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Original Article
Extended trial equation method for nonlinear coupled Schrodinger Boussinesq partial differential equations
Khaled A. Gepreelab*
a Mathematics Department, Faculty of Sciences, Zagazig University, Zagazig, Egypt b Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia
Received 18 March 2015; revised 7 August 2015; accepted 12 August 2015 Available online xxx
Keywords
Nonlinear partial differential equations; Extend trial equation method;
Traveling wave solutions; Soliton solutions; Jacobi elliptic functions
Abstract In this paper, we improve the extended trial equation method to construct the exact solutions for nonlinear coupled system of partial differential equations in mathematical physics. We use the extended trial equation method to find some different types of exact solutions such as the Jacobi elliptic function solutions, soliton solutions, trigonometric function solutions and rational, exact solutions to the nonlinear coupled Schrodinger Boussinesq equations when the balance number is a positive integer. The performance of this method is reliable, effective and powerful for solving more complicated nonlinear partial differential equations in mathematical physics. The balance number of this method is not constant as we have in other methods. This method allows us to construct many new types of exact solutions. By using the Maple software package we show that all obtained solutions satisfy the original partial differential equations.
2010 Mathematics Subject Calssification: 34A36; 34L05; 47A70
Copyright 2015, Egyptian Mathematical Society. Production and hosting by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
The effort in finding exact solutions to nonlinear differential equations is important for the understanding of most nonlinear * Tel.: 00966595374598. physical phenomena. For instance, the nonlinear wave phenom-
E-mail address: kagepreel@yahoo.com ena observed in fluid dynamics, plasma and optical fibers are
Peer review under responsibility of Egyptian Mathematical Society. often modeled by the bell shaped sech solutions and the kink
shaped tanh solutions. In recent years, the exact solutions of nonlinear PDEs have been investigated by many authors (see for example [1-30]) who are interested in non-linear physical
S1110-256X(15)00072-3 Copyright 2015, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article
under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
http://dx.doi.org/10.1016/j.joems.2015.08.007
phenomena. Many powerful methods have been presented by those authors such as the inverse scattering transform [1], the Backlund transform [2], Darboux transform [3], the generalized Riccati equation [4,5], the Jacobi elliptic function expansion method [6,7], Painleve expansionsmethod [8], the extended tanh-function method [9,10], the F-expansion method [11,12], the exp-function expansion method [13,14], the sub-ODE method [15,16], the extended sinh-cosh and sine-cosine methods [17,18], the (G'/G)-expansion method [19,20] and so on. Also, there are many methods for finding the analytic approximate solutions for nonlinear partial differential equations such as the homotopy perturbation method [21,22], Adomain decomposition method [23,24], Variation iteration and homotopy analysis method [27,28]. There are many other methods for solving the nonlinear partial differential equations (see for example [29-37]). Recently Khan et al. [38-43] implemented the modified simple equation method and enhanced (G'/G) expansion method to construct the traveling wave solutions for nonlinear evolution equations in mathematics physics. Also Khan and Akbar [44] and Akter et al. [45] used the exp(-$(g)) expansion method to find exact solutions for nonlinear partial differential equations. More recently Gurefe et al. [46] have presented a direct method, namely the extended trial equation method for solving the nonlinear partial differential equations. The main objective of this paper is to modify the extended trial equation method to construct a series of some new analytic exact solutions for some nonlinear partial differential equations in mathematical physics via nonlinear coupled Schrodinger Boussinesq equations. In this present paper, we will construct the exact solutions in many different types of the roots of the trial equation. We will obtain many different kinds of exact solutions in hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic functions solutions and rational solutions.
2. Description of the extended trial equation method
Suppose that we have a nonlinear partial differential equation in the following form:
F(u, Ut, ux, Utt, uxt, uxx, ...) = 0,
where u = u(x, t) is an unknown function, F is a polynomial in u = u(x, t) and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved. Let us now give the main steps for solving Eq. (2.1) using the extended trial equation method as [46-48]:
Step 1. The traveling wave variable
where Y satisfies the following nonlinear auxiliary equation:
<Y> * = A(Y \ = = Y ° + gfl-1Y g-1 +--- + g1Y + go
*(Y) ZsYs + Zs-Ys-1 +•••+?! Y + Zo
where Ti, gi, Zj are constants to be determined later. Using (2.4) and (2.5), we have
u (g) = --riY
2^ 2 (Y ) *(YHtX' ( ) '
where $(Y), ^(Y) are polynomials in Y.
Step 3. Balancing the highest derivative term with the nonlinear term we can find the relations between S, 0 and s. We can calculate some values of S, 0 and s.
Step 4. Substituting Eqs. (2.4)-(2.6) into (2.3) yields a polynomial as follows
Q(y) = psYs + •••+ pi Y + po = 0
Step 5. Setting the coefficients of the polynomial Œ(y) yield to be zero, and then we have a set of algebraic equations
pi = 0, i = 0, ..., s.
Solving this system of algebraic equations to determine the values of ge,ge-1,..., gu go, Zs , Zs-1.....Z1, Zo and
Step 6. Reduce Eq. (2.5) in the elemental integral form:
±(i - n0 ) =
f dY f ¡*(Y) JY
J yx^ = 7 i&(Y)dY.
u(x, t ) = u(g), g = x — a>t,
where no is an arbitrary constant. Using a complete discrimination system for the polynomial to classify the roots of $( Y), we solve (2.9) with the help of software packages such as Maple or Mathematica and classify the exact solutions to Eq. (2.3). In addition, we can write the exact traveling wave solutions to (2.1).
3. Extended trial equation method for nonlinear coupled Schrodinger Boussinesq equations
We consider the coupled nonlinear Schrodinger Boussinesq equations
where rn is a nonzero constant, the transformation (2.2) permits us reducing Eq. (2.1) to an ODE for u = u(g) in the following form
P(u, —mu', u, a> u", —mu", u", ...) = 0,
where P is a polynomial of u = u(g) and its total derivatives. Step 2. Suppose the trial solution is of the form:
iEt + Exx + PiE - NE = 0,
3Ntt - Nxxxx + 3(N2)xx + PiNxx - (\E|2)xx = 0,
where p1, p2 are real constants and N(x, t ) is a real function while E(x, t) is a complex function. The system (3.1) is known to describe various physical processes in Laser and plasma, such as formation, Langmuir field amplitude and intense electromagnetic waves and modulation instabilities [49,50]. The traveling wave variable
u(g) = Y, TiYi,
(2.4) E (x, t ) = u(x, t )ein, n = kx + ct + c0
where u(x, t) is a real function and k, c, c0 are real arbitrary constants, permits us to convert (3.1) into the following nonlinear system of partial differential equations:
ut + 2kux = 0,
uxx — (c + k2 — ßi )u — Nu = 0,
3N„ — Nxxxx + 3(N2 )xx + ß2Nxx — (u2 )xx = 0,
We suppose that the solutions of the system (3.3)-(3.5) has the form:
u = ф(1), N = f(I), I = x — 2kt,
where 0, ^ are arbitrary functions of §. The transformations (3.6) lead to write Eqs. (3.3)—(3.5) in the following form:
ф" — (c + k2 — ß1 )ф — фf = 0, —f " + (12k2 + ß2 )f + 3f2 — ф2 + A = 0.
Suppose that the exact solutions of Eqs. (3.7) and (3.8) can be rewritten in the form:
T = —Ш2 M — 4ßZ2 + 16k2Zq2 — 5Ç2 + ß2?02],
T = Tгь
To = —[ — 6k2?o2 — 5Ç2 + 6ctf — 6ßiZo2 — ß2?o2],
§3 = — r1 -t 2
§1 = 25^^ + [(12k2 - 12c) + 12A]^2+36^2
25§ 2
+ (72k2 - 72c)A--41 + (36k4+36c2 - 72k2c)} (3.13)
where f0, §2 and t1 are arbitrary constants. Substituting Eqs. (3.13) into Eqs. (2.5) and (2.9), we have
±(I — no ) = L
Y 3 4_ |2 Y 2 _L- Il Y 41 + I3 1 + I3 1 + I3
(3.14)
¿1 ¿2 ф(1) = J2?!, f(i) = J2 ty,
where t¡, T are arbitrary constants to be determined later, Y satisfies the auxiliary Eq. (2.5) and 51, S2 are arbitrary positive integers. Balancing the highest order derivative terms with the nonlinear terms in (3.7) and (3.8), we get the relations between ¿i, S2, 0 and s as follows
¿1 = S2 = в — e — 2
(3.10)
Eq. (3.10) has infinity solutions, consequently we suppose some of these solutions as follows:
Case 1. If 0 = 3 and s = 0, we get ¿1 = S2 = 1. In this case we have:
ф(1) = To + T1 Y,
r2(|3Y3 + I2Y2 + I1Y + |o)
(ф )2 = Ф" =
t^iY2 + 2I2Y + h) 2Zo
(3.11)
where L = .
To integrate Eq. (3.14), we must discuss the following families:
Family 1. If Y3 + | Y2 + | Y + ,canbe written in the following form:
Y3 W2|2 y2 3^2
Y +^^ Y — Щ
ß22 + [(12k2 — 12c) + 12ft] ß2 + 36ß2
+ (72k2 — 72c)ß1--12 + (36k4 + 36c2 — 72k2 c)
, 3V!|^ 3
+ ^-¿2 = (Y — a1 ) 2Т1 Kq2
(3.15)
where a1 is an arbitrary nonzero constant.
From equating the coefficients of Y in both sides of Eq. (3.15), we get a system of algebraic equations in §0, §2, f0, t1 and c, which can be solved by using the Maple software package to get the following results:
c = ß1 + k2 + - ß2, Io = —a^Ko, I2 = —3«1fo, T1 = —-.
(3.16)
f(I) = To + T1 Y,
2 T12(|3Y3 + I2Y2 + I1Y + Io)
(f ) = -
T1 (3|э Y2 + 2I2Y + IQ 2?o '
(3.12)
Substituting Eqs. (3.11, (3.12) into Eqs. (3.7), (3.8) and setting the coefficients of Y to be zero, we get a system of algebraic equations which can be solved by using the Maple software package to obtain the following results:
297 4 -/ 49 6 6 \ 1 , 2 ,, ^ = 25"+ k\-25^2 + 25- 25^ + 25 - 2^2)
+ 25£2(A - c) + 25c (c - 2A ),
Eqs. (3.16), (3.13) and (3.14) lead to get:
A = ——ß22 — 12k4 — 2ß2k2, To = —2k2 — 6 ß2 — 2 a1,
3\f2a1
T1 = 2, I1 = 3?oa2, I3 = Ko, To = — 2 where Кч is an arbitrary constant and
±(I — no) =
Y = a1 +
(Y — a1 )3/2 VY—a ■
( x — 2 kt — n o ) 2
(3.17)
(3.18)
(3.19)
Substituting Eqs. (3.16), (3.17) and (3.19) into (3.11) and (3.12), we deduce that exact solutions of Eqs. (3.7) and (3.8) have the following form:
Ф(Ю =
(x — 2kt — no )
f(D = —2k — - ß2 +
6 (x — 2kt — n0 )
(3.20)
(3.21)
Hence the exact solutions of nonlinear Schrodinger Boussinesq equations (3.1) take the following form:
Y = a2 + (a2 — a1 )tan2
Va2 — a1 2
(4 — no )
a2 > ai.
(3.28)
Substituting Eqs. (3.28), (3.26) and (3.25) into (3.11) and (3.12), we get the exact solutions of Eqs. (3.7) and (3.8) take the form:
V2 3V2
Ф(!) = ~(ai + 2a2) + —
a2 + (a2 — a1 )tan2
Va2 — ai 2
(x — 2kt — n0 )
(3.29)
E(x, t) = and
N(x, t) =
(x — 2kt — n0 )
f'^kx^^ ß1+k2+ 6 ß2j t+C0]
—2k2 —- ß2 + -у .
6 (x — 2kt — n0 )2
(3.22)
(3.23)
Family 2. If Y3 + ^ Y2 + ^ Y + I° ,canbe written in the following form:
Y 3 + Y 2_ЪЛ
2т1 ?2 50т2
ß22 + [(12k2 — 12c) + 12ft] ß2 + 36ß2
+ (72k2 — 72c)ß1 — 25112 + (36k4 + 36c2 — 72k2 c)
2T1 ?02
= (Y — a1 )2 (Y — a2 )
(3.24)
where a1 and a2 are arbitrary nonzero constants. From equating the coefficients of Y in both sides of Eq. (3.24), we get a system of algebraic equations in go, g2, Zo, T and c which can be solved by using the Maple software package to get the following results:
c = ß1 + k2 + 1 ß2 + 5 (a2 — a1 ), I0 = —0^02^0, 66
I2 = — 2a1^0 — a2^0, T1 =
3\fl ~.
(3.25)
1 ,9 3 5 3 f(I) = — ß2 — 2k2 — - a1 — - a2 + -6 2 2 2
a2 + (a2 — a1 )tan2
V07—a 2
( x — 2 kt — n 0 )
(3.30)
Hence the exact solutions of nonlinear Schrodinger Boussinesq equations (3.1) take the following form:
, V2 3V2
E (x, t ) = {— — a + 2a2 ) + —
(x—2kt — n0 )
a2 + (a2 — a1 )tan2
kx+ß ßi+k2+6 ß2) t+c0
(3.31)
N (x, t ) = —-ß2 — 2k2 — - 01 — - 02 6 2 2
a2 + (a2 — a1 )tan2
«J02 — 01
( x — 2kt — n 0 )
(3.32)
Also, when a1 > a2, we have
Y = a1 + (a1 — a2 )csch2
V01 — 02
(I — n0 )
01 > 02.
(3.33)
Eqs. (3.13), (3.14) and (3.25) lead to get:
A = 12 (022 + a2 ) — Z 0201 — — ß22 — 12k4 — 2ß2 k2
T0 = — ß2 — 2k2 — 3 01 — 5 02, T1 = 3, 6 2 6 2
I1 = Z0 (01 + 202 )01, I3 = Z0, T0 = — — (01 + 202 ),
(3.26)
Substituting Eqs. (3.33), (3.26) and (3.25) into (3.11) and (3.12), we get the exact solutions of Eqs. (3.7) and (3.8) take the form:
V2 3V2
0(g) = - — («1 + 2«2) + —
01 + (01 — 02 )csch2
-J 01 — 02 2
(I — n0)
(3.34)
where f0 is an arbitrary constant and if 02 > 01, we have
± (I — n0 ) =
(Y — 01V Y — 02 V02 — 01
, , 02 >01.
02 — 01
(3.27)
1 ,2 3 5
f(I) = — ß2 — 2k2 — - 01 — 02 6 2 2
01 + (01 — 02)csch2
(I — n0 )
(3.35)
Hence the exact solutions of coupled nonlinear Schrodinger Boussinesq equations (3.1) take the form:
i V2 3V2
E (x, t ) = - — («1 + 2a2 ) + —
a1 + (a1 — a2 )csch
va - a2 2
(x - 2kt - no )
ei[kx+(^1+k2+ 6 £2 )t+C0]
(3.36)
N (x, t ) = -- £2 - 2k2 - - a1 — a2 6 2 2
a1 + (a1 - a2 )csch2
var-a-2
(x - 2kt - n0 )
(3.37)
Family 3. If Y3 + f2 Y2 + f Y + ^ ,canbe written in the follow-
J §3 §3 §3
ing form:
Y3 + f2 Y2 -
2T1 Z2
£2 + [(12k2 - 12c) + 12ft]£2 + 36£2
25 f 2
+ (72k2 - 72c)£1 - + (36k4 + 36c2 - 72k2c)
2T1 Z02
= (Y - a1 )(Y - a2)(Y - a3)
(3.38)
where a1, a2anda3 are nonzero arbitrary constants. Equating the coefficients of Y in both sides of Eq. (3.38), we get a system of algebraic equations in f0, f2, f0, t1 and c which can be solved by using the Maple software package to get the following results:
c = Di(i = 1, 2), §0 = -a^a^,
§2 = -a1 Z0 - a2?0 - a3&, T1 = ,
(3.39)
where Di(i = 1, 2)are the roots of equation 36Z2 + ( - 72£1 - 12£2 - 72k2)Z + 25a2a3
+ 25a1a3 + 25a1a2 + 36£j2 + 36k4 - 25a2 - 25a2 - 25a2
+ 12k2 £2 + £22 + 12£1 £2 + 72k2 £1 = 0, which take the following form
(3.40)
D1 = £1 + ^£2 + k2 + + a2 + a2 - a2a3 - aa - a1a2,
D2 = £1 + 6£2 + k2 - 7^2 + a2 + a2 - a2a3 - a1a3 - a^2.
Eqs. (3.39), (3.13) and (3.14) lead to get:
A = -12k4 - 2£2k2 - 12 £2 - 12 (a2a3 + a1a3 + aa )
+ 112 (a2 + a22 + a2 ), 2 2 8 2 1 1 3
T = -5Di + 5£1 - 5k2 - 2 (a1 + a2 + a3)- — £2, T1 = 2,
§1 = Z0 (a2a3 + aa + a1a2 ), §3 = Z0,
T0 = (-6k2 + 5(a1 + a2 + a3) + 6Dt - 6£1 - £2), (3.42)
where f0 is an arbitrary constant and
± (§ - n0 ) =
V(Y - a1 )(Y - a2)(Y - a3) y/a3 - a1
EllipticF
V Y - a1 Ia1 - a2
_ v*a2 - a^ V a1 - a3
Y = a1 + (a2 - a1 ^n"2
Va3 - a1 2
(n - n0X
a1 - a2 a1 - a3
(3.43)
. (3.44)
Substituting Eqs. (3.39), (3.42) and (3.44) into (3.11) and (3.12), we get the exact solutions of Eqs. (3.7) and (3.8) take the form:
V2 2 3V2
0(§) = -To (-6k2 + 5(a1 + «2 + «3) + 6Di-6£1 -£2) + —
a1 + (a2 - a1 )sn
Va3 - ai a - a2 -»-(f - n0),
a1 - a3
(3.45)
2 2 8 2 1 1 3
■f(f) = -5Di + 5£1 - 5k2 - 2(a1 + a2 + a3) - — £2 + 2
a1 + (a2 - a1 )sn'
Va3-ai ja1 - a2 -»-(f - n0),
a1 - a3
(3.46)
Hence the exact solutions of the nonlinear Schrodinger Boussi-nesq equations (3.1) have the following form:
E(x, t) = \- j2(-6k2 + 5(«1 + «2 + a3) + 6Di - 6£1 - £2
a1 + (a2-a1 )sn
VÔ3-ÔT 2
(x - 2kt - n0),
a1 - a2 a1 - a3
>i[kx+D;-t+c0]
(3.47)
2 2 8 2 1 1 N (x, t ) = -5 Di + 5 £1 - 5 k2 - 2 (a1 + a2 + a3 ) - — £
a1 + (a2 - a1 )sn'
VaT-a 2
(x - 2kt - n0),
a1 - a2 a1 - a3
(3.48)
Family 4. If Y3 + ^ Y2 + | Y + ,canbe written in the follow-
(3.41) ingform:
{£2 + [(12k2 - 12c)
+ 12£1]£2 + 36£2 + (72k2 - 72c)£1
25f2 , „¿,,4 , ,¿„2 , ^^
+ (36k4 + 36c2 - 72k c)}Y +
2t1 Z02
= (Y - a1 )(Y - (N1 + iN2)(Y - (N1 - iN2)
(3.49)
where a1, N1, N2 are nonzero real numbers. From equating the coefficients of Y in both sides of Eq. (3.49), we get a system of algebraic equations in §0, §2, f0, t1 and c which can be
solved by using the Maple software package to get the following results:
c = Di(i = 1, 2), g0 = -«1?0(N12 + N22),
g2 = -«1^0 - 2N1Z0, T1 =
(3.50)
(3.51)
where Di(i = 1, 2) are the roots of equation
36Z2 + (-72fa - 12fa - 72k1 )Z + 72N22 + 5o«1N1 - 25N2 + 36^2 + 36k4 + 12k1p1 + fa2 - 25a2 + 72k2 fo1 + 12fo1fo2 = o,
which take the following form:
D1 = fa + 1 fa + k2 + ^a! + N2 - 3N3 - 2a1N1,
D2 = fa + 6fa + k2 - a2 + N - 3N2 - 2a1N1. (3.52)
Eqs. (3.5o), (3.13) and (3.14) lead to get:
A = -12k4-2P2k2--P22"N22 — aN + — N2 + —a2, T0 =
2 2 8 , 1 1
- Di + - p1--k2--a1 - N1--p
5 ' + 5P1 5 2 1 1 10P
T1 = ^, g1 = Z0 (N22 + 2«1N1 + N2 ), g3 = Z0, T0 = (-6k2 + 5«1 + 10N1 + 6Di - 6P1 - P2),
(3.53)
where Zo is an arbitrary constant. With the help of Maple software package the integration of Eq. (3.14) in this family takes the following form:
±(g - n0) =
^^Y—al)(J2—2nytnî+nî)
EllipticF
VN1 + iN2 - a1 '
N1 - iN2 - a1
VN1 - iN2 - a1 y N1 + iN2 - a1
(3.54)
Y = a1 + (N1 - iN2 - a1 )sn2
VN1 + iN2 - a1 2
(g - n0),
N1 - iN2 - a1 N1 + iN2 - a1
(3.55)
Substituting Eqs. (3.5o), (3.53) and (3.55) into (3.11) and (3.12), we get the exact solutions of Eqs. (3.7) and (3.8) take the form:
0(g) = -To (-6k + 5a1 + 1oN1 + 6Dt - 6P1 - fa)
a1 + (N1 - iN2 - a1 )sn2
VN1 + iN2 - a1
*(g - n0),
N1 - iN2 - a1 N1 + iN2 - a1
(3.56)
2 2 8 2 1 1 f(g) = -A + ^p1 - 8k2 - 2a - N1 - ^p
a1 + (N1 - iN2 - a1 )sn2
VN1 + iN2 - a1 2
x(g - m),
N1 - iN2 - a1 N1 + iN2 - a1
(3.57)
Hence the exact solutions of nonlinear Schrodinger Boussinesq equations (3.1) take the following form:
E(x, t) = \ (-6k2 + 5a1 + 1oN1 + 6Di - 6fa - fa)
a1 + (N1 - iN2 - a1 )sn:
x(x - 2kt - n0),
N1 - iN2 - a1 N1 + iN2 - a1
VN1 + iN2 - a1 2
>i[kx+Dit+c0]
(3.58)
N (x, t ) = -- Di + - P1 — k2 — «1 - N1 - — P2
a1 + (N1 - iN2 - a1 )sn
VN1 + iN2 - a1
x(x - 2kt - n0),
N1 - iN2 - a1 N1 + iN2 - a1
(3.59)
Case 2. If e = 0 and 0 = 4, we get 51 = S2 = 2. In this case, we have
0(g) = T0 + T1Y + x2Y2,
(T1 + 2T2Y )2 (g4Y4 + g3Y3 + g2Y2 + g1 Y + g0 ) (0) =---,
T1 (4g4Y3 + 3g3 Y2 + 2g2 Y + g1 ) 2?0
+ ^ (6g4Y4 + 5g3 Y3 + 4g2 Y2 + 3g1 Y + 3g0 ) (3.60)
f(g) = T0 + T1Y + T2Y2,
2 (T1 + 2T2Y )2 (g4Y4 + g3 Y3 + g2Y2 + g1 Y + g0 ) f (g) = -
f " (g) =
T1 (4g4Y3 + 3g3 Y2 + 2g2 Y + g1 )
(6g4Y4 + 5g3 Y3 + 4g2 Y2 + 3g1 Y + 3g0 ), (3.61)
Substituting Eqs. (3.60), (3.61) into Eqs. (3.7), (3.8) and setting the coefficient Y to be zero, we get a system of algebraic equations which can be solved to obtain the following results:
A = -15k + k1 ~P2 ^P1 " ^ + ^ (3P2 - 2P2)
+25P2(P1 - c) + ^^c (c - 2P1 ),
V2 1 , V2 V2
T0 = -y-T0 - 5 (P1 +P2-c+11 k2 ), T1 ^^ T1, T2 ^^ T2,
480т23
— 30V2t2t2to + 2 rf + 36k2r2t2
+6T—ß2T2 + 36T—ß1T2 — 36т—c2 — 288k2T0T— — 288ß1T0T
+288ctqt22 + 1 20V2t2t22 — 48r0r22ß2
1 20т22
T1 f^30T2r2r0--22 T— — 36k2 т2 — 6ß2T2
—36ß 1 т2 + 36ct2
40т,,
Zo{52r T2 — 1 0T2T2TQ — 1 2k2Т2 — 2ß2T2
— 1 2ß 1 т2 + 1 2ct2
Ь = —— T1 Ko , Ç4 = TT T2 fo , 6 1 2
(3.62)
where Zo, t2, t1 and t0 are arbitrary constants. Substituting these results (3.62) into Eqs. (2.5) and (2.9), we have
±(| — no ) = L
VY 4 +1Y 3 +I—Y 2 +IY +1 '
(3.63)
where L = . To integrate Eq. (3.63), we must discuss the following families:
Family 5. If Y4 + | Y3 + | Y2 + | Y + | can be written in the following form:
2t1 T2
Y4 + ^ Y3 + "(д2t2 — 10T2T2T0
— 12k2T2 — 2ß2T2 — 12ß1T2 + 12ct2
20т23
t^3öV2t2t0 — T2 — 36k2T2 — 6ß2T2
—36ß1T2 + 36ct2
)]y + s|[(+ -t
+36k2T2T2 + 6r2ß2r2 + 36T12ß1T2 — 36т 2c2 — 288k2 т0т
—288ß 1т0т22 + 288ctqt22 + 1 2OV2t02t2 — 48T0T22ß2
= (Y — a1 )4.
(3.64)
where f0 is an arbitrary constant and
±(П — no) =
Y = a1 T
(Y — a1)2 Y — a1
( x — 2 kt — n 0 )
(3.67)
(3.68)
Substituting (3.68), (3.66) and (3.65) into (3.60) and (3.61), we get the exact solutions of Eqs. (3.7) and (3.8) take the form:
Ф(1) =
( x — 2kt — n 0 ) 2
f(l) = —2k2 — - ß2 + ----^.
6 (x — 2kt — n0 )2
(3.69)
(3.70)
Hence the exact solutions of nonlinear Schrodinger Boussinesq equations (3.1) have the following form:
E (x, t ) =
( x — 2kt — n 0 ) 2
,i[kx+(ß1+k2+1 ß2 )t+co]
N (x, t ) = —2k2 — - ß2 + ----^,
6 (x — 2kt — n0 )2
(3.71)
(3.72)
Note that the solutions (3.71) and (3.72) are the same solutions (3.22) and (3.23) in Family 1.
Family 6. If Y4 + f3 Y3 + f22 Y2 + f1 Y + f0 can be written in the
f4 f4 f4 f4
following form:
2t1 T2
Y4 + hl y 3 +
20т22
("у2 т2 — 10V2t2To — 12k2 Т2
— 2ß2T2 — 12ß1 T2 + 12ct^ Y2 + T ^OV—T—To — 2 т2
— 36k2 т2 — 6ß2T2 — 36ß1T2 + 36ct2
80т24
3öV2t2t2t0 + 2 T4 + 36k2T2
+ 6T—ß2T2 + 36t 2ß 1 т2 — 36t 2c2 — 288k2 т0т22 — 288ß 1т0т22 + 288ct0t22
+ 1 2^V2t02T-2 — 48ToT22ß2
= (Y — a 1)2 (Y — a— )2 (3.73)
where a 1 is an arbitrary constant. From equating the coefficients of Y in both sides of Eq. (3.64), we get a system of algebraic equations in Z0, t2, t1 , t0 and c which can be solved by using the Maple software package to get the following results:
c = ß1 + k2 + 1 ß—, tq = 6 Via—, 6
t1 = — 12\Î2a1: т2 = 6V2.
Eqs. (3.65), (3.62) and (3.63) lead to get:
(3.65)
A =--ß22 — 12k4 — 2ß2k2, T0 = —2k2 — ß2 + 6a—,
T = —12a1, T— = 6 |o = Zq a4, h = —4foaJ, I— = 6?oa—, I3 = — 4foab I4 = Zo,
(3.66)
where a1, a2 are arbitrary constants and a1 = a2. From equating the coefficients of Y in both sides of Eq. (3.73), we get a system of algebraic equations in Z0, t2, t1 , t0 and c which can be solved by using the Maple software package to get the following results:
c = ß1 + k2 + 1 ß— + - (a— — a1 )2, To = 6V—a1a2, 66
t1 = —6V2(a1 + a2 ), т2 = 6V2. Eqs. (3.74), (3.62) and (3.63) lead to get:
A= T0 =
_ J_ ß— — 12k4 — 2ß—k2 + (a— — a1 )4,
;2 1 1 2 1 2 17 -2k2 — - ß— + - a2 + - a— + — a—a1, 6 6 6 6
(3.74)
T1 = -6(«1 + «2 ), T2 = 6, g0 = f0a2a2,
g1 = -2a1«2 (a1 + a2 )&, g2 = Z0 (a2 + 4a1«2 + aj2 ),
g3 = -2Z0 (a1 + a2 ), g4 = Z0, (3.75)
where f0 is an arbitrary constant and
±(g - n0) =
(Y — a1 )(Y — a2) a1 - a2
Y - a1
Y - a2
-a1 + a2e±(a1 —a2>«-»0 >
-1 + e±(a1-a2)(g-n0) .
(3.76)
(3.77)
Substituting (3.77), (3.75) and (3.74) into (3.6o) and (3.61), we get the exact solutions of Eqs. (3.7) and (3.8) take the form:
0(g) = 6\f2a1 a2 - 6V2(a1 + a2)
-a1 + a2e
±(a1 -"2)(g-n0 )
-1 + e±(a1-a2)(g-n0 )
-a1 + a2e±(°1—a2>«-»0 > -1 + e±(a1 -a2)(g-n0 )
(3.78)
;2 1 1 2 1 2 17 f(g) = -2k2 - -P2 + -a;2 + -a2 + —-a2a1 - 6(a1 + a2)
6 6 6 6
-a1 + a2e±(°1—a2>«-»0 >
-1 + e±(a1-a2)(g-n0 )
-a1 + a2e±(a1-a2><g-n0 ) -1 + e±(a1 -a2)(g-n0 )
(3.79)
Hence the exact solutions of nonlinear Schrodinger Boussinesq equations (3.1) have the following form:
E(x, t)
6V2a1a2 - 6\f2(a1 + a2 )
-a1 + a2e
±(a1—a2)(x-2kt-n0) "
- 1 + e±(al-a2)(x-2kt-n0)
-a1 + a2e
±(a\—a-2){x-2kt-n0 )
- 1 + e±(a1-a2)(x-2kt-n0 )
xei[kx+(P1+k2+^ P2 + § (0-2 - a )2 ) t+c0]
(3.80)
2 1 1 2 1 2 17
N(x, t) = -2k2 - -P2 + -a2 + -a2 + — a2a1 - 6(a1 + a2) 6 6 6 6
-a1 + a2e
±(a1 -a2)(x-2kt-n0 )
- 1 + e±(a1-a2)(x-2kt-,0 )
;±(a1-a2)(g-,0) I2
-a1 + a2e
-1 + e
±(a1-a2)(x-2kt-n0 )
(3.81)
Family 7. If Y4 + | Y3 + g2 Y2 + | Y + | can be written in the following form:
Y4 + ^ Y3 + T2
t2 - 10V2r2T0 - 12k2T2
-2P2 t2 - 12P1r2 + 12ct2
r^30V2r2r0 - 522 t2 - 36k2t2
-6P2t2 - 36P1T2 + 36ct2
^ - 30V2r2r2r0 + T4 + 36k2r2r2 + 6r2P2r2 +36t12P1t2 - 36t2c2 - 288k2T0T22
-288P1r0r22 + 288ct0T22 + 120V2T02T22 - 48T0T22P2 = (Y - a1 )(Y - a2)(Y - a3)(Y - a4)
(3.82)
where a1, a2, a3,a4 are arbitrary different constants. From equating the coefficients of Y in both sides of Eq. (3.82), we get a system of algebraic equations in f0, t2, t1 , t0 and c which can be solved by using the Maple software package to get the following results:
c = Di, (i = 1, 2), a1 = a2 + a3 - a4,
T0 = —— (—20a3a4 + 20a2 - 20a2a4 - 30a2a3
-5a2 - P2 - 6k2 - 5a32 + 6Dt - 6P1 ), T1 = - 6\f2(a2 + a3 ), t2 = 6V2,
(3.83)
Di = P1 + k2 + ^P2 ± - (a34 - 4a3a4 + 16a4 - 32a2a4
+20a2a2 - 4a4a2 + - 32a3a34 + 14a2a2 + 20a2a2
-28a2a4a2 + 56a2a3a2 - 28a4a3a2) Eqs. (3.62), (3.63) and (3.83) lead to get:
(3.84)
A = --12 P2 - 12k4 - 2P2k2 + 3a2 - 3 (as + a2 a
5^2 2 14 \1 33 2 2 + 3a41 a2 + a2 + — asa2\ - 3a4 (a3 +a2 + 7a3a2 + 7a2a2 )
+ 12 a4 + a2 + 14a32a2 ),
2 8 2 1 2 2
T0 = -5Di - 5k - 10P2 + 5P1 + 2a3a4 - 2a4 + 2a2a4
1 2 1 2
+3a2a3 + 2 a2 + 2 a3,
T1 =-6(a2 + a3), T2 = 6, g0 = f0a2a3a4(a2 + a3 - a4), g1 = -(asa4 - a2 + a2a4 + a2a3 )(a2 + a3 )Z0, g2 = Z0(3a2a3 + a2a4 + a2 + 0304 - a2 + a2), g3 = -2Z0 (a2 + a3 ), g4 = Z0, where f0 is an arbitrary constant and
(3.85)
±(g - n0)
vTY-(02+a7—a4))(T—a2)(T—as)(T—a4)
(a2 - a4 )
ElliplticF
(a2 - a4)(Y - a4) (a2 + a3 - 2a4)( Y - a3)'
(a2 - a3 )(a2 + a3 - 2a4 )
(a4 - a2)2
(3.86)
a— — a—a4 + (a—a3 + a— — 2a3a4)sn2(j(a2 — a4)(I — no), y^
(a——a3 )(a2+a3 —2a4)
(a4—a2)
a4 — a— + (a— + a3 — 2a4)sn2(^—(a2 — a4)(I — no), y^
(a2—a3 )(a2+a3 —)
(a4 —a2)-
(3.87)
Substituting (3.83), (3.87) and (3.85) into (3.60) and (3.61), we get the exact solutions of Eqs. (3.7) and (3.8) take the following form:
0(f ) = —10 (-20a3a4 + 20a2 - 20a2a4 - 30a2a3 - 5a2 - £2 - 6k2 - 5a2 + 6Di - 6 £1 )
— 6\f—(a2 + a3 )
■ a—a4 + (a—a3 + a— — 2a3a4)sn2^— (a— — a4)(I — no), y^
(a——a3 )(a2+a3 — 2a4)
a4 — a— + (a— + a3 — 2a4)sn2(^—(a2 — a4)(I — no), y^
fe^ )(a2+a3 —2^4) («4 —a— )2 y
a— — a—a4 + (a—a3 + a— — 2a3a4)sn2(j(a2 — a4)(I — no), y^ a4 — a— + (a— + a3 — 2a4)sn2(— (a2 — a4)( I — no), y^
fe^ )(a2+a3 —2а4)
(a4—a2)
(^2—^3 )(a2+a3 — 2a4 )
(a4 —a2)
(3.88)
f(I ) = — Di — - k2 — 10 ß— + — ß— + - ß1 + 2a3a4 — 2a4— + 2a—a4 + 3a—a3 + - a—2 + - a—
— 6(a2 + a3 )
a— — a—a4 + (a—a3 + a— — 2a3a4)sn2(j(a2 — a4)(I — no), y^
(a2—a3 )(a2+a3 —2a4)
(a4—a2)
a4 — a— + (a— + a3 — 2a4)sn2(j (a— — a4)(I — no), y^
fe^ )(a2+a3 —2^4)
(a4 —a2)
a— — a—a4 + (a—a3 + a— — 2a3a4)sn2(j (a— — a4)(I — no), y^
(a2—a3 )(a2+a3 —2а4)
(an—a— )2
a4 — a— + (a— + a3 — 2a4)sn2(j(a2 — a4)(I — no), y^
fe^ )(a2+a3 —2^4)
(a4 —a2)
(3.89)
Family 8. If Y4 + | Y3 + I- Y2 + Y + can be written in the following form:
Y4 + y 3 +
т— — 10V2t2t0 — 12k2T2 — 2ß2T2 — 12ß1T2 + 12ct;
^ ^30V2t2t0--1— т12 — 36k2 т2 — 6ß2T2 — 36ß1T2 + 36ct2
(—30V2t—t2t0 + t 4 + 36k2 t —т2 + 6T—ß2T2 + 36T—ß1T2 — 36т—c2 — 288k2T0T22
—288ß1ToT22 + 288ctot22 + 1 20V2t,2t22 — 48то T22ß— )] = (Y — (N1 + iN, ))( Y — (N1 — iN— ))(Y — (N3 + iN4 ))(Y — (N3 — iN )),
(3.90)
where Ni, (i = 1 , 2, 3, 4) are arbitrary constants. From equating the coefficients of Y in both sides of Eq. (3.90), we get a system of algebraic equations in Z0, t2, t1, t0 and c which can be solved by using the Maple software package to get the following results:
c = Di, (i = 1, 2), N1 = N3,
To = (ß— + 6 ß1 + 6k2 — 6Di + 20(3N34 + N44 + N—4 )), t1 = — 12V2N3, т2 = 6V2,
where Di = 1 ß2 + ß1 + k2 ± f ^N4 + N4 — N2N42. Eqs. (3.91), (3.62) and (3.63) lead to get:
(3.91)
A = — ß— — 12k4 — 2ß—k2 — - N—N— + 4 N44 + - N—4
2 2 8,, 1
T0 = - ß1 — D,- — k2--ß
5ß1 5 '5 10ß
: + 2(3N— + N— + N— ), I4 = Zo,
T1 = —12N3, T, = 6, Io = Zo (N—N— + N—N— + N,2n2 + N34 ),
I1 = —2(2N32 + N— + N,2 N3Z0, I, = (6N32 + N— + N— )Zo, I3 = —4N3Z0,
(3.92)
where f0 is an arbitrary constant and
±(g - n0) =
J(Y2 - 2N3Y + N2 + N2)(Y2 - 2N3Y + N2 + N42)
(N2 - N4 )
ElliplticF
(N2 - N4)(—Y + N3 + iN4) (N2 + N4) (N2 + N4)(—Y + N3 - iN4), (N2 - N4)
(3.93)
(N4 - N2)(N + iN4) + (N4 + N2)(N - iN4)sn2{2(N2 - N4)(g - n)
(N2+N4 ) (N2—N4 )
(N4 - N2) + (N4 + N2)sn2(2(N2 - N4)(g - n0)
(N2+N4 ) (N2—N4 ))
(3.94)
Substituting (3.94), (3.92) and (3.91) into (3.60) and (3.61), we get the exact solutions of Eqs. (3.7) and (3.8) have the form:
0(g) = TÔ(P2 + 6P1 + 6k2 - 6Di + 20(3N34 + N44 + N4))
- 12V2N
(N4 - N2)(N3 + iN4) + (N4 + N2)(N3 - iN4)sn^1 (N2 - N4)(g - n0), N—N))
(N2+N4 ) (N2—N4 )
(N4 - N2) + (N4 + N2)sn2(2(N2 - N4)(g - n0), (N4 - N2)(N + iN4) + (N4 + N2)(N3 - iN4)sn2{2(N2 - N4)(g - n) (N4 - N2) + (N4 + N2)sn2(2(N2 - N4)(g - n0), N—N))
(N2+N4 ) (N2-N4 )
2 2 8 2 1
f(g) = - P1 - - Di - - k2--P
f(g) 5P1 5 ' 5 10P
+ 2(3 N2 + N42 + N22 )
— 12N3
(N4 - N2)(N3 + iN4) + (N4 + N2)(N3 - iN4)sn2^f (N2 - N4)(g - n0),
(N2+N4 ) (N2-N4 )
(N2+N4 ) (N2-N4 )
(N4 - N2) + (N4 + N2)sn2(2(N2 - N4)(g - n0) (N4 - N2)(N3 + iN4) + (N4 + N2)(N3 - iN4)sn2{2(N2 - N4)(g - n0) (N4 - N2 ) + (N4+N2 )sn2(2 (N2 - N4 )(g - n0 ), N+N) )
(N2+N4 ) (N2-N4 )
(3.95)
(3.96)
4. Conclusion
In this paper, we used the extended trial equation method to construct a series of some new analytic exact solutions for some nonlinear partial differential equations in mathematical physics when the balance number is a positive integer. We constructed the exact solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Ja-cobi elliptic functions solutions and rational solutions for the nonlinear coupled nonlinear Schrodinger Boussinesq equations. This method is more powerful than other method for solving the nonlinear partial differential equations. This method can be used to solve many nonlinear partial differential equations in mathematical physics.
Acknowledgments
The author expresses his sincere thanks to the referees for their valuable suggestions and comments.
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