# Separation transformation and a class of exact solutions to the higher-dimensional Klein-Gordon-Zakharov equation.

1. Introduction

The Klein-Gordon equation (sometimes called Klein-Gordon-Fock equation) [1] is a relativistic version of the Schrodinger equation. Its nonlinear counterpart is the nonlinear Klein-Gordon equation [2]:

[[partial derivative].sup.2]u/[partial derivative][t.sup.2] - [[partial derivative].sup.2]u/[partial derivative][x.sup.2] + [alpha]u - [beta][[absolute value of u].sup.2]u = 0, (1)

where [alpha] and [beta] are constants, which has important applications in various fields. For example, it is attributed to the classical [u.sup.4] field theory in the physics of elementary particles and fields, and it can describe the propagation of dislocations within crystals and the propagation of magnetic flux on a Josephson line, and so on. One extension of the nonlinear Klein-Gordon equation is the (1 + 1)-dimensional Klein-Gordon-Zakharov (KGZ) equation [3, 4]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

with u(t, x) as a complex function and v(t, x) as a real function, which is a classical model describing the interaction of the Langmuir wave and the ion acoustic wave in plasma [3, 4]. The variable u(t, x) denotes the fast time scale component of electric field raised by electrons and the variable v(t, x) denotes the deviation of ion density from its equilibrium. In recent years, some authors applied analytical and numerical methods [5-8] to solve the (1 + 1)-dimensional KGZ equation (2) and obtained many exact and numerical solutions.

The high-dimensional extension of KGZ equation is important in real applications, so in this paper we would like to investigate the (n + 1)-dimensional KGZ equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [DELTA] = [[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.1] + [[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.2] + ... + [[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.n] = [[summation].sup.n.sub.j=1]([[partial derivative].sup.2]/[partial derivative][x.sup.2.sub.j]) is the Laplacian operator and x [member of] [R.sup.n]. This equation is the generalization of the KGZ equation (2) and we will show that it has many exact solutions with an arbitrary function.

More recently, Wang [9] extended the separation transformation method proposed in [10-12] to the (N + 1)dimensional coupled nonlinear Klein-Gordon equations. Then Liu et al. [13] and we [14] further extended the separation transformation method to various high-dimensional nonlinear soliton equations and obtained explicitly many exact solutions with arbitrary functions.

In this paper, by means of the separation transformation approach [9-14] we derive the exact solutions of the (n + 1)-dimensional KGZ equation (3). The rest of this paper is organized as follows. In Section 2, a separation transformation is presented and the (n + 1)-dimensional KGZ equation (3) is reduced to a set of partial differential equations and two nonlinear ordinary differential equations. In Section 3, the two nonlinear ordinary differential equations are solved and some special exact solutions of the (n + 1)-dimensional KGZ equation (3) are constructed explicitly. Conclusions are presented in Section 4.

2. Separation Transformation and Its Application

The following proposition reveals the relationship between the exact solutions of the (n + 1)-dimensional KGZ equation (3) and two nonlinear ordinary differential equations (ODEs) along with a set of partial differential equations (PDEs).

Proposition 1. The functions u(t; [x.sub.1], ..., [x.sub.n]) = U[[omega](t; [x.sub.1],..., [x.sub.n])] exp[iX(t; [x.sub.1], ..., [x.sub.n])] and v(t; [x.sub.1], ..., [x.sub.n]) = V[[omega](t; [x.sub.1], ..., [x.sub.n])] solve the (n + 1)-dimensional KGZ equation (3) if the functions [omega] = [omega](t; [x.sub.1], ..., [x.sub.n]), X = X(t; [x.sub.1], ..., [x.sub.n]), U([omega]) = U[[omega](t; [x.sub.1], ..., [x.sub.n])], and V([omega]) = V[[omega](t; [x.sub.1], ..., [x.sub.n])] solve the following set of differential equations:

[[omega].sub.tt] - [DELTA][omega] = 0, (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where U" = [d.sup.2]U/d[[omega].sup.2], V" = [d.sup.2]V/d[[omega].sup.2], and [K.sub.1] > 0 and [K.sub.2] are constants.

Proof. Assume that the (n + 1)-dimensional KGZ equation (3) has the following solution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where [omega](t; [x.sub.1], ..., [x.sub.n]) and X(t; [x.sub.1], ..., [x.sub.n]) are functions to be determined.

Substituting (6) into the (n + 1)-dimensional KGZ equation (3) yields two ODEs:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

If asking [omega](t; [x.sub.1], ..., [x.sub.n]) and X(t; [x.sub.1], ..., [x.sub.n]) to solve

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where [K.sub.1] and [K.sub.2] are auxiliary constants, then (7) are reduced to two nonlinear ODEs of functions U([omega]) and V([omega]) as

V"([omega]) + [gamma]([U.sup.2]([omega]))" = 0,

[k.sub.1]u"([omega]) + ([alpha] - [K.sub.2])U([omega]) + U([omega])v([omega]) - [beta][U.sub.3]([omega]) = 0. (9)

This finishes the proof of the proposition.

We see that under the separation transformation (6) the (n + 1)-dimensional KGZ equation (3) is separated into two sets of differential equations, namely, the PDEs in (8) and ODEs in (9). If we can obtain the exact solutions of the differential equations (8) and (9), the explicitly exact solutions of the (n + 1)-dimensional KGZ equation (3) can be built immediately. In what follows, we solve the PDEs in (8) firstly.

When n = 1, the PDEs of functions w(t; [x.sub.1]), X(t; [x.sub.1]) in (8) become

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

which has the following general solutions:

[omega](t; [x.sub.1]) = [c.sub.1]([x.sub.1] + t) - [K.sub.1]/4[c.sub.1]([x.sub.1] - t) + [c.sub.2],

X(t; [x.sub.1]) = [d.sub.0]t + [d.sub.1][x.sub.1] + [d.sub.2], (11)

where [d.sup.2.sub.0] = [d.sup.2.sub.1] + [K.sub.2], [d.sub.0](4[c.sup.2.sub.1] + [K.sub.1]) + [d.sub.1]([K.sub.1] - 4[c.sup.2.sub.1]) = 0, and [c.sub.2], d2 are integral constants.

When n [greater than or equal to] 2, the PDEs in 8) have the following general solutions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where f = f(*) is an arbitrary function, [c.sub.3], [c.sub.4], and [d.sub.3] are constants, and [k.sub.0], [k.sub.j], [[lambda].sub.0], [[lambda].sub.j], [l.sub.0], and [l.sub.j] (j = 1, ..., n) are constants satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Thus we conclude in this section that the exact solution of the (n + 1)-dimensional KGZ equation (3) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where U([omega]) and V([omega]) satisfy ODEs (9) and w(t; [x.sub.1], ..., [x.sub.n]), X(t; [x.sub.1], ..., [x.sub.n]) are functions given by (11) for n = 1 and (12) with (13) for n [greater than or equal to] 2.

It is remarked that when n [greater than or equal to] 2, there is an arbitrary function f([[summation].sup.n.sub.j=1] [k.sub.j][x.sub.j] + [k.sub.0]t + [c.sub.3]) in each exact solution of the (n + 1)-dimensional KGZ equation (3), which may reveal abundant nonlinear structures in this nonlinear equation.

3. New Exact Solutions of the (n+1)-Dimensional KGZ Equation (3)

In this section, we search for the exact solutions of the nonlinear ODEs in (9) by means of the F-expansion method proposed by Wang et al. [15-17]. Based on the explicit solutions of the ODEs in (11), many exact solutions of the (n + 1)-dimensional KGZ equation (3) are obtained explicitly via the separation transformation (6).

Integrating the first equation in (9) we have

V([omega]) = -[gamma][U.sup.2]([omega]). (15)

Thus the second equation in (9) becomes an ODE of U([omega]) as

[K.sub.1]U"([omega]) + ([alpha] - [K.sub.2])U([omega]) - [gamma][U.sup.3]([omega]) - [beta][U.sup.3]([omega]) = 0. (16)

In what follows, we solve the ODE (16) by using the extended F-expansion method proposed by Wang et al. [15-17]. In doing so, assume that the solution of ODE (16) is

U([omega]) = [a.sub.0] + [a.sub.1]F([omega]) + [b.sub.1]/F([omega]), (17)

where [a.sub.0], [a.sub.1], and [b.sub.1] are constants to be determined and F([omega]) satisfies the elliptic equation [18]:

[(dF([omega])/d[omega]).sup.2] = r + [qF.sup.2]([omega]) + [pF.sup.4]([omega]), (18)

whose solutions in Jacobi elliptic function forms [18]are listed in Table 1 in the Appendix.

Substituting (17) with 18) into ODE 16), we find that the variables [a.sub.0], [a.sub.1], [b.sub.1], and [alpha] have two groups of solutions.

Group 1. Consider that

[a.sub.0] = 0, [b.sub.1] = 0, [alpha] = [K.sub.2] - [K.sub.1]q, [a.sub.1] = [square root of 2[K.sub.1]p/[gamma] + [beta]]. (19)

Group 2. Consider that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Combining the separation transformation (6) with (15), (17), (19), or (20) along with (11)-(13) and the solutions of elliptic equation (18) in Table 1 yields the special exact solutions of the (n + 1)-dimensional KGZ equation (3) as follows.

Solution 1. Jacobi elliptic sn-function solution is as follows:

u(t; [x.sub.1], ..., [x.sub.n]) = m[square root of 2[K.sub.1]/[gamma] + [beta]]sn([omega], m)[e.sup.iX],

v(t; [x.sub.1], ..., [x.sub.n]) = -[2[K.sub.1][m.sup.2][gamma]/[gamma] + [beta]][sn.sup.2]([omega], m), (21)

where sn([omega], m) is Jacobi elliptic sn-function with modulus 0 [less than or equal to] m [less than or equal to] 1 and the functions w and X are given by (11) for n = 1 and (12) with 13) for n [greater than or equal to] 2. When m [right arrow] 1, we have the soliton solution of the (n + 1)-dimensional KGZ equation (3) as

u(t; [x.sub.1], ..., [x.sub.n]) = [square root of 2[K.sub.1]/[gamma] + [beta]]tanh([omega])[e.sup.iX],

v(t; [x.sub.1], ..., [x.sub.n]) = -[2[K.sub.1][gamma]/[gamma] + [beta]][tanh.sup.2]([omega]). (22)

Solution 2. Jacobi elliptic cn-function solution is as follows:

u(t; [x.sub.1], ..., [x.sub.n]) = m[square root of -2[K.sub.1]/[gamma] + [beta]]cn([omega], m)[e.sup.iX],

v(t; [x.sub.1], ..., [x.sub.n]) = [2[K.sub.1][m.sup.2][gamma]/[gamma] + [beta]][cn.sup.2]([omega], m), (23)

where cn([omega], m) is Jacobi elliptic cn-function with modulus 0 [less than or equal to] m [less than or equal to] 1 and the functions [omega] and X are given by (11) for n = 1 and (12) with 13) for n [greater than or equal to] 2. When m [right arrow] 1, we have the soliton solution of the (n + 1)-dimensional KGZ equation (3 as

u(t; [x.sub.1], ..., [x.sub.n]) = [square root of -2[K.sub.1]/[gamma] + [beta]]sech([omega])[e.sup.iX],

v(t; [x.sub.1], ..., [x.sub.n]) = [2[K.sub.1][gamma]/[gamma] + [beta]][dn.sup.2]([omega], m), (24)

Solution 3. Jacobi elliptic dn-function solution is as follows:

u(t; [x.sub.1], ..., [x.sub.n]) = [square root of 2[K.sub.1]/[gamma] + [beta]]dn([omega], m)[e.sup.iX],

v(t; [x.sub.1], ..., [x.sub.n]) = [2[K.sub.1]/[gamma] + [beta]][dn.sup.2]([omega], m)

where dn([omega], m) is Jacobi elliptic dn-function with modulus 0 [less than or equal to] m [less than or equal to] 1 and the functions [omega] and X are given by (11) foi n = 1 and (12) with 13) for n [greater than or equal to] 2.

Solution 4. Jacobi elliptic ns-cs-function solution is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

where ns([omega], m) = 1/sn([omega], m) and cs([omega], m) = cn([omega], m)/sn([omega], m) are Jacobi elliptic functions with modulus 0 [less than or equal to] m [less than or equal to] 1 and the functions [omega] and X are given by (11) for n = 1 and (12) with 13) for n [greater than or equal to] 2.

Solution 5. Jacobi elliptic nc-sc-function solution is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

where nc([omega], m) = 1/cn([omega], m) and sc([omega], m) = sn([omega], m)/cn([omega], m) are Jacobi elliptic functions with modulus 0 [less than or equal to] m [less than or equal to] 1 and the functions [omega] and X are given by (11) for n = 1 and (12) with (13) for n [greater than or equal to] 2. Note that m [not equal to] 1 here.

Solution 6. Jacobi elliptic ns-ds-function solution is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

where ds([omega], m) = dn([omega], m)/sn([omega], m) is a Jacobi elliptic function with modulus 0 [less than or equal to] m [less than or equal to] 1 and the functions [omega] and X are given by (11) for n = 1 and (12) with 13) for n [greater than or equal to] 2.

Solution 7. Jacobi elliptic sn-ns-function solution is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

where the functions w and X are given by (11) for n = 1 and (12) with 13) for n [greater than or equal to] 2.

Solution 8. Jacobi elliptic cn-nc-function solution is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where the functions w and X are given by (11) for n = 1 and (12) with (13) for n [greater than or equal to] 2.

Solution 9. Jacobi elliptic dn-nd-function solution is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (31)

where nd([omega], m) = 1/dn([omega], m) is a Jacobi elliptic function with modulus 0 [less than or equal to] m [less than or equal to] 1 and the functions [omega] and X are given by (11) for n = 1 and (12) with 13) for n [greater than or equal to] 2.

Remark 2. It is noted that we can also list many other types of exact solutions for the (n + 1)-dimensional KGZ equation (3) by using the exact solutions of the elliptic equation (18) in Table 1.

When n = 2, denote [x.sub.1] = x and [x.sub.2] = y; we find that 3) becomes the (2 + 1)-dimensional KGZ equation as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

which has separation solution of the form

u(t, x, y) = U[[omega](t, x, y)]exp[iX (t, x, y)],

v(t, x, y) = V[[omega](t, x, y)], (33)

where U[[omega](t, x, y)] and V[[omega](t, x, y)] are expressed by Solutions 1-9 above and [omega](t, x, y) and X(t, x, y) satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where [c.sub.3], [c.sub.4], and [d.sub.3] are constants and [k.sub.0], [k.sub.1], [k.sub.2], [[lambda].sub.0], [[lambda].sub.1], [[lambda].sub.2], [l.sub.0], [l.sub.1], and [l.sub.2] are constants satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

In what follows, we take Solution 2 in (23)-(24) as an example to study the novel nonlinear structures in (2 + 1)-dimensional KGZ equation. To do so, choose a set of special solution of (35) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

Because f([k.sub.1]x + [k.sub.2]y + [k.sub.0]t + [c.sub.3]) = f([eta]) is an arbitrary function of [eta], we can select special form of f(q) to describe the nonlinear excitations of the (2 + 1)-dimensional KGZ equation.

Figures 1 and 2 show particular nonlinear wave structures at time t = 0, given by 33)-(34), (23), and (24) with 36) and the function selection

f([eta]) = [square root of [eta] + 1] (37)

and the choice of other parameters

[K.sub.1] = -1, [gamma] = 1, [beta] = 1,

[c.sub.3] = 0, [c.sub.4] = 0, [d.sub.3] = 1. (38)

Figure 3 demonstrates a particular nonlinear wave structure at time t = 0, given by (33)-(34) and (24) with parameters in (36) and 38) and the function selection

f([eta]) = sin([eta] + 1). (39)

It is observed from Figure 3 that the special choice of arbitrary function f([eta]) reveals a nonlinear soliton-like structure in the (2 + 1)-dimensional KGZ equation, which is periodic in the direction of plane [k.sub.1]x + [k.sub.2]y.

4. Conclusion

In conclusion, we have derived some exact Jacobi elliptic function solutions and soliton solutions of the (n + 1)-dimensional Klein-Gordon-Zakharov equation by separation transformation method proposed in [9, 10, 12]. It is shown that, for the high-dimensional case, that is, n [greater than or equal to] 2, there is an arbitrary function in every exact solution of the (n + 1)dimensional Klein-Gordon-Zakharov equation. For the case of n = 2 we demonstrate some novel nonlinear structures by choosing the arbitrary function f([k.sub.1] x + [k.sub.2]y + [k.sub.0]t + [c.sub.3]) specially. The separation transformation method may also be useful to solve other nonlinear wave models to explain the nonlinear excitations and localized nonlinear wave structures [19-23] in the physics of elementary particles and fields.

Appendix

In the Appendix, we present the relationships between the values of (p, q, r) and the corresponding solutions of the elliptic equation (18) in Table 1; see also 18].

In the following table, the functions sn([omega], m), cn([omega], m), and dn([omega], m) are three basic Jacobi elliptic functions with modulus 0 [less than or equal to] m [less than or equal to] 1. As seen in Table 1, the other Jacobi elliptic functions are the combinations of these three Jacobian elliptic functions. For example, as shown before we have cs([omega], m) = cn([omega], m)/sn([omega], m). When the modulus m = 1, the Jacobi elliptic functions degenerate as the hyperbolic functions: sn([omega], 1) = tanh([omega]), cn([omega], 1) = sech([omega]), and dn([omega], 1) = sech([omega]). When the modulus m = 0, they degenerate as the trigonometric functions or constant: sn([omega], 0) = sin([omega]), cn([omega], 0) = cos([omega]), and dn([omega], 0) = 1.

http://dx.doi.org/10.1155/2014/974050

Received 8 February 2014; Accepted 30 March 2014; Published 24 April 2014

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by Talent Youth Program of Beijing Municipal Commission of Education (Grant no. YETP0984), Scientific Research Common Program of Beijing Municipal Commission of Education (Grant no. SQKM201211232017), and Excellent Talent Program of Beijing (Grant no. 2012D005007000005).

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Jing Chen, (1) Ling Liu, (2) and Li Liu (3)

(1) School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China

(2) School of Science, Beijing Information Science and Technology University, Beijing 100192, China

(3) China Petroleum Engineering and Construction Corp., Beijing 100028, China

Correspondence should be addressed to Jing Chen; chenjingcufe@163.com
```
TABLE 1: Exact solutions of [(df(w)/dw).sup.2] = r +
[qF.sup.2] + [pF.sup.4]([omega])([omega]).

r                   q                   P

1            1           -(1 + [m.sup.2])        [m.sup.2]
2      1 - [m.sup.2]      2[m.sup.2] - 1        -[m.sup.2]
3      [m.sup.2] - 1       2 -[m.sup.2]             -1
4        [m.sup.2]       -(1 + [m.sup.2])            1
5       -[m.sup.2]        2[m.sup.2] - 1       1 - [m.sup.2]
6           -1                 2 2-m           [m.sup.2] - 1
7            1             2 - [m.sup.2]       1 - [m.sup.2]
8            1            2[m.sup.2] - 1        -[m.sup.2]
(1 - [m.sup.2])
9      1 - [m.sup.2]        2-[m.sup.2]              1
10           1           -(1 + [m.sup.2])        [m.sup.2]
11      -[m.sup.2]        2[m.sup.2] - 1             1
(1 - [m.sup.2])
12       [m.sup.2]       -(1 + [m.sup.2])            1
13          1/4          (1-2[m.sup.2])/2           1/4
14   (1 - [m.sup.2])/4   (1 + [m.sup.2])/2   (1 - [m.sup.2])/4
15      [m.sup.4]/4      ([m.sup.2] - 2)/2          1/4

F([omega])

1                    sn([omega], m)
2                    cn([omega], m)
3                    dn([omega], m)
4      ns([omega], m) = [[sn([omega],m)].sup.-1]
5       nc([omega], m) = [[cn([omega],m)].sup.1]
6      nd([omega], m) = [[dn([omega],m)].sup.-1]
7    sc([omega], m) = sn([omega], m)/cn([omega], m)
8    sd([omega], m) = sn([omega], m)/dn([omega], m)

9    cs([omega], m) = cn([omega], m)/sn([omega], m)
10   cd([omega], m) = cn([omega], m)/dn([omega], m)
11   ds([omega], m) = dn([omega], m)/sn([omega], m)

12   dc([omega], m) = dn([omega], m)/cn([omega], m)
13          ns([omega], m) + cs([omega], m)
14          nc([omega], m) + sc([omega], m)
15          ns([omega], m) + ds([omega], m)
```