# New Exact Jacobi Elliptic Function Solutions for the Coupled Schrödinger-Boussinesq Equations

Authors: Baojian Hong; Dianchen Lu
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

A general algebraic method based on the generalized Jacobi elliptic functions expansion
method, the improved general mapping deformation method, and the extended auxiliary function method
with computerized symbolic computation is proposed to construct more new exact solutions for coupled
Schrödinger-Boussinesq equations. As a result, several families of new generalized Jacobi elliptic function
wave solutions are obtained by using this method, some of them are degenerated to solitary wave
solutions and trigonometric function solutions in the limited cases, which shows that the general method
is more powerful than plenty of traditional methods and will be used in further works to establish more
entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical
physics.


---

## Body

## 1. Introduction

Nonlinear partial differential equations (NLPDEs) are widely used to describe complex physical phenomena arising in the world around us and various fields of science. The investigation of exact solutions of NLPDEs plays an important role in the study of these phenomena such as the nonlinear dynamics and the mechanism behind the phenomena. With the development of soliton theory, many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as homotopy perturbation method [1], nonperturbative method [2], homogeneous balance method [3], Bäcklund transformation [4], Darboux transformation [5], extended tanh-function method [6], extended F-expansion method [7], G′/G method [8], exp-function method [9], sine-cosine method [10], Jacobi elliptic function method [11], extended Riccati equation rational expansion method [12], extended auxiliary function method [13], and  other methods [14, 15].In [16, 17], Hong proposed a generalized Jacobi elliptic functions expansion method to obtain generalized exact solutions of NLPDEs. In [18], Hong and Lu proposed an improved general mapping deformation method which is more general than many other algebraic expansion methods [19, 20]. The solution procedure of this method, by the help of Matlab or Mathematica, is of utmost simplicity, and this method can be easily extended to all kinds of NLPDEs. In this work, we will propose the general algebraic method which contained the two methods [16–18] to obtain several new families of exact solutions for the coupled Schrödinger-Boussinesq equations.


## 2. Summary of the General Algebraic Method

Consider a given nonlinear evolution equation with one physical fieldu(x,t) in two variables x and t(1)P(u,ut,ux,uxx,…)=0.
We seek the following formal solutions of the given system by a new intermediate transformation:
(2)u(ξ)=∑i=0n‍Aiϕi(ξ),
where Ai are constants to be determined later. ξ=ξ(x,t) are arbitrary functions with the variables x and t. The parameter n can be determined by balancing the highest order derivative terms with the nonlinear terms in (1). And ϕ(ξ) is a solution of the following ordinary differential equation:
(3)ϕ′2(ξ)=∑i=04‍aiϕi(ξ).
Substituting (3) and (2) into (1) and setting the coefficients of ϕi(ξ)(i=0,1,2,…) and ϕj(ξ)∑i=04‍aiϕi(ξ)(j=…,-2,-1,0,1,2,…) to zero yield a set of algebraic equations for Ai, ai, and ξ. Using the Mathematica to solve the algebraic equations and substituting each of the solutions of the set, that is, each of the expressions of ϕ(ξ) into (2), we can get the solutions of (1). In order to obtain some new general solutions of (3), we assume that (3) has the following solutions:
(4)ϕ(ξ)=c0+c1e(ξ)+c2f(ξ)+c3g(ξ)+c4h(ξ),
where ci=ci(t)(i=0,…,4) are functions of t to be determined later and the four functions e=e(ξ),f=f(ξ),g=g(ξ), and h=h(ξ) are expressed as follows:
(5)e=1p+qF+rF2+lF′,f=Fp+qF+rF2+lF′,g=F2p+qF+rF2+lF′,h=F′p+qF+rF2+lF′,
where p,q,r, and l are arbitrary constants which ensure the denominator unequal to zero, and F=F(ξ) is a solution of the following equations:
(6)F′2=A+BF2+CF4+2DF+2EF3,F′′=BF+2CF3+D+3EF2,
where “′” denotes d/dξ, “′′” denotes d2/dξ2,  A,B,C,D, and E are arbitrary constants, and the four functions e,f,g, and h satisfy the following relations:
(7)e′=-qeh-2rfh-l(De2+Bef+2Cfg+3Ef2),f′=peh-rgh+l(Ae2+Def-Cg2-Efg),g′=qgh+2pfh+l(2Aef+3Df2+Bfg+Eg2),h′=(Dp-Aq)e2+(Bp-Dq-2Ar)ef+(2Cp+Eq-Br)fg+3(Ep-Dr)f2+(Cq-Er)g2,f2=eg,h2=Ae2+Bf2+Cg2+2Def+2Efg,pe+qf+rg+lh=1.
And e,f,g, and h satisfy one of the following relations at the same time.Family 1.
Whenp=0(8a)(Cl2-r2)h2=-C+2Clh-Br(1-lh-qf)e-Ae2r2-2Dr2ef+(2Cq-2Er)f+(2lEr-2Clq)fh+(2Eqr-Cq2)f2.Family 2.
Whenq=0(8b)(Cl2-r2)h2=2C(lh+pe-pleh)+2Er(lh-1)f-Br(1-lh-pe)e-(Cp2+Ar2)e2+(2Epr-2Dr2)ef-C.Family 3.
Whenr=0(8c)Cl2g2=1-2El2fg-2pe+(p2-Al2)e2-2qf+2(pq-Dl2)ef+(q2-Bl2)eg.Family 4.
Whenl=0(8d)r2h2=C-2Cpe+(2Er-2Cq)f+(Cp2+Ar2)e2+(Cq2-2Eqr+Br2)eg+(2Cpq-2Epr+2Dr2)ef.Substituting (4), (7) along with (8a)–(8d) into (3) separately yields four families of polynomial equations for e,f,g, and h. Setting the coefficients of ei,eif,eig,eih,eifg,eifh, and eigh(i=0,1,2,…) to zero yields a set of overdetermined differential equations (ODEs) in p,q,r,l,ai, and ci(i=0,1,2,3,4),A,B,C,D,E, and ξ(x,t), solving the ODEs by Mathematica and Wu elimination, we can obtain many exact solutions of (3) according to (4), (5), (6). If we let c0=c1=c3=c4=0, c2=1, p=1, q=r=l=0,  a0=A, a1=2D, a2=B, a3=2E, and a4=C, we have φ(ξ)=F(ξ); our method contains the improved general mapping deformation method [18].Remark 1.
Our method proposed here is more general than theG′/G method [8], the extended Riccati equation rational expansion method [12], the extended auxiliary function method [13], the generalized Jacobi elliptic functions expansion method [16, 17], and many other algebra expansion methods [6, 7, 11, 18–21].Remark 2.
Equations (2) and (3) can be extended to the following forms:
(9)u(ξ)=∑i=0n‍Ai(t)ϕi(ξ)+∑i=-n-1‍An-i(t)ϕi(ξ)+∑i=-nn‍Bi(t)ϕi(ξ)ϕ′(ξ),ϕ′2(ξ)=∑i=0N‍ai(t)ϕi(ξ),
where N can be an arbitrary positive integer. n is usually a positive integer. If n is a fraction or a negative integer, we make the following transformation: (a)
whenn=d/c is a fraction, we let u(ξ)=vd/c(ξ), then return to determine the balance constant n again;(b)
whenn is a negative integer, we suppose u(ξ)=vn(ξ), then return to determine the balance constant n again.Remark 3.
Notice that(10)F1(ξ)|(A,B,C,D,E)1⟶φ1(ξ)|(a0,a1,a2,a3,a4)1=F2(ξ)|(A,B,C,D,E)2⟶φ2(ξ)|(a0,a1,a2,a3,a4)2=F3(ξ)|(A,B,C,D,E)3⟶⋯.
We find a meanful conclusion that this general method implies a BT of (1) with the compatible conditions (4), (5), (6), (7), and (8a)–(8d).In the following, we will use this method to solve the Schrödinger-Boussinesq equations.


## 3. Exact Solutions to the Coupled Schrödinger-Boussinesq Equations

We consider the coupled Schrödinger-Boussinesq equations [21–28]
(11)iEt+Exx+αE-NE=0,3Ntt-Nxxxx+3(N2)xx+βNxx-(|E|2)xx=0.
These equations are known to describe various physical processes in laser and plasma, such as formation, Langmuir field amplitude, intense electromagnetic waves, and modulational instabilities [22–25]. The problem of the complete integrability of this system has been studied by Chowdhury et al. from the point of view of Painlevé analysis [25]. The solitary wave solutions for system (11) have been obtained in [26, 27]. The Jacobi doubly periodic wave solutions and a range of other solutions for this system have been investigated in [21, 28]. We are interested in searching new generalized Jacobi elliptic function solutions for (11) by using our method.We consider the following transformations:(12)E(x,t)=E(ξ)=u(ξ)ei(kx+ωt+ξ00),N(x,t)=N(ξ)=v(ξ),(13)ξ=ax-2akt+ξ0,
where a and k are constants to be determined later and ξ0 and ξ00 are arbitrary constants.Substituting (12) and (13) into (11) and integrating the second equation of system (11) twice, we have
(14)a2u′′+(α-ω-k2)u-uv=0,a2v′′+u2-(12k2+β)v-3v2=0.
By balancing the highest derivative term with the nonlinear terms in (14), we obtain n=2. Therefore, we assume that (14) have the following solutions:
(15)u(ξ)=A0+A1φ(ξ)+A2φ2(ξ),v(ξ)=B0+B1φ(ξ)+B2φ2(ξ).
Substituting (3), (13), and (15) into (14) and setting the coefficients of ϕi(ξ)(i=0,1,2,…) and φj(ξ)∑i=04‍aiφi(ξ)(j=…,-2,-1,0,1,2,…) to zero yield a set of over-determined equations (ODEs) for Ai,k,ω, and ai. After solving the ODEs by Mathematica, we could determine the following solutions.Family 1.
(16)
A
0
=
±
a
2
-
a
1
a
3
-
8
a
0
a
4
2
,
A
1
=
±
2
a
2
a
4
a
3
-
2
a
1
a
3
-
16
a
0
a
4
,
A
2
=
0
,
B
0
=
0
,
B
1
=
a
2
a
3
,
B
2
=
2
a
2
a
4
,
ω
=
-
a
2
(
a
3
4
-
80
a
1
a
3
a
4
2
-
128
a
0
a
4
3
)
48
a
3
2
a
4
+
α
+
β
12
,
k
=
ε
4
a
2
(
a
3
4
+
16
a
1
a
3
a
4
2
-
128
a
0
a
4
3
)
-
4
a
3
2
a
4
β
3
a
3
2
a
4
,
ε
2
=
1
,
a
2
=
a
3
2
4
a
4
+
2
a
1
a
4
a
3
.Family 2.
(17)
A
0
=
±
2
B
0
,
A
1
=
0
,
A
2
=
±
6
2
a
2
a
4
,
B
0
=
const
,
B
1
=
0
,
B
2
=
6
a
2
a
4
,
a
1
=
a
3
=
0
,
ω
=
1
12
(
44
a
2
a
2
-
22
B
0
+
12
α
+
β
)
,
k
=
ε
4
a
2
a
2
-
2
B
0
-
β
12
,
ε
2
=
1
.Substituting (4), (5), (6), (7) along with (8a)–(8d) and (16) into (14) separately yields an ODEs; after solving the ODEs by Mathematica and Wu elimination, we can obtain the following solutions of (14) according to (4), (5), (6), (16).Case 1.
(18)
A
=
a
0
1
=
1
4
(
1
-
m
2
)
,
B
=
a
2
1
=
2
-
m
2
,
C
=
a
4
1
=
1
,
D
=
1
2
a
1
1
=
1
2
(
m
2
-
1
)
,
E
=
1
2
a
3
1
=
-
1
,
0
≤
m
≤
1
,
p
=
1
,
q
=
r
=
l
=
0
,
c
0
=
c
1
=
c
3
=
c
4
=
0
,
c
2
=
1
,
φ
1
(
ξ
1
)
=
F
1
(
ξ
1
)
=
cn
ξ
1
±
1
±
±
sn
ξ
1
+
cn
ξ
1
,
ξ
1
=
a
x
-
2
a
k
1
t
+
ξ
0
.Case 2.
(19)
A
=
a
0
2
=
1
4
,
B
=
a
2
2
=
2
-
m
2
,
C
=
a
4
2
=
1
-
m
2
,
D
=
1
2
a
1
2
=
1
2
,
E
=
1
2
a
3
2
=
1
-
m
2
,
0
≤
m
≤
1
,
p
=
1
,
q
=
r
=
l
=
0
,
c
0
=
c
1
=
c
3
=
c
4
=
0
,
c
2
=
1
,
φ
2
(
ξ
2
)
=
F
2
(
ξ
2
)
=
sn
ξ
2
±
1
-
sn
ξ
2
±
±
cn
ξ
2
,
ξ
2
=
a
x
-
2
a
k
2
t
+
ξ
0
.Case 3.
(20)
A
=
a
0
3
=
-
(
c
0
2
-
1
)
(
1
+
(
c
0
2
-
1
)
m
2
)
,
B
=
a
2
3
=
(
2
-
6
c
0
2
)
m
2
-
1
,
C
=
a
4
3
=
-
m
2
,
D
=
1
2
a
1
3
=
c
0
-
2
c
0
m
2
+
2
c
0
3
m
2
,
E
=
1
2
a
3
3
=
2
c
0
m
2
,
0
≤
m
≤
1
,
p
=
1
,
q
=
r
=
l
=
0
,
c
0
=
const
,
c
1
=
c
3
=
c
4
=
0
,
c
2
=
1
,
φ
3
(
ξ
3
)
=
F
3
(
ξ
3
)
=
c
0
+
cn
2
ξ
3
,
ξ
3
=
a
x
-
2
a
k
3
t
+
ξ
0
.Case 4.
(21)
A
=
1
,
B
=
-
m
2
-
1
,
C
=
m
2
,
D
=
E
=
0
,
F
=
sn
ξ
,
0
≤
m
≤
1
,
a
0
4
=
1
,
a
1
4
=
∓
4
2
(
1
+
m
)
m
,
a
3
4
=
±
2
2
(
1
+
m
)
m
(
1
+
6
m
+
m
2
-
4
(
1
+
m
)
m
)
,
a
2
4
=
12
(
1
+
m
)
m
-
6
m
-
m
2
-
1
,
a
4
4
=
8
m
(
m
+
1
)
2
-
2
(
1
+
m
)
(
1
+
m
2
+
6
m
)
m
,
p
=
1
,
q
=
±
2
(
1
+
m
)
m
,
r
=
m
,
l
=
0
,
c
0
=
c
1
=
c
3
=
c
4
=
0
,
c
2
=
1
,
φ
4
(
ξ
4
)
=
sn
ξ
4
1
±
2
(
1
+
m
)
m
sn
ξ
4
+
m
sn
2
ξ
4
,
ξ
4
=
a
x
-
2
a
k
4
t
+
ξ
0
.Case 5.
(22)
A
=
1
-
m
2
,
B
=
2
m
2
-
1
,
C
=
-
m
2
,
D
=
E
=
0
,
0
≤
m
≤
1
,
F
=
cn
ξ
a
0
5
=
1
,
a
1
5
=
-
4
,
a
2
5
=
8
-
4
m
2
,
a
3
5
=
8
m
2
-
8
,
a
4
5
=
4
-
4
m
2
,
p
=
0
,
q
=
1
,
r
=
0
,
l
=
±
1
,
c
0
=
c
1
=
c
3
=
c
4
=
0
,
c
2
=
1
,
φ
5
(
ξ
5
)
=
cn
ξ
5
cn
ξ
5
∓
sn
ξ
5
dn
ξ
5
,
ξ
5
=
a
x
-
2
a
k
5
t
+
ξ
0
.Case 6.
(23)
A
=
m
2
-
1
,
B
=
2
-
m
2
,
C
=
-
1
,
D
=
E
=
0
,
0
≤
m
≤
1
,
F
=
dn
ξ
a
0
6
=
1
,
a
1
6
=
-
4
m
,
a
2
6
=
8
m
2
-
4
,
a
3
6
=
8
m
-
8
m
3
,
a
4
6
=
4
m
4
-
4
m
2
,
p
=
0
,
q
=
m
,
r
=
0
,
l
=
±
1
,
c
0
=
c
1
=
c
3
=
c
4
=
0
,
c
2
=
1
,
φ
6
(
ξ
6
)
=
dn
ξ
6
dn
ξ
6
∓
m
2
sn
ξ
6
cn
ξ
6
,
ξ
6
=
a
x
-
2
a
k
6
t
+
ξ
0
.Case 7.
(24)
A
=
C
1
C
3
q
-
5
C
3
2
q
3
+
ε
(
C
1
+
3
C
3
q
2
)
C
3
(
3
C
3
q
2
-
2
C
1
)
4
C
3
,
B
=
0
,
C
=
0
,
2
D
=
C
1
,
2
E
=
C
3
,
F
=
℘
(
C
3
2
ξ
,
-
4
C
1
C
3
,
-
4
M
C
3
)
,
a
0
7
=
0
,
a
1
7
=
C
3
,
a
2
7
=
-
3
C
3
q
,
a
3
7
=
C
1
+
3
C
3
q
2
,
a
4
7
=
(
{
C
3
(
3
C
3
q
2
-
2
C
1
)
}
-
3
C
1
C
3
q
-
9
C
3
2
q
3
+
ε
(
C
1
+
3
C
3
q
2
)
C
3
(
3
C
3
q
2
-
2
C
1
)
)
×
(
4
C
3
)
-
1
,
ε
=
sgn
[
C
1
+
3
C
3
q
2
]
,
p
=
0
,
q
=
const
,
r
=
1
,
l
=
0
,
c
0
=
c
1
=
c
3
=
c
4
=
0
,
c
2
=
1
,
φ
7
(
ξ
7
)
=
1
q
+
℘
(
(
C
3
/
2
)
ξ
7
,
-
4
C
1
/
C
3
,
-
4
M
/
C
3
)
,
ξ
7
=
a
x
-
2
a
k
7
t
+
ξ
0
.We obtain the following solutions of (11) according to (4), (5), (12), (13), (15), and (16) and Cases 1–7:
(25)Ei(x,t)=[±a2-a1ia3i-8a0ia4i2±2a2a4ia3i-2a1ia3i-16a0ia4iφi(ξi){-a1ia3i-8a0ia4i2}]×ei(kix+ωit+ξ00),Ni(x,t)=a2a3iφi(ξi)+2a2a4iφi2(ξi),ξi=ax-2akit+ξ0,(i=1,…,7),
where a is an arbitrary constant,  a0i,a1i,a2i,a3i, and a4i are defined as Cases 1–7, and ωi and ki are defined as follows:
(26)ωi=-a2(a3i4-80a1ia3ia4i2-128a0ia4i3)48a3i2a4i+α+β12,ki=ε4a2(a3i4+16a1ia3ia4i2-128a0ia4i3)-4a3i2a4iβ3a3i2a4i,ε2=1,(i=1,…,7).With the similar process,substituting (4), (5), (6), and (7) along with (8a)–(8d) and (17) into (14) separately yields ODEs; after solving the ODEs by Mathematica and Wu elimination, we can obtain the following solutions of (11) according to (4), (5), (12), (13), (15), and (17) and the solutions of (3) mentioned in [29]:
(27)Ej(x,t)=[±2B0±62a2a4jφj2(ξj)]ei(kjx+ωjt+ξ00),(j=0,1,2,…),Nj(x,t)=B0+6a2a4jφj2(ξj),ξj=ax-2akjt+ξ0,ωj=112(44a2a2j-22B0+12α+β),kj=ε4a2a2j-2B0-β12,ε2=1.Remark 4.
If we leta01.1=1-m2,  a21.1=2m2-1,  a41.1=-m2,  a11.1=a31.1=0, and φ1.1(ξ1.1)=cnξ1.1, then we have
(28)E1.1(x,t)=[±2B0∓62a2m2cn2(ξ1.1)]ei(k1x+ω1t+ξ00),N1.1(x,t)=B0-6a2m2cn2(ξ1.1),ξ1.1=ax-2ak1.1t+ξ0,ω1.1=112(44a2(2m2-1)-22B0+12α+β),k1.1=ε4a2(2m2-1)-2B0-β12,ε2=1.
If we let a=(1/2)b0,  m=1, or a=(1/2)(1/2m2-1)b0, then solution (E1.1(x,t),N1.1(x,t)) is in full agreement with the solution E1,  N1 and E2,  N2 mentioned in [21].Remark 5.
The seven types of explicit solutions we obtained here to (11) are not shown in the previous literature to our knowledge. They are new exact solutions of (11). Notice that snξ→tanhξ,  cnξ→sechξ, and dnξ→sechξ when the modulus m→1, and snξ→sinξ,  cnξ→cosξ, and dnξ→1 when the modulus m→0. Solutions Ei,Ni(i=1,…,6) are degenerated to solitary wave solutions when the modulus m→1 and to triangular functions solutions when the modulus m→0.We can give the numerical simulation ofE6 to show their physical properties (see Figures 1, 2, and 3).The real part ofE6 when a=α=1,β=-1,ξ0=0, and m=0.9 and a plane graph when t=0.
(a)
(b)The imaginary part ofE6 when a=α=1,β=-1,ξ0=0, and m=0.9 and a plane graph when t=0.
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(b)The modulus ofE6 when a=α=1,  β=-1,  ξ0=0, and m=0.9 and a plane graph when t=0.
(a)
(b)


## 4. Conclusion

In this paper, we succeed to propose a general algebraic method approach for finding new exact solutions of the nonlinear evolution equations. By using this method and computerized symbolic computation, we have found abundant new exact solutions for the coupled Schrödinger-Boussinesq equations (11). More importantly, our method is much simpler and powerful to find new solutions to various kinds of nonlinear evolution equations, such as KdV equation, Boussinesq equation, and zakharov equation. We believe that this method should play an important role for finding exact solutions in the mathematical physics.



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