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0Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 170372, 5 pages http://dx.doi.org/10.1155/2013/170372
Research Article
The Investigation of Solutions to the Coupled Schrodinger-Boussinesq Equations
Xin Huang1,2
1 College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
2 Department of Basic Courses, Sichuan Finance and Economics Vocational College, Chengdu 610101, China
Correspondence should be addressed to Xin Huang; huangxinnv@163.com Received 12 May 2013; Accepted 2 June 2013 Academic Editor: Shaoyong Lai
Copyright © 2013 Xin Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The (G'/G)-expansion method and the symbolic computation system Mathematica are employed to investigate the coupled Schrodinger-Boussinesq equations. The hyperbolic function solutions, trigonometric function solutions, and rational function solutions to the equations are obtained. The decaying properties of several solutions are analyzed.
1. Introduction
In laser and plasma physics, the important problems under interactions between a nonlinear complex Schrodinger field and a real Boussinesq field have been raised. In particular, the study of the coupled Schroodinger-Boussinesq equations has attracted much attention of mathematicians and physicists (see [1-3]). The existence of the global solution of the initial-boundary problem for the equations was investigated in [1]. The existence of a periodic solution for the equations was considered in [2]. Kilicman and Abazari [3] used the (G'/G)-expansion method to construct periodic and soliton solutions for the Schrodinger-Boussinesq equations iut + uxx -auv = 0, vtt - vxx + vxxxx - b(M2)xx = 0, where a and b are real constants. The investigation of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena (see [4-7]).
In this paper, we consider the following coupled Schroodinger-Boussinesq equations:
iEt + Exx + ß1E = NE, 3Ntt - Nxxxx + 3(N2)xx + ß2Nxx = (\E\2)xx,
where E(x, t) is a complex unknown function, N(x, t) is a real unknown function, and p1 and p2 are real positive constants. System (1) is known to describe various physical
processes in laser and plasma physics, such as formation, Langmuir field amplitude, intense electromagnetic waves, and modulational instabilities (see [8]). The approximate solutions and conservation law for the coupled system (1) have been studied in [9]. In [10], Chen and Xu used the F-expansion method to obtain a number of periodic wave solutions expressed by various Jacobi elliptic functions for (1). Cai et al. [11] studied same equations by the modified F-expansion method.
In the present paper, we use the (G'/G)-expansion method and the symbolic computation system Mathematica to investigate the coupled Schrodinger-Boussinesq system (1). Here, we state that the previous works do not obtain the solutions presented in this paper.
The layout of this paper is as follows. In Section 2, we give the description of the generalized (G'/G)-expansion method. In Section 3, we apply this method to solve (1). A conclusion will be obtained in Section 4.
2. Brief Description of the (G'/G)-Expansion Method
To make our presentation self-contained, we recall the (G'/G)-expansion method. The details can be found in Wang et al.'s work [12].
Step 1. For a given PDE with two independent variables t and
...) = 0,
P (m, Mt, M„, M^, M we convert it into an ODE
' ,...) = 0.
n i ' '' ''' P ( M, M , M , M
Using travelling transformation w(x, i) = w(£), £ = x - fci. Equation (3) can be integrated as long as all terms contain derivatives where integration constants are considered to be
where m(x, i) is a real function, fc, / are constants to be determined, and is an arbitrary constant. Substituting (7) into (1) yields
Mt + 2fcMx = 0, (8)
mxx + -^i)M = NM, (9)
- + 3(N2)^ + ^ = (M2)_. (10)
Step 2. Suppose that the solution of (3) can be expressed as a polynomial in (G'/G)
■«>=i,(|).
where G = G(£) satisfies the second-order ODE with respect to Namely
G" + AG' + ^G = 0,
where «j,..., an = 0, A, and ^ are constants to be determined later. The positive integer n can be determined by balancing the highest-order derivatives with highest-order nonlinear terms appearing in (3). It is easy to check that (5) admits three types of solutions
V« /q sinh (1/2)V^> + c2 cosh (1/2) A
2 Vq cosh (1/2)V«^ + c2 sinh (1/2) V«^/ 2'
« > 0,
v-«/'-c1 sin (1/2)v—5^+c2 cos (1/2)V~«^ A
2 V c1 cos (1/2>V^^+c2 sin (1/2>V^W 2
a < 0,
a = 0,
Ci + C2^ 2
in which a = A2 - 4^.
Step 3. By substituting (4) into (3) and using (5), collecting all terms with the same order of (G'/G) together, the left-hand side of (3) can be written as a polynomial in each coefficient of this polynomial be zero yields a system of algebraic equations for a^..., a„, fc, A, and
Step 4. Since the general solutions of (5) have been known, substituting a„, fc, A, and ^ into (4), we can obtain
travelling wave solutions of the nonlinear evolution equation (2).
3. Solutions of the Coupled
Schrodinger-Boussinesq Equations
Following the procedure described in Section 2, we adopt the ansatz solution of (1) in the form
We take
m (x, i) = m (£) = m (x - 2fci + ,
where is an arbitrary constant. Substituting (11) into (9), one gets
NM^-C^-A)-
Suppose that
= v (£> = v (x-2fa +
It follows from (9), (10), (11), and (13) that
m'' - (/ + fc2 - ) m - MV =0, -v'' + (l2fc2 + ^2) v + 3v2 - m2 = 0.
Balancing w'' with wv in (14) and v'' with w2 in (15) leads to m = 1, n = 2. Thus we can search for the solutions of (14) and (15) in the following forms:
M(O = a0 + fli ( fWf i >
v (ç>=&o+b ( gg ( gg
fl2 = 0, (16) b2 = 0. (17)
Substituting (16) and (17) into (14) and (15), using (5), and setting the coefficients of (G'/G)' (î = 0,..., 4) to be zero,
we obtain the algebraic system
6a2 - = 0, 2a1 + \0a2X - a2b1 - a1b2 = 0, 8a2^ + 3a1A + 4a2X2 - lb2 - k2b2
+ №2 - a2b0 - 0^1 - a0b2 = 0,
6a2X^ + 2a1^ + a1X2 - lb1 - k2b1 + ^1b1 - a1b0 - a0b1 = 0, 2a2y2 + a1X^ - lb0 - k2b0 + p1b0 - a0b0 = 0, 6b2 - + a2 = 0, 2b1 + 10b2X - 6b1 b2 + 2a1a2 = 0, 8b2^ + 3b1X + 4b2X2 - 12k\ + p2b2 - 3bf + 6b0b2 + af + 2a0a2 = 0, 6b2X^ + 2b1^ + b1X2 - \2k2b1 - p2b1 - 6b0b1 + 2a0a1 = 0,
2b2y2 + b1X^ - \2k2b0 - ß2b0 - 3b£ + a0 = 0.
Solving this system with the Mathematica, we find a0 = ±(^2X2 + a1 = ±6^2X, a2 = ±6^2, b0 = X2 + b1 = 6X, b2 = 6,
k = ±^3(?2 + X2 - 4 I = 1 (12^1 + p2 + X2 + 12V2X2 -4p± 48V2p) ,
a0 = ±6^2^, a1 = ±6^2X, a2 = ±6^2, b0 = 6^, b1 = 6X, b2 = 6,
k = ±6^3(02 -V + 4rf,
I = 12 (12^1 + -X2 ± 12^2X2 +4p + 48^2p) , where X and ^ are arbitrary constants.
By using (19) and (20), the solutions (16) and (17) are written as
u(Ç) = ±(^2X2 +2^)±6^2x( — ) ± 6^2,( — ) ,
where Ç = x± (\/3)^3(ß2 + X2 - 4p)t + ^ or u(Ç) = ±6^2p±6^2x(^) ±6^2(^
V (S) = 6l4+6\(^ )+6Î^ ) ,
where Ç = x ± (1/3)^3(p2 -X2 + 4p)t +
Substituting general solutions of (5) into (21) and (22), we obtain three types of travelling wave solutions of the coupled Schrodinger-Boussinesq equations as follows.
3.1. The Hyperbolic Function Solutions to (1) If a = X2-4p > 0. Consider
E1 (x,t) = ±-
3. q sinh (l/2)^a<; + c2 cosh (1/2) ^
x e N1 (x, t)
c1 cosh (1/2) + c2 sinh (1/2) -^a^ J
i(kx+lt+Ç0)
3. q sinh (1/2)^a£, + c2 cosh (1/2) ) 1 c1 cosh (1/2) + c2 sinh (1/2) -JœÇ J
where k = ±(1/6)^3(p2 + a), I = (1/12)(12^1 + ¡32 + X2 + 12^2X2 -4p± 48^2^, c1, c2 are arbitrary constants, and
E2 (x, t) 3^2a
c1 sinh (1/2)^a^ + c2 cosh (1/2) ^a^)2 1 q cosh (1/2) ^x^ + c2 sinh (1/2) ^al; )
N2 (x, t)
q sinh (1/2) + c2 cosh (1/2) 2 q cosh (1/2) ^ + c2 sinh (1/2) ^x^ )
where k = ±(1/6)y3(ft -a), I = (1/12)(12ft + ft - A2 ± where k = ±(1/6)y3(ft - a), I = (1/12)(12ft + ft - A2 ±
12V2A2 +4^ + 48V2^).
Remark 1. When £ ^ œ>, we find |E1(%, i)| ^ N1(x,i) ^ a, |£2(x,f)| ^ 0,N"2(%,i) ^ 0.
±V2a,
Remark 2. If c1 =0, c2 = 0, A > 0, ^ = 0, we find envelope solitary wave solutions for (1). Namely, £1 (x, i) and N1 (x, i) become
El (x,i) = ±^A2 (3tanh21) e'^^,
N1 (x,i) = iA2 (3tanh2 A?-1). £2(x, i) and N"2(%, i) are turned into
£2 (x,i) = ±^A2 (tanh2 A^-l)e N2 (x;î) = 2a2 (tanh2 2ç-l).
i(fcx+it+ç0)
3.2. The Trigonometric Function Solutions to (1) if a < 0. Consider
£3 (x, i)
V2a = ±-
31 -Ci sin (1/2) + cos (1/2) a/-^ . + I
x e N3 (x, i)
ci cos (i/2> y-a^ + c2 sin (i/2> y-a^
i(fcx+lt+ç0)
3(-ci sin(1/2) y-a£ + c2 cos (1/2) y-«£)2 i V ci cos (1/2) y—+ c2 sin (1/2) y—/
where fc = ±(1/6)y3(ft + a), I = (1/12)(12ft + ft + A2 +
12y2A2 - ± 48^), and
£4 (x, i)
x e N4 (x, i)
-ci sin(1/2)y—+ c2 cos(1/2)V— ci cos(1/2)y—a£ + c2sin(1/2)y—/
i(fcx+lt+?0)
-ci sin(1/2)y—+ c2 cos(1/2)V—2 ci cos(1/2)y-«£ + c2sin(1/2)y—/
12V2A2 + 4^ + 48^).
3.3. The Rational Function Solutions to (1) fa = 0. We obtain
£5 (x,0 = ± 6V2C2 2e'(fc*+te+^ (ci + c2^
N5 (x, i) =
(ci + c2^
(25) where fc = ±(1/6)y3(ft + a), I = (1/12)(12ft + ft + A2 +
12v2A2 - ± 48^), and
£ (x, i) = £9>io (x, i) = ±
6y2c22 ei(fcx+;t+^) (ci + c2^)2
N6 (x,i) = N9,io (x,f) =
(ci + c2^
fc = ±1^3(^2 -a), (31)
Z = (l2ft + ft + A2 ± 12y2A2 + 4^ + 48^) . (32)
4. Conclusion
The (G'/G)-expansion method is effectively employed to deal with the coupled Schrodinger-Boussinesq equations. The hyperbolic function solutions, the trigonometric function solutions, and the rational function solutions to the equations in the case of a > 0, a < 0, and a = 0 are obtained. In particular, the well-known soliton solutions are only the special case of the hyperbolic-type solutions. We find several properties of solutions when ? ^ œ>.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11171241, 11071177, 11226162), the Key Project of Chinese Ministry of Education (Grant no. 211162), and the Sichuan Province Science Foundation for Youths (no. 2012JQ0011).
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