-axis.

4.Conclusions

In this paper, we have studied the KP-type equation. By using auxiliary equation method, some traveling wave solutions in terms of theta functions are successfully obtained with the aid of symbolic computation for the first time, they should be meaningful to explain some physics phenomena. It is shown that the auxiliary equation method is a very effective and powerful mathematical tool for solving nonlinear evolution equations in mathematics and physics. Moreover, with the aid of computer symbolic systems (Mathematica or Maple), the method can be conveniently operated .

Acknowledgements

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

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