- Email: [email protected]
Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine–cosine and the exp-function methods
Applied Mathematics and Computation 218 (2011) 2259–2268
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine–cosine and the exp-function methods E.M.E. Zayed ⇑, M.A.M. Abdelaziz Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
a r t i c l e
i n f o
Keywords: The generalized nonlinear Schrödinger equation with variable coefficients Generalized extended tanh-function method Sine–cosine method Exp-function method Exact solutions
a b s t r a c t In this article we find the exact traveling wave solutions of the generalized nonlinear Schrödinger (GNLS) equation with variable coefficients using three methods via the generalized extended tanh-function method, the sine–cosine method and the exp-function method. The main objective of this article is to compare the efficiency of these methods by delivering the exact traveling wave solutions of the proposed nonlinear equation. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In the nonlinear science, many important phenomena in various fields can be described by the nonlinear evolution equations (NLEES). Searching for exact soliton solutions of NLEEs plays an important and a significant role in the study on the dynamics of those phenomena. With the development of soliton theory, many powerful methods for obtaining the exact solutions of NLEES have been presented, such as the extended tanh-function method [1–5], the tanh-sech method [6–8], the sine–cosine method [9–11], the homogeneous balance method [12,13], the exp-function method [14–17], the Jacobi elliptic function method [18–21], the F-expansion method [22], the homotopy perturbation method [23,24], the variational iteration method [25], the inverse scattering transformation method [26], the Bäcklund transformation method [27], the Hirota bilinear method [28,29] and so on. To our knowledge, most of the aforementioned methods are related to constant coefficients models. Recently, much attention has been paid to the variable-coefficient nonlinear equations which can describe many nonlinear phenomena more realistically than their constant-coefficient ones. In the present article, we discuss two important objectives. Firstly, we apply the generalized extended tanh-function method, the sine–cosine method and the exp-function method to find the exact solutions of the following generalized nonlinear Schrödinger (GNLS) equation with variable coefficients [30]:
1 iux þ bðxÞutt þ aðxÞujuj2 icðxÞu ¼ 0; 2
ð1:1Þ
pffiffiffiffiffiffiffi where u = u(x, t) is a real or complex valued function of x, t and i ¼ 1. Secondly, we are comparing the efficiency of these methods. The coefficients a(x), b(x) and c(x) in Eq. (1.1) are functions of the indicated variable x, satisfying the condition a(x)b(x) / c(x)2 in order that Eq. (1.1) has the given solutions presented in this paper.
⇑ Corresponding author. E-mail addresses: [email protected] (E.M.E. Zayed), [email protected] (M.A.M. Abdelaziz). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.07.043
2260
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
2. Description of the generalized extended tanh-function method Suppose that a nonlinear evolution equation is given by
Fðu; ut ; ux ; utt ; uxx ; . . .Þ ¼ 0;
ð2:1Þ
where u = u(x, t) is an unknown function, F is a polynomial in u and its partial derivatives, in which the highest order derivatives and non-linear terms are involved. In the following, we give the main steps of this method: Step 1. Using the generalized wave transformation
uðx; tÞ ¼ uðnÞ;
n ¼ pðxÞt þ qðxÞ;
ð2:2Þ
where p(x) and q(x) are differentiable functions of x to be determined. Then, Eq. (2.1) is reduced to the following ODE:
Qðu; pðxÞu0 ðnÞ; ½p0 ðxÞt þ q0 ðxÞu0 ðnÞ; pðxÞ2 u00 ðnÞ; . . .Þ ¼ 0;
ð2:3Þ
where Q is a polynomial in u and its total derivatives, while 0 denotes the derivative with respect to the indicated variable. Step 2. We suppose that Eq. (2.3) has the following formal solution:
uðnÞ ¼ a0 þ
N h i X ak Y k ðnÞ þ ak Y k ðnÞ ;
ð2:4Þ
i¼0
where N is a positive integer, and a0, ak, ak are constants, while Y(n) is given by
YðnÞ ¼ tanhðnÞ:
ð2:5Þ
The independent variable (2.5) leads to the following derivatives:
d d ¼ ð1 Y 2 Þ ; dn dY " # 2 2 d d 2Þ 2 d þ ð1 Y Þ 2 ; ¼ ð1 Y 2Y dY dn2 dY " # 3 2 3 d d 2Þ 2 2 d 2 2 d þ 6Yð1 Y Þ 2 þ ð1 Y Þ ¼ ð1 Y ð6Y 2Þ ; 3 dY dn3 dY dY
ð2:6Þ
and so on. Step 3. Determine the positive integer N in (2.4) by balancing the highest order derivatives and nonlinear terms in Eq. (2.3). Step 4. Substituting (2.4) along with (2.6) into (2.3) and equating the coefficients of tjYs(n) to zero, we get a system of algebraic equations. Step 5. Solving these algebraic equations by Maple or Mathematica, we get the values of a0, ak, ak, p(x) and q(x). Step 6. Substituting these values into (2.4) and (2.2), we can obtain the exact traveling wave solutions of Eq. (2.1). 3. On using the generalized extended tanh-function method to solve Eq. (1.1) Assume that the solution of Eq. (1.1) can be written in the form
uðx; tÞ ¼ tðx; tÞ exp½ihðx; tÞ;
ð3:1Þ
where t(x, t) and h(x, t) are amplitude and phase functions respectively. Substituting (3.1) into (1.1) and separating the real and imaginary parts, we obtain
1 thx þ bðxÞ½ttt tðht Þ2 þ aðxÞt3 ¼ 0 2
ð3:2Þ
and
1 2
tx þ bðxÞ½2tt ht þ thtt cðxÞt ¼ 0:
ð3:3Þ
Balancing ttt and t3 in Eq. (3.2), we have N = 1. We assume that Eqs. (3.2) and (3.3) have the following formal solutions
tðnÞ ¼ a0 þ a1 YðnÞ þ a1 Y 1 ðnÞ; 2
hðx; tÞ ¼ f ðxÞt þ gðxÞt þ hðxÞ:
ð3:4Þ ð3:5Þ
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
2261
where f(x), g(x) and h(x) are differentiable functions to be determined. From (2.6) and (3.4) we have the derivatives
t0 ¼ ½a1 Y 2 ðnÞ þ a1 Y 2 ðnÞ þ a1 þ a1 ; t00 ¼ 2½a1 Y 3 ðnÞ þ a1 Y 3 ðnÞ 2½a1 YðnÞ þ a1 Y 1 ðnÞ; t000 ¼ 6½a1 Y 4 ðnÞ þ a1 Y 4 ðnÞ þ 8½a1 Y 2 ðnÞ þ a1 Y 2 ðnÞ 2½a1 þ a1 ;
ð3:6Þ ð3:7Þ ð3:8Þ
and so on. Substituting (3.4)–(3.8) along with (2.2) into (3.2) and (3.3), collecting the coefficients of tjYs(n), (i = 0, 1, 2, s = 3, 2, 1, 0, 1, 2, 3) and setting each coefficient to zero, we obtain the following system of algebraic equations:
t 0 Y 3 : a1 ½bðxÞpðxÞ2 þ a31 aðxÞ ¼ 0; t 0 Y 3 : a1 ½bðxÞpðxÞ2 þ a31 aðxÞ ¼ 0; t 0 Y 2 : 3a0 a21 aðxÞ ¼ 0; t0 Y
2
a1 ½q0 ðxÞ þ bðxÞpðxÞgðxÞ ¼ 0;
: 3a0 a21 aðxÞ ¼ 0;
a1 ½q0 ðxÞ þ bðxÞpðxÞgðxÞ ¼ 0; 1 0 t 0 Y 1 : a1 ½h ðxÞ þ bðxÞpðxÞ2 þ bðxÞgðxÞ2 3aðxÞða1 a1 þ a20 Þ ¼ 0; : a1 ½bðxÞf ðxÞ cðxÞ ¼ 0; 2 1 0 2 0 1 t Y : a1 ½h ðxÞ þ bðxÞpðxÞ þ bðxÞgðxÞ2 3aðxÞða1 a1 þ a20 Þ ¼ 0; : a1 ½bðxÞf ðxÞ cðxÞ ¼ 0; 2 1 0 t 0 Y 0 : a0 ½h ðxÞ þ bðxÞgðxÞ2 aðxÞð6a1 a1 þ a20 Þ ¼ 0; : ða1 þ a1 Þ½q0 ðxÞ þ bðxÞpðxÞgðxÞ þ a0 ½bðxÞf ðxÞ cðxÞ ¼ 0; 2 t 1 Y 2 : a1 ½p0 ðxÞ þ 2bðxÞpðxÞf ðxÞ ¼ 0; t 1 Y 2 : a1 ½p0 ðxÞ þ 2bðxÞpðxÞf ðxÞ ¼ 0; t 1 Y 1 : a1 ½g 0 ðxÞ þ 2bðxÞgðxÞf ðxÞ ¼ 0; t 1 Y 1 : a1 ½g 0 ðxÞ þ 2bðxÞgðxÞf ðxÞ ¼ 0; t 1 Y 0 : a0 ½g 0 ðxÞ þ 2bðxÞgðxÞf ðxÞ ¼ 0;
: ða1 þ a1 Þ½p0 ðxÞ þ 2bðxÞpðxÞf ðxÞ ¼ 0;
t 2 Y 1 : a1 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 ¼ 0; t 2 Y 1 : a1 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 ¼ 0; t 2 Y 0 : a0 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 ¼ 0: ð3:9Þ Solving the algebraic Eq. (3.9) by Maple, we have the following cases of solutions: Case1.
a0 ¼ 0; hðxÞ ¼
a1 ¼ 0;
a1 ¼ a1 ;
1 2 ðc þ 2c22 Þf ðxÞ þ c4 ; 4 1
pðxÞ ¼ c2 f ðxÞ;
cðxÞ ¼ bðxÞf ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2 c2 bðxÞf ðxÞ2 aðxÞ ¼ 2 2 ; a1
qðxÞ ¼
gðxÞ ¼ c1 f ðxÞ; ð3:10Þ
where ci(i = 15) are arbitrary constants, such that c2 – 0 and f(x) is given by
Z f ðxÞ 2 bðxÞdx þ c5 ¼ 1:
ð3:11Þ
In this case, the exact solution of Eq. (1.1) has the from:
1 uðnÞ ¼ a1 cothðnÞ exp if ðxÞ t 2 þ c1 t þ ðc21 þ 2c22 Þ : 4
ð3:12Þ
Case2.
a0 ¼ 0; hðxÞ ¼
a1 ¼ a1 ;
a1 ¼ 0;
1 2 ðc þ 2c22 Þf ðxÞ þ c4 ; 4 1
pðxÞ ¼ c2 f ðxÞ;
cðxÞ ¼ bðxÞf ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2 c2 bðxÞf ðxÞ2 aðxÞ ¼ 2 2 : a1
qðxÞ ¼
gðxÞ ¼ c1 f ðxÞ; ð3:13Þ
In this case, the exact solution of Eq. (1.1) has the from:
1 uðnÞ ¼ a1 tanhðnÞ exp ifðxÞ t2 þ c1 t þ ðc21 þ 2c22 Þ : 4
ð3:14Þ
2262
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
Case3.
a0 ¼ 0; hðxÞ ¼
a1 ¼ a1 ;
a1 ¼ a1 ;
1 2 ðc 4c22 Þf ðxÞ þ c4 ; 4 1
pðxÞ ¼ c2 f ðxÞ;
cðxÞ ¼ bðxÞf ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2 2 c bðxÞf ðxÞ2 aðxÞ ¼ 2 2 : a1 qðxÞ ¼
gðxÞ ¼ c1 f ðxÞ; ð3:15Þ
In this case, the exact solution of Eq. (1.1) has the from:
1 uðnÞ ¼ a1 ½tanhðnÞ cothðnÞ exp if ðxÞ t 2 þ c1 t þ ðc21 4c22 Þ : 4
ð3:16Þ
which can be written in the form
ip 1 uðnÞ ¼ 2ia1 sech 2n þ exp if ðxÞ t2 þ c1 t þ ðc21 4c22 Þ ; 4 2
ð3:17Þ
where
n¼
1 c2 f ðxÞðc1 þ 2tÞ þ c3 : 2
ð3:18Þ
Remark 1. The following observations are a consequence of the remarks in the [31]: 1. We note that n given by (3.18) contains an arbitrary constant c3. So, it is always possible to phase shift n by an arbitrary constant amount. 2. We note the following identities: (i) tanhh + cothh = 2coth (2h), (ii) tanh h coth h ¼ 2isechð2h þ i2pÞ, (iii) coth (ih) = icoth, (iv) tanh (ih) = itanh, (v) sech (ih) = sech. 3. The multiplier of the exp-function in the expressions (3.12), (3.14) and (3.17) can be written in the form AY where A is an arbitrary constant and Y = cothn in case 1, Y = tanhn in case 2 and Y = sechn in case 3 having the identity (ii) and the transformation c2 ! c22 . 4. If we use the identities (iii), (iv) and (v) and the transformation c2 ? ic2, then Y can also be cotn, tann and secn respectively. Remark 2. From cases 1, 2 and 3 we deduce that a(x) / b (x)f(x)2. But c(x) = b(x)f(x), then we conclude that
aðxÞbðxÞ / cðxÞ2 :
ð3:19Þ
4. Description of the sine–cosine method In this method, we admit the use of the relation (3.1), where t (n) has the form
tðnÞ ¼ kðxÞ sinm ðnÞ; jnj <
p 2
ð4:1Þ
or
p tðnÞ ¼ kðxÞ cosm ðnÞ; jnj < ; 2
ð4:2Þ
where m is a constant while k(x) is a differentiable function of x to be determined later. From (4.1) and (4.2) we have
tx ¼
dkðxÞ dpðxÞ dqðxÞ m m1 sin ðnÞ þ mkðxÞ tþ sin ðnÞ cosðnÞ; dx dx dx
tt ¼ mkðxÞpðxÞ sinm1 ðnÞ cosðnÞ; ttt ¼ mkðxÞpðxÞ2 ½m sinm ðnÞ þ ðm 1Þ sinm2 ðnÞ
ð4:3Þ
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
2263
or
tx ¼
dkðxÞ dpðxÞ dqðxÞ cosm1 ðnÞ sinðnÞ; cosm ðnÞ mkðxÞ tþ dx dx dx
tt ¼ mkðxÞpðxÞ cosm1 ðnÞ sinðnÞ;
ttt ¼ mkðxÞpðxÞ2 m cosm ðnÞ þ ðm 1Þ cosm2 ðnÞ :
ð4:4Þ
From (3.5) we have
df ðxÞ 2 dgðxÞ dhðxÞ t þ tþ ; dx dx dx ht ¼ 2f ðxÞt þ gðxÞ;
hx ¼
ð4:5Þ
htt ¼ 2f ðxÞ:
ð4:7Þ
ð4:6Þ
Substituting (4.1), (4.3) and (4.5)–(4.7) along with (3.5) or (4.2), (4.4) and (4.5)–(4.7) along with (3.5) into (3.2) and (3.3) a trigonometric equations are obtained with either tnsinR(n) or tncosR(n) terms so that the parameter R can be determined by comparing exponents. Equating the coefficients of tnsinR(n) and tncosR(n) to zero, we get a system of algebraic equations. Solving these algebraic equations by Maple or Mathematica, we get the values of m, k(x), p(x), q(x), f(x), g(x), and h(x). Substituting these values into (3.1), we can obtain the exact traveling wave solutions of Eq. (1.1). 5. On using the sine–cosine method to solve Eq. (1.1) In this section, we shall derive the exact solutions of Eq. (1.1) using the sine–cosine method. To this end, we first apply the cosine method to Eq. (1.1). Substituting (4.2), (4.4) and (4.5)–(4.7) along with (3.5) into (3.2) and (3.3), we have
dkðxÞ dpðxÞ dqðxÞ cosm ðnÞ mkðxÞ tþ cosm1 ðnÞ sinðnÞ mkðxÞbðxÞpðxÞ½2f ðxÞt þ gðxÞ cosm1 ðnÞ sinðnÞ dx dx dx þ kðxÞbðxÞf ðxÞ cosm ðnÞ kðxÞcðxÞ cosm ðnÞ ¼ 0
ð5:1Þ
df ðxÞ 2 dgðxÞ 1 1 t cosm ðnÞ kðxÞ t cosm ðnÞ m2 kðxÞbðxÞpðxÞ2 cosm ðnÞ þ mðm 1ÞkðxÞbðxÞpðxÞ2 cosm2 ðnÞ dx dx 2 2 dhðxÞ 1 2 2 2 m m m kðxÞ cos ðnÞ 2kðxÞbðxÞf ðxÞ t cos ðnÞ kðxÞbðxÞgðxÞ cos ðnÞ 2kðxÞbðxÞf ðxÞgðxÞt cosm ðnÞ dx 2 þ kðxÞ3 aðxÞ cos3m ðnÞ ¼ 0;
ð5:2Þ
and
kðxÞ
which are satisfied only if the following conditions hold:
8 dkðxÞ > > > dx þ kðxÞbðxÞf ðxÞ kðxÞcðxÞ ¼ 0; < > > > :
mkðxÞ dqðxÞ mkðxÞbðxÞpðxÞgðxÞ ¼ 0; dx mkðxÞ dpðxÞ dx
ð5:3Þ
2mkðxÞbðxÞpðxÞf ðxÞ ¼ 0
and
8 kðxÞ dfdxðxÞ 2kðxÞbðxÞf ðxÞ2 ¼ 0; > > > > > > > > kðxÞ dgðxÞ 2kðxÞbðxÞf ðxÞgðxÞ ¼ 0; > dx > > > > dhðxÞ < kðxÞ dx 12 m2 kðxÞbðxÞpðxÞ2 > > > 12 kðxÞbðxÞgðxÞ2 ¼ 0; > > > > > 1 > > mðm 1ÞkðxÞbðxÞpðxÞ2 þ kðxÞ3 aðxÞ ¼ 0; > 2 > > : m 2 ¼ 3m:
ð5:4Þ
Similar results are also obtained by using the sine method which are omitted here. Solving the algebraic Eqs. (5.3) and (5.4) by Maple, we have the following cases of solutions:
2264
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
Case1.
Z qðxÞ ¼ c1 c2 bðxÞdx þ c5 ; f ðxÞ ¼ 0; Z 1 gðxÞ ¼ c1 ; hðxÞ ¼ ðc21 þ c22 Þ bðxÞdx þ c4 ; cðxÞ ¼ cðxÞ; 2 2 R c2 2 cðxÞdx aðxÞ ¼ bðxÞe ; bðxÞ ¼ bðxÞ; m ¼ 1; c3 kðxÞ ¼ c3 e
R
cðxÞdx
;
pðxÞ ¼ c2 ;
ð5:5Þ
where ci(i = 15) are arbitrary constants, such that c1, c2 and c3 – 0. In this case, the exact solutions of Eq. (1.1) have the forms:
Z Z 1 uðnÞ ¼ c3 secðnÞ exp i c1 t ðc21 þ c22 Þ bðxÞdx þ c4 þ cðxÞdx ; 2
ð5:6Þ
where
Z n ¼ c2 t c1 bðxÞdx þ c5 :
ð5:7Þ
Case2.
kðxÞ ¼
qffiffiffiffiffiffiffiffiffi R f ðxÞe cðxÞdx ;
pðxÞ ¼ c2 f ðxÞ; qðxÞ ¼
1 hðxÞ ¼ ðc21 þ c22 Þf ðxÞ þ c4 ; R4 aðxÞ ¼ c22 bðxÞf ðxÞe2 cðxÞdx ; bðxÞ ¼ bðxÞ; gðxÞ ¼ c1 f ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2
cðxÞ ¼ cðxÞ
ð5:8Þ
m ¼ 1;
where ci(i = 14) are arbitrary constants, such that c1, c2 – 0 and f(x) has the same form (3.11). In this case, the exact solutions of Eq. (1.1) have the forms:
uðnÞ ¼
Z qffiffiffiffiffiffiffiffiffi 1 f ðxÞ secðnÞ exp if ðxÞ t 2 þ c1 t þ ðc21 þ c22 Þ þ cðxÞdx ; 4
ð5:9Þ
where n is given by (3.18). Remark 3 (i) When k(x) is a constant we deduce that the solution (5.9) of case 2 reduces to the solution (3.16) of case 3 in Section 3. R To show this, we see that if k(x) is a constant then, we deduce from the conditions (5.8) that f ðxÞ / e2 cðxÞdx ; aðxÞ / bðxÞf 2 ðxÞ and c(x) = b(x)f(x). Thus the conditions (5.8) of case 2 agree with the conditions (3.15) of case 3 in Section 3. (ii) By the use of the transformation c2 ? ic2 we get sec (n) ? sech (n). 6. Description of the exp-function method and its use to solve Eq. (1.1) The exp-function method was first proposed by He and Wu [14]. Using the generalized wave transformation (2.2), Eqs. (3.2) and (3.3) become
1 1 0 tðnÞft2 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 þ t½g 0 ðxÞ þ 2bðxÞf ðxÞgðxÞ þ bðxÞgðxÞ2 þ h ðxÞg þ bðxÞpðxÞ2 t00 ðnÞ þ aðxÞt3 ðnÞ ¼ 0 2 2
ð6:1Þ
t0 ðnÞftp0 ðxÞ þ q0 ðxÞ þ ½2tf ðxÞ þ gðxÞbðxÞpðxÞg þ tðnÞ½bðxÞf ðxÞ cðxÞ ¼ 0:
ð6:2Þ
and
In this method, we admit the use of the relation (3.1), where t (n) has the form
Pd
tðnÞ ¼ Pqn¼c
an expðnnÞ ; expðmnÞ
m¼p bm
ð6:3Þ
where c, d, p and q are positive integers which are unknown to be determine later, an and bm are unknown constants. Eq. (6.3) can be written in the form
tðnÞ ¼
ac expðcnÞ þ . . . þ ad expðdnÞ : ap expðpnÞ þ . . . þ aq expðqnÞ
ð6:4Þ
2265
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
In order to determine values of c and p, we balance the linear term of the highest order derivatives t00 with the highest order nonlinear term t3 in Eq. (6.1). By simple calculations, we have
c1 exp½ðc þ 3pÞn þ . . . ; c2 exp½4pn þ c3 exp½ð3c þ pÞn þ . . . ¼ ; c4 exp½4pn þ
t00 ¼
ð6:5Þ
t3
ð6:6Þ
where ci are coefficients for simplicity. By balancing the highest order of exp-function in Eqs. (6.5) and (6.6), we have c + 3p = 3c + p which leads to
p ¼ c:
ð6:7Þ
Similarly, from the ansatz (6.4), we have
. . . þ d1 exp½ðd þ 3qÞn ; þ d2 exp½4qn þ d3 exp½ð3d þ qÞn ¼ ; þ d4 exp½4qn
t00 ¼
ð6:8Þ
t3
ð6:9Þ
where di are coefficients for simplicity. By balancing the lowest order of exp-function in Eqs. (6.8) and (6.9), we have (d + 3q) = (3d + q) which leads to
q ¼ d:
ð6:10Þ
Choosing p = c = 1 and q = d = 1, Eq. (6.4) becomes
tðnÞ ¼
a1 expðnÞ þ a0 þ a1 expðnÞ : b1 expðnÞ þ b0 þ b1 expðnÞ
ð6:11Þ
Substituting Eq. (6.11) into Eqs. (6.1) and (6.2), we get 3 1 X ðri þ t 2 qi þ t ei Þ ¼ 0; A i¼3
2 1X ðli þ t xi Þ ¼ 0; B i¼2
ð6:12Þ
where
A ¼ ðb1 expðnÞ þ b0 þ b1 expðnÞÞ3 ;
B ¼ ðb1 expðnÞ þ b0 þ b1 expðnÞÞ2 ;
dhðxÞ 1 2 a1 b1 bðxÞgðxÞ2 þ a31 aðxÞ ¼ 0; dx 2 dhðxÞ 1 1 1 2 dhðxÞ 2 2 ¼ a0 b1 2a1 b1 b0 a1 b1 b0 bðxÞgðxÞ2 a0 b1 bðxÞgðxÞ2 þ 3a21 a0 aðxÞ a1 b1 b0 bðxÞpðxÞ2 þ a0 b1 bðxÞpðxÞ2 ¼ 0; dx dx 2 2 2 dhðxÞ dhðxÞ 1 2 dhðxÞ 2 dhðxÞ 2 ¼ a1 b0 a1 b1 2a0 b1 b0 2a1 b1 b1 a1 b0 bðxÞgðxÞ2 dx dx dx dx 2 1 1 1 2 2 a1 b1 bðxÞgðxÞ2 a1 b1 b1 bðxÞgðxÞ2 a0 b1 b0 bðxÞgðxÞ2 a0 b1 b0 bðxÞpðxÞ2 þ a1 b0 bðxÞpðxÞ2 2a1 b1 b1 bðxÞpðxÞ2 2 2 2
r3 ¼ a1 b21 r2 r1
2
þ 2a1 b1 bðxÞpðxÞ2 þ 3a21 a1 aðxÞ þ 3a1 a20 aðxÞ ¼ 0;
r0 ¼ 2a0 b1 b1
dhðxÞ dhðxÞ dhðxÞ 1 2 dhðxÞ 2 2 2a1 b1 b0 2a1 b1 b0 a0 b0 a0 b0 bðxÞgðxÞ dx dx dx dx 2 2
2
2
a1 b1 b0 bðxÞgðxÞ a0 b1 b1 bðxÞgðxÞ a1 b1 b0 bðxÞgðxÞ þ a30 aðxÞ þ 6a0 a1 a1 aðxÞ 3 3 2 2 2 3a0 b1 b1 bðxÞpðxÞ þ a1 b1 b0 bðxÞpðxÞ þ a1 b1 b0 bðxÞpðxÞ ¼ 0; 2 2 dhðxÞ dhðxÞ 1 2 dhðxÞ 2 dhðxÞ 2 2 r1 ¼ 2a1 b1 b1 2a0 b1 b0 a1 b1 a1 b0 a1 b1 bðxÞgðxÞ dx dx dx dx 2 1 1 2 2 a1 b0 bðxÞgðxÞ2 a1 b1 b1 bðxÞgðxÞ2 a0 b1 b0 bðxÞgðxÞ2 þ 3a1 a21 aðxÞ þ 3a1 a20 aðxÞ þ a1 b0 bðxÞpðxÞ2 2 2 1 2 a0 b0 b1 bðxÞpðxÞ2 þ 2a1 b1 bðxÞpðxÞ2 2a1 b1 b1 bðxÞpðxÞ2 ¼ 0; 2 dhðxÞ dhðxÞ 1 2 r2 ¼ a0 b21 2a1 b1 b0 a1 b1 b0 bðxÞgðxÞ2 a0 b1 bðxÞgðxÞ2 þ 3a0 a21 aðxÞ dx dx 2 1 1 1 2 2 dhðxÞ 2 a1 b1 b0 bðxÞpðxÞ2 þ a0 b1 bðxÞpðxÞ2 ¼ 0; r3 ¼ a1 b1 a1 b1 bðxÞgðxÞ2 þ a31 aðxÞ ¼ 0; 2 2 dx 2 df ðxÞ 2 df ðxÞ 2 2 df ðxÞ 2 2 2 2 2a1 b1 bðxÞf ðxÞ ¼ 0; t 2 q2 ¼ a0 b1 2a1 b1 b0 2a0 b1 bðxÞf ðxÞ 4a1 b1 b0 bðxÞf ðxÞ ¼ 0; t 2 q3 ¼ a1 b1 dx dx dx
2266
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
df ðxÞ df ðxÞ df ðxÞ 2 df ðxÞ 2 2 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞ 4a0 b1 b0 bðxÞf ðxÞ dx dx dx dx df ðxÞ 2 2 2 df ðxÞ 2a1 b1 bðxÞf ðxÞ2 2a1 b0 bðxÞf ðxÞ2 ¼ 0; t 2 q0 ¼ a0 b0 2a1 b1 b0 dx dx df ðxÞ df ðxÞ 2 2 2 2a1 b1 b0 2a0 b0 bðxÞf ðxÞ 4a1 b1 b0 bðxÞf ðxÞ 4a1 b1 b0 bðxÞf ðxÞ2 4a0 b1 b1 bðxÞf ðxÞ2 ¼ 0; 2a0 b1 b1 dx dx df ðxÞ df ðxÞ 2 df ðxÞ 2 df ðxÞ 2 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞ ¼ a1 b1 dx dx dx dx 2 2 4a0 b1 b0 bðxÞf ðxÞ2 2a1 b1 bðxÞf ðxÞ2 2a1 b0 bðxÞf ðxÞ2 ¼ 0; 2
t2 q1 ¼ a1 b1
t2 q1
df ðxÞ df ðxÞ 2 2 2 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞ 2a0 b1 bðxÞf ðxÞ ¼ 0; dx dx 2 df ðxÞ 2 2 dgðxÞ 2 t2 q3 ¼ a1 b1 2a1 b1 bðxÞf ðxÞ2 ¼ 0; te3 ¼ a1 b1 2a1 b1 bðxÞf ðxÞgðxÞ ¼ 0; dx dx dgðxÞ 2 dgðxÞ 2 te2 ¼ a0 b1 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞgðxÞ 2a0 b1 bðxÞf ðxÞgðxÞ ¼ 0; dx dx dgðxÞ dgðxÞ 2 dgðxÞ 2 dgðxÞ te1 ¼ a1 b1 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞgðxÞ 4a0 b1 b0 bðxÞf ðxÞgðxÞ dx dx dx dx dgðxÞ dgðxÞ 2 2 2 dgðxÞ 2a1 b1 bðxÞf ððxÞgðxÞ 2a1 b0 bðxÞf ððxÞgðxÞ ¼ 0; t e0 ¼ a0 b0 2a1 b1 b0 2a0 b1 b1 dx dx dx dgðxÞ 2 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞgðxÞ 4a1 b1 b0 bðxÞf ðxÞgðxÞ 4a0 b1 b1 bðxÞf ðxÞgðxÞ 2a0 b0 bðxÞf ðxÞgðxÞ ¼ 0; dx dgðxÞ dgðxÞ 2 dgðxÞ 2 dgðxÞ te1 ¼ a1 b1 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞgðxÞ 4a0 b1 b0 bðxÞf ðxÞgðxÞ dx dx dx dx dgðxÞ 2 2 2 dgðxÞ 2a1 b1 bðxÞf ðxÞgðxÞ 2a1 b0 bðxÞf ðxÞgðxÞ ¼ 0; t e2 ¼ a0 b1 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞgðxÞ dx dx 2 2 dgðxÞ 2 2a0 b1 bðxÞf ðxÞgðxÞ ¼ 0; t e3 ¼ a1 b1 2a1 b1 bðxÞf ðxÞgðxÞ ¼ 0; l2 ¼ a1 b1 bðxÞf ðxÞ a1 b1 cðxÞ ¼ 0; dx dqðxÞ dqðxÞ l1 ¼ a1 b0 a0 b1 þ a1 b0 bðxÞgðxÞpðxÞ a0 b1 bðxÞgðxÞpðxÞ þ a0 b1 bðxÞf ðxÞ þ a1 b0 bðxÞf ðxÞ a1 b0 cðxÞ a0 b1 cðxÞ ¼ 0; dx dx dqðxÞ dqðxÞ l0 ¼ 2a1 b1 2a1 b1 þ a1 b1 bðxÞf ðxÞ þ a1 b1 bðxÞf ðxÞ þ a0 b0 bðxÞf ðxÞ þ 2a1 b1 bðxÞgðxÞpðxÞ dx dx dqðxÞ dqðxÞ a1 b0 þ a0 b1 bðxÞgðxÞpðxÞ 2a1 b1 bðxÞgðxÞpðxÞ a1 b1 cðxÞ a1 b1 cðxÞ a0 b0 cðxÞ ¼ 0; l1 ¼ a0 b1 dx dx a1 b0 bðxÞgðxÞpðxÞ þ a0 b1 bðxÞf ðxÞ þ a1 b0 bðxÞf ðxÞ a0 b1 bðxÞ a1 b0 cðxÞ ¼ 0; 2
t2 q2 ¼ a0 b1
l2 ¼ a1 b1 bðxÞf ðxÞ a1 b1 cðxÞ ¼ 0;
dpðxÞ dpðxÞ a0 b1 þ 2a1 b0 bðxÞf ðxÞpðxÞ 2a0 b1 bðxÞf ðxÞpðxÞ ¼ 0; dx dx dpðxÞ dpðxÞ 2a1 b1 þ 4a1 b1 bðxÞf ðxÞpðxÞ 4a1 b1 bðxÞf ðxÞpðxÞ ¼ 0; t x0 ¼ 2a1 b1 dx dx dpðxÞ dpðxÞ t x1 ¼ a0 b1 a1 b0 þ 2a0 b1 bðxÞf ðxÞpðxÞ 2a1 b0 bðxÞf ðxÞpðxÞ ¼ 0: dx dx
t x1 ¼ a1 b0
ð6:13Þ
Solving the system (6.13) by Maple, we have the following cases of solutions: Case1.
a0 ¼ a0 ;
a1 ¼ a1 ;
a1 ¼ 0;
b0 ¼
a0 b1 ; a1
1 c1 c2 f ðxÞ þ c3 ; gðxÞ ¼ c1 f ðxÞ; 2 2 2 0 c b f ðxÞ aðxÞ ¼ 2 1 2 ; cðxÞ ¼ bðxÞf ðxÞ; 8a1 qðxÞ ¼
b1 ¼ b1 ;
hðxÞ ¼
b1 ¼ 0;
pðxÞ ¼ c2 f ðxÞ;
1 2 c þ 2c21 f ðxÞ þ c4 ; 8 2
ð6:14Þ
where ci(i = 14) are arbitrary constants, such that c1, c2 – 0 and f(x) has the same form (3.11). In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a1 ða1 en þ a0 Þ 1 2 2 2 ; c þ c t þ þ 2c f ðxÞ þ c exp i t 1 4 1 b1 ða1 en a0 Þ 8 2
which can be written in the form
ð6:15Þ
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
uðnÞ ¼
a1 nþ/ 1 exp i t 2 þ c1 t þ c22 þ 2c21 f ðxÞ þ c4 ; coth 2 8 b1
2267
ð6:16Þ
where e/ ¼ aa10 . Case2.
a0 ¼ 0;
a1 ¼ a1 ;
a1 ¼ a1 ;
b0 ¼ 0;
1 c1 c2 f ðxÞ þ c3 ; gðxÞ ¼ c1 f ðxÞ; 2 2 c2 b f 0 ðxÞ aðxÞ ¼ 2 1 2 ; cðxÞ ¼ bðxÞf ðxÞ: 2a1 qðxÞ ¼
b1 ¼ b1 ; hðxÞ ¼
b1 ¼
a1 b1 ; a1
pðxÞ ¼ c2 f ðxÞ;
1 2 c þ 2c22 f ðxÞ þ c4 ; 4 1
ð6:17Þ
In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a1 ða1 en þ a1 en Þ 1 2 2 2 þ c t þ þ 2c ; c f ðxÞ þ c exp i t 1 4 2 b1 ða1 en a1 en Þ 4 1
ð6:18Þ
which can be written in the form
uðnÞ ¼
a1 1 cothðn þ /Þ exp i t2 þ c1 t þ c21 þ 2c22 f ðxÞ þ c4 ; 4 b1
ð6:19Þ
where e2/ ¼ aa11 . Case3.
a0 ¼ a0 ; a1 ¼ 0; a1 ¼ 0; b0 ¼ 0; b1 ¼ b1 ; b1 ¼ b1 ; pðxÞ ¼ c2 f ðxÞ; 1 1 qðxÞ ¼ c1 c2 f ðxÞ þ c3 ; gðxÞ ¼ c1 f ðxÞ; hðxÞ ¼ c21 c22 f ðxÞ þ c4 ; 2 4 2b b c2 f 0 ðxÞ aðxÞ ¼ 1 12 2 ; cðxÞ ¼ bðxÞf ðxÞ: a0
ð6:20Þ
In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a0 1 2 2 2 ; exp i t þ c t þ 2c c f ðxÞ þ c 1 4 1 2 4 b1 en þ b1 en
ð6:21Þ
which can be written in the form
a0 1 uðnÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi sechðn þ /Þ exp i t 2 þ c1 t þ c21 2c22 f ðxÞ þ c4 ; 4 2 b1 b1
ð6:22Þ
where e2/ ¼ bb11 . Case4.
a0 ¼ a0 ;
a1 ¼ a1 ;
pðxÞ ¼ c2 f ðxÞ; hðxÞ ¼
a1 ¼ a1 ;
qðxÞ ¼
b0 ¼
1 c1 c2 f ðxÞ þ c3 ; 2
1 2 c2 þ 2c21 f ðxÞ þ c4 ; 8
b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a20 4a1 a1 a1
;
b1 ¼ b1 ;
b1 ¼
a1 b1 ; a1
2
gðxÞ ¼ c1 f ðxÞ;
aðxÞ ¼
c22 b1 f 0 ðxÞ ; 8a21
ð6:23Þ
cðxÞ ¼ bðxÞf ðxÞ:
In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a1 ða1 en þ a0 þ a1 en Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
exp i t2 þ c1 t þ c22 þ 2c21 f ðxÞ þ c4 ; 8 b1 a1 en a20 4a1 a1 a1 en
ð6:24Þ
which can be written in the form
uðnÞ ¼ where e/ ¼
a1 nþ/ 1 exp i t 2 þ c1 t þ c22 þ 2c21 f ðxÞ þ c4 ; coth 2 8 b1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
a0
a0 4a1 a1 2a1
and n is given by (3.18).
ð6:25Þ
2268
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
Remark 4. From cases 1–4, we deduce that a(x) / f0 (x). But c(x) = b(x)f(x) where f(x) is given by (3.11). Differentiating (3.11), we get f0 (x) / b (x)f(x)2. Consequently we have
aðxÞbðxÞ / cðxÞ2 ;
ð6:26Þ
which is equivalent to (3.19). Remark 5. Each of these solutions is just a disguised version of one or other of the solutions of Section 3. Remark 6. All solutions of this article have been checked with Maple by putting them back into the original Eq. (1.1).
7. Conclusions In this article, the generalized extended tanh-function method, the sine–cosine method and the exp-function method have been applied to find the exact traveling wave solutions of the generalized nonlinear Schrödinger (GNLS) equation with variable coefficients. We have shown in remarks 2 and 4 that if the coefficients a(x), b (x) and c(x) in Eq. (1.1) satisfying the condition a (x)b(x) / c(x)2 then Eq. (1.1) has the solutions given in this article. On comparing the solutions constructed in Sections 3, 5 and 6 using the above three methods respectively, we have noted that, when k(x) is not a constant, the solution of Eq. (1.1) in Section 5 are different from those obtained in Sections 3 and 6. But, when k(x) is a constant we have shown in remark 3 that, the solution of case 2 in Section 5 reduces to the solution of case 3 in Section 3. Finally, we have noted under certain transformations that the solutions obtained by the exp-function method are just a disguised version of one or other of the solutions obtained by using the tanh-function method. Acknowledgment The authors thank the referee for his comments on this paper. References [1] S.A. El-Wakil, M.A. Abdou, New exact traveling wave solutions using modified extended tanh-function method, Chaos Solitons Fract. 31 (2007) 840– 852. [2] E. Fan, Extended tanh-function method, and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [3] A.M. Wazwaz, The tanh-method for traveling wave solutions of nonlinear wave equations, Appl. Math. Comput. 187 (2007) 1131–1142. [4] E.M.E. Zayed, H.M. AbdelRahman, The extended tanh-method for finding traveling wave solutions of nonlinear PDEs, Nonlinear Sci. Lett. A 1 (2) (2010) 193–200. [5] E.M.E. Zayed, M.A.M. Abdelaziz, The tanh-function method using a generalized wave transformation for nonlinear equations, Int. J. Nonlinear Sci. Numer. Simul. 11 (8) (2010) 595–601. [6] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) 650–654. [7] W. Malfliet, W. Hereman, The tanh-method part I. Exact solutions of nonlinear evolution and wave equations, Phys. Script. 54 (1996) 563–568. [8] A.M. Wazwaz, The tanh-method for traveling wave solutions of nonlinear equations, Appl. Math. Comput. 154 (2004) 713–723. [9] M. Yaghobi Moghaddam, A. Asgari, H. Yazdani, Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by extended tanh–coth, sine–cosine and Exp-function methods, Appl. Math. Comput. 210 (2009) 422–435. [10] A.M. Wazwaz, A sine–cosine method for handling nonlinear wave equations, Math. Comput. Model. 40 (2004) 499–508. [11] Y. Yang, Z.L. Tao, F.R. Austin, Solutions of the generalized KdV equation with time-dependent damping and dispersion, Appl. Math. Comput. 216 (2010) 1029–1035. [12] E. Fan, H. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. [13] M.L. Wang, Exact solutions of a compound Kdv-Burgers equations, Phys. Lett. A 213 (1996) 279–287. [14] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fract. 30 (2006) 700–708. [15] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos Solitons Fract. 34 (2007) 1421–1429. [16] S. Zhang, Application of exp-function method to high dimensional evolution equation, Chaos Solitons Fract. 38 (2008) 270–276. [17] S.D. Zhu, Exp-function method for the discrete mKdV lattice, Int. J. Nonlinear Sci. Numer. Simul. 8 (2007) 465–469. [18] C.Q. Dai, J.F. Zhang, Jacobian elliptic function method for nonlinear differential-difference equations, Chaos Solitons Fract. 27 (2006) 1042–1047. [19] E. Fan, J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A 305 (2002) 383–392. [20] S. Liu, Z.Fu. S. Liu, Q. Zhao, Jacobi elliptic function method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001) 69–74. [21] X.Q. Zhao, H.Y. Zhi, H.Q. Zhang, Improved Jacobi-function method with symbolic computation to construct new double-periodic solutions for the generalized Ito system, Chaos Solitons Fract. 28 (2006) 112–116. [22] J.L. Zhang, M.L. Wang, Y.M. Wang, Z.D. Fang, The improved F-expansion method and its applications, Phys. Lett. A 350 (2006) 103–109. [23] J.H. He, New interpretation of homotopy perturbation method, Int. J. Mod. Phys. B 20 (2006) 2561–2568. [24] J.H. He, Asymptology by homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul. 6 (2005) 207–208. [25] J.H. He, X.H. Wu, Construction of solitary solutions and compact on like solution by variational iteration method, Chaos Solitons Fract. 29 (2006) 108– 113. [26] M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. [27] R. Hirota, Direct method of finding exact solutions of nonlinear evolution equations, in: R. Bullough, P. Caudrey (Eds.), Backlund Transformation, Springer, Berlin, 1980. [28] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004. [29] A.M. Wazwaz, On multiple soliton solutions for coupled KdV-mKdV equation, Nonlinear Sci. Lett. A 1 (3) (2010) 289–296. [30] J.L. Zhang, B.A. Li, M.L. Wang, The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients, Chaos Solitons Fract. 39 (2009) 858–865. [31] E.J. Parkes, Observation on the tanh-coth expansion method for finding solutions to nonlinear evolution equations, Appl. Math. Comput. 217 (2010) 1749–1754.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine–cosine and the exp-function methods E.M.E. Zayed ⇑, M.A.M. Abdelaziz Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
a r t i c l e
i n f o
Keywords: The generalized nonlinear Schrödinger equation with variable coefficients Generalized extended tanh-function method Sine–cosine method Exp-function method Exact solutions
a b s t r a c t In this article we find the exact traveling wave solutions of the generalized nonlinear Schrödinger (GNLS) equation with variable coefficients using three methods via the generalized extended tanh-function method, the sine–cosine method and the exp-function method. The main objective of this article is to compare the efficiency of these methods by delivering the exact traveling wave solutions of the proposed nonlinear equation. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In the nonlinear science, many important phenomena in various fields can be described by the nonlinear evolution equations (NLEES). Searching for exact soliton solutions of NLEEs plays an important and a significant role in the study on the dynamics of those phenomena. With the development of soliton theory, many powerful methods for obtaining the exact solutions of NLEES have been presented, such as the extended tanh-function method [1–5], the tanh-sech method [6–8], the sine–cosine method [9–11], the homogeneous balance method [12,13], the exp-function method [14–17], the Jacobi elliptic function method [18–21], the F-expansion method [22], the homotopy perturbation method [23,24], the variational iteration method [25], the inverse scattering transformation method [26], the Bäcklund transformation method [27], the Hirota bilinear method [28,29] and so on. To our knowledge, most of the aforementioned methods are related to constant coefficients models. Recently, much attention has been paid to the variable-coefficient nonlinear equations which can describe many nonlinear phenomena more realistically than their constant-coefficient ones. In the present article, we discuss two important objectives. Firstly, we apply the generalized extended tanh-function method, the sine–cosine method and the exp-function method to find the exact solutions of the following generalized nonlinear Schrödinger (GNLS) equation with variable coefficients [30]:
1 iux þ bðxÞutt þ aðxÞujuj2 icðxÞu ¼ 0; 2
ð1:1Þ
pffiffiffiffiffiffiffi where u = u(x, t) is a real or complex valued function of x, t and i ¼ 1. Secondly, we are comparing the efficiency of these methods. The coefficients a(x), b(x) and c(x) in Eq. (1.1) are functions of the indicated variable x, satisfying the condition a(x)b(x) / c(x)2 in order that Eq. (1.1) has the given solutions presented in this paper.
⇑ Corresponding author. E-mail addresses: [email protected] (E.M.E. Zayed), [email protected] (M.A.M. Abdelaziz). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.07.043
2260
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
2. Description of the generalized extended tanh-function method Suppose that a nonlinear evolution equation is given by
Fðu; ut ; ux ; utt ; uxx ; . . .Þ ¼ 0;
ð2:1Þ
where u = u(x, t) is an unknown function, F is a polynomial in u and its partial derivatives, in which the highest order derivatives and non-linear terms are involved. In the following, we give the main steps of this method: Step 1. Using the generalized wave transformation
uðx; tÞ ¼ uðnÞ;
n ¼ pðxÞt þ qðxÞ;
ð2:2Þ
where p(x) and q(x) are differentiable functions of x to be determined. Then, Eq. (2.1) is reduced to the following ODE:
Qðu; pðxÞu0 ðnÞ; ½p0 ðxÞt þ q0 ðxÞu0 ðnÞ; pðxÞ2 u00 ðnÞ; . . .Þ ¼ 0;
ð2:3Þ
where Q is a polynomial in u and its total derivatives, while 0 denotes the derivative with respect to the indicated variable. Step 2. We suppose that Eq. (2.3) has the following formal solution:
uðnÞ ¼ a0 þ
N h i X ak Y k ðnÞ þ ak Y k ðnÞ ;
ð2:4Þ
i¼0
where N is a positive integer, and a0, ak, ak are constants, while Y(n) is given by
YðnÞ ¼ tanhðnÞ:
ð2:5Þ
The independent variable (2.5) leads to the following derivatives:
d d ¼ ð1 Y 2 Þ ; dn dY " # 2 2 d d 2Þ 2 d þ ð1 Y Þ 2 ; ¼ ð1 Y 2Y dY dn2 dY " # 3 2 3 d d 2Þ 2 2 d 2 2 d þ 6Yð1 Y Þ 2 þ ð1 Y Þ ¼ ð1 Y ð6Y 2Þ ; 3 dY dn3 dY dY
ð2:6Þ
and so on. Step 3. Determine the positive integer N in (2.4) by balancing the highest order derivatives and nonlinear terms in Eq. (2.3). Step 4. Substituting (2.4) along with (2.6) into (2.3) and equating the coefficients of tjYs(n) to zero, we get a system of algebraic equations. Step 5. Solving these algebraic equations by Maple or Mathematica, we get the values of a0, ak, ak, p(x) and q(x). Step 6. Substituting these values into (2.4) and (2.2), we can obtain the exact traveling wave solutions of Eq. (2.1). 3. On using the generalized extended tanh-function method to solve Eq. (1.1) Assume that the solution of Eq. (1.1) can be written in the form
uðx; tÞ ¼ tðx; tÞ exp½ihðx; tÞ;
ð3:1Þ
where t(x, t) and h(x, t) are amplitude and phase functions respectively. Substituting (3.1) into (1.1) and separating the real and imaginary parts, we obtain
1 thx þ bðxÞ½ttt tðht Þ2 þ aðxÞt3 ¼ 0 2
ð3:2Þ
and
1 2
tx þ bðxÞ½2tt ht þ thtt cðxÞt ¼ 0:
ð3:3Þ
Balancing ttt and t3 in Eq. (3.2), we have N = 1. We assume that Eqs. (3.2) and (3.3) have the following formal solutions
tðnÞ ¼ a0 þ a1 YðnÞ þ a1 Y 1 ðnÞ; 2
hðx; tÞ ¼ f ðxÞt þ gðxÞt þ hðxÞ:
ð3:4Þ ð3:5Þ
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
2261
where f(x), g(x) and h(x) are differentiable functions to be determined. From (2.6) and (3.4) we have the derivatives
t0 ¼ ½a1 Y 2 ðnÞ þ a1 Y 2 ðnÞ þ a1 þ a1 ; t00 ¼ 2½a1 Y 3 ðnÞ þ a1 Y 3 ðnÞ 2½a1 YðnÞ þ a1 Y 1 ðnÞ; t000 ¼ 6½a1 Y 4 ðnÞ þ a1 Y 4 ðnÞ þ 8½a1 Y 2 ðnÞ þ a1 Y 2 ðnÞ 2½a1 þ a1 ;
ð3:6Þ ð3:7Þ ð3:8Þ
and so on. Substituting (3.4)–(3.8) along with (2.2) into (3.2) and (3.3), collecting the coefficients of tjYs(n), (i = 0, 1, 2, s = 3, 2, 1, 0, 1, 2, 3) and setting each coefficient to zero, we obtain the following system of algebraic equations:
t 0 Y 3 : a1 ½bðxÞpðxÞ2 þ a31 aðxÞ ¼ 0; t 0 Y 3 : a1 ½bðxÞpðxÞ2 þ a31 aðxÞ ¼ 0; t 0 Y 2 : 3a0 a21 aðxÞ ¼ 0; t0 Y
2
a1 ½q0 ðxÞ þ bðxÞpðxÞgðxÞ ¼ 0;
: 3a0 a21 aðxÞ ¼ 0;
a1 ½q0 ðxÞ þ bðxÞpðxÞgðxÞ ¼ 0; 1 0 t 0 Y 1 : a1 ½h ðxÞ þ bðxÞpðxÞ2 þ bðxÞgðxÞ2 3aðxÞða1 a1 þ a20 Þ ¼ 0; : a1 ½bðxÞf ðxÞ cðxÞ ¼ 0; 2 1 0 2 0 1 t Y : a1 ½h ðxÞ þ bðxÞpðxÞ þ bðxÞgðxÞ2 3aðxÞða1 a1 þ a20 Þ ¼ 0; : a1 ½bðxÞf ðxÞ cðxÞ ¼ 0; 2 1 0 t 0 Y 0 : a0 ½h ðxÞ þ bðxÞgðxÞ2 aðxÞð6a1 a1 þ a20 Þ ¼ 0; : ða1 þ a1 Þ½q0 ðxÞ þ bðxÞpðxÞgðxÞ þ a0 ½bðxÞf ðxÞ cðxÞ ¼ 0; 2 t 1 Y 2 : a1 ½p0 ðxÞ þ 2bðxÞpðxÞf ðxÞ ¼ 0; t 1 Y 2 : a1 ½p0 ðxÞ þ 2bðxÞpðxÞf ðxÞ ¼ 0; t 1 Y 1 : a1 ½g 0 ðxÞ þ 2bðxÞgðxÞf ðxÞ ¼ 0; t 1 Y 1 : a1 ½g 0 ðxÞ þ 2bðxÞgðxÞf ðxÞ ¼ 0; t 1 Y 0 : a0 ½g 0 ðxÞ þ 2bðxÞgðxÞf ðxÞ ¼ 0;
: ða1 þ a1 Þ½p0 ðxÞ þ 2bðxÞpðxÞf ðxÞ ¼ 0;
t 2 Y 1 : a1 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 ¼ 0; t 2 Y 1 : a1 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 ¼ 0; t 2 Y 0 : a0 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 ¼ 0: ð3:9Þ Solving the algebraic Eq. (3.9) by Maple, we have the following cases of solutions: Case1.
a0 ¼ 0; hðxÞ ¼
a1 ¼ 0;
a1 ¼ a1 ;
1 2 ðc þ 2c22 Þf ðxÞ þ c4 ; 4 1
pðxÞ ¼ c2 f ðxÞ;
cðxÞ ¼ bðxÞf ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2 c2 bðxÞf ðxÞ2 aðxÞ ¼ 2 2 ; a1
qðxÞ ¼
gðxÞ ¼ c1 f ðxÞ; ð3:10Þ
where ci(i = 15) are arbitrary constants, such that c2 – 0 and f(x) is given by
Z f ðxÞ 2 bðxÞdx þ c5 ¼ 1:
ð3:11Þ
In this case, the exact solution of Eq. (1.1) has the from:
1 uðnÞ ¼ a1 cothðnÞ exp if ðxÞ t 2 þ c1 t þ ðc21 þ 2c22 Þ : 4
ð3:12Þ
Case2.
a0 ¼ 0; hðxÞ ¼
a1 ¼ a1 ;
a1 ¼ 0;
1 2 ðc þ 2c22 Þf ðxÞ þ c4 ; 4 1
pðxÞ ¼ c2 f ðxÞ;
cðxÞ ¼ bðxÞf ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2 c2 bðxÞf ðxÞ2 aðxÞ ¼ 2 2 : a1
qðxÞ ¼
gðxÞ ¼ c1 f ðxÞ; ð3:13Þ
In this case, the exact solution of Eq. (1.1) has the from:
1 uðnÞ ¼ a1 tanhðnÞ exp ifðxÞ t2 þ c1 t þ ðc21 þ 2c22 Þ : 4
ð3:14Þ
2262
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
Case3.
a0 ¼ 0; hðxÞ ¼
a1 ¼ a1 ;
a1 ¼ a1 ;
1 2 ðc 4c22 Þf ðxÞ þ c4 ; 4 1
pðxÞ ¼ c2 f ðxÞ;
cðxÞ ¼ bðxÞf ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2 2 c bðxÞf ðxÞ2 aðxÞ ¼ 2 2 : a1 qðxÞ ¼
gðxÞ ¼ c1 f ðxÞ; ð3:15Þ
In this case, the exact solution of Eq. (1.1) has the from:
1 uðnÞ ¼ a1 ½tanhðnÞ cothðnÞ exp if ðxÞ t 2 þ c1 t þ ðc21 4c22 Þ : 4
ð3:16Þ
which can be written in the form
ip 1 uðnÞ ¼ 2ia1 sech 2n þ exp if ðxÞ t2 þ c1 t þ ðc21 4c22 Þ ; 4 2
ð3:17Þ
where
n¼
1 c2 f ðxÞðc1 þ 2tÞ þ c3 : 2
ð3:18Þ
Remark 1. The following observations are a consequence of the remarks in the [31]: 1. We note that n given by (3.18) contains an arbitrary constant c3. So, it is always possible to phase shift n by an arbitrary constant amount. 2. We note the following identities: (i) tanhh + cothh = 2coth (2h), (ii) tanh h coth h ¼ 2isechð2h þ i2pÞ, (iii) coth (ih) = icoth, (iv) tanh (ih) = itanh, (v) sech (ih) = sech. 3. The multiplier of the exp-function in the expressions (3.12), (3.14) and (3.17) can be written in the form AY where A is an arbitrary constant and Y = cothn in case 1, Y = tanhn in case 2 and Y = sechn in case 3 having the identity (ii) and the transformation c2 ! c22 . 4. If we use the identities (iii), (iv) and (v) and the transformation c2 ? ic2, then Y can also be cotn, tann and secn respectively. Remark 2. From cases 1, 2 and 3 we deduce that a(x) / b (x)f(x)2. But c(x) = b(x)f(x), then we conclude that
aðxÞbðxÞ / cðxÞ2 :
ð3:19Þ
4. Description of the sine–cosine method In this method, we admit the use of the relation (3.1), where t (n) has the form
tðnÞ ¼ kðxÞ sinm ðnÞ; jnj <
p 2
ð4:1Þ
or
p tðnÞ ¼ kðxÞ cosm ðnÞ; jnj < ; 2
ð4:2Þ
where m is a constant while k(x) is a differentiable function of x to be determined later. From (4.1) and (4.2) we have
tx ¼
dkðxÞ dpðxÞ dqðxÞ m m1 sin ðnÞ þ mkðxÞ tþ sin ðnÞ cosðnÞ; dx dx dx
tt ¼ mkðxÞpðxÞ sinm1 ðnÞ cosðnÞ; ttt ¼ mkðxÞpðxÞ2 ½m sinm ðnÞ þ ðm 1Þ sinm2 ðnÞ
ð4:3Þ
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
2263
or
tx ¼
dkðxÞ dpðxÞ dqðxÞ cosm1 ðnÞ sinðnÞ; cosm ðnÞ mkðxÞ tþ dx dx dx
tt ¼ mkðxÞpðxÞ cosm1 ðnÞ sinðnÞ;
ttt ¼ mkðxÞpðxÞ2 m cosm ðnÞ þ ðm 1Þ cosm2 ðnÞ :
ð4:4Þ
From (3.5) we have
df ðxÞ 2 dgðxÞ dhðxÞ t þ tþ ; dx dx dx ht ¼ 2f ðxÞt þ gðxÞ;
hx ¼
ð4:5Þ
htt ¼ 2f ðxÞ:
ð4:7Þ
ð4:6Þ
Substituting (4.1), (4.3) and (4.5)–(4.7) along with (3.5) or (4.2), (4.4) and (4.5)–(4.7) along with (3.5) into (3.2) and (3.3) a trigonometric equations are obtained with either tnsinR(n) or tncosR(n) terms so that the parameter R can be determined by comparing exponents. Equating the coefficients of tnsinR(n) and tncosR(n) to zero, we get a system of algebraic equations. Solving these algebraic equations by Maple or Mathematica, we get the values of m, k(x), p(x), q(x), f(x), g(x), and h(x). Substituting these values into (3.1), we can obtain the exact traveling wave solutions of Eq. (1.1). 5. On using the sine–cosine method to solve Eq. (1.1) In this section, we shall derive the exact solutions of Eq. (1.1) using the sine–cosine method. To this end, we first apply the cosine method to Eq. (1.1). Substituting (4.2), (4.4) and (4.5)–(4.7) along with (3.5) into (3.2) and (3.3), we have
dkðxÞ dpðxÞ dqðxÞ cosm ðnÞ mkðxÞ tþ cosm1 ðnÞ sinðnÞ mkðxÞbðxÞpðxÞ½2f ðxÞt þ gðxÞ cosm1 ðnÞ sinðnÞ dx dx dx þ kðxÞbðxÞf ðxÞ cosm ðnÞ kðxÞcðxÞ cosm ðnÞ ¼ 0
ð5:1Þ
df ðxÞ 2 dgðxÞ 1 1 t cosm ðnÞ kðxÞ t cosm ðnÞ m2 kðxÞbðxÞpðxÞ2 cosm ðnÞ þ mðm 1ÞkðxÞbðxÞpðxÞ2 cosm2 ðnÞ dx dx 2 2 dhðxÞ 1 2 2 2 m m m kðxÞ cos ðnÞ 2kðxÞbðxÞf ðxÞ t cos ðnÞ kðxÞbðxÞgðxÞ cos ðnÞ 2kðxÞbðxÞf ðxÞgðxÞt cosm ðnÞ dx 2 þ kðxÞ3 aðxÞ cos3m ðnÞ ¼ 0;
ð5:2Þ
and
kðxÞ
which are satisfied only if the following conditions hold:
8 dkðxÞ > > > dx þ kðxÞbðxÞf ðxÞ kðxÞcðxÞ ¼ 0; < > > > :
mkðxÞ dqðxÞ mkðxÞbðxÞpðxÞgðxÞ ¼ 0; dx mkðxÞ dpðxÞ dx
ð5:3Þ
2mkðxÞbðxÞpðxÞf ðxÞ ¼ 0
and
8 kðxÞ dfdxðxÞ 2kðxÞbðxÞf ðxÞ2 ¼ 0; > > > > > > > > kðxÞ dgðxÞ 2kðxÞbðxÞf ðxÞgðxÞ ¼ 0; > dx > > > > dhðxÞ < kðxÞ dx 12 m2 kðxÞbðxÞpðxÞ2 > > > 12 kðxÞbðxÞgðxÞ2 ¼ 0; > > > > > 1 > > mðm 1ÞkðxÞbðxÞpðxÞ2 þ kðxÞ3 aðxÞ ¼ 0; > 2 > > : m 2 ¼ 3m:
ð5:4Þ
Similar results are also obtained by using the sine method which are omitted here. Solving the algebraic Eqs. (5.3) and (5.4) by Maple, we have the following cases of solutions:
2264
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
Case1.
Z qðxÞ ¼ c1 c2 bðxÞdx þ c5 ; f ðxÞ ¼ 0; Z 1 gðxÞ ¼ c1 ; hðxÞ ¼ ðc21 þ c22 Þ bðxÞdx þ c4 ; cðxÞ ¼ cðxÞ; 2 2 R c2 2 cðxÞdx aðxÞ ¼ bðxÞe ; bðxÞ ¼ bðxÞ; m ¼ 1; c3 kðxÞ ¼ c3 e
R
cðxÞdx
;
pðxÞ ¼ c2 ;
ð5:5Þ
where ci(i = 15) are arbitrary constants, such that c1, c2 and c3 – 0. In this case, the exact solutions of Eq. (1.1) have the forms:
Z Z 1 uðnÞ ¼ c3 secðnÞ exp i c1 t ðc21 þ c22 Þ bðxÞdx þ c4 þ cðxÞdx ; 2
ð5:6Þ
where
Z n ¼ c2 t c1 bðxÞdx þ c5 :
ð5:7Þ
Case2.
kðxÞ ¼
qffiffiffiffiffiffiffiffiffi R f ðxÞe cðxÞdx ;
pðxÞ ¼ c2 f ðxÞ; qðxÞ ¼
1 hðxÞ ¼ ðc21 þ c22 Þf ðxÞ þ c4 ; R4 aðxÞ ¼ c22 bðxÞf ðxÞe2 cðxÞdx ; bðxÞ ¼ bðxÞ; gðxÞ ¼ c1 f ðxÞ;
1 c1 c2 f ðxÞ þ c3 ; 2
cðxÞ ¼ cðxÞ
ð5:8Þ
m ¼ 1;
where ci(i = 14) are arbitrary constants, such that c1, c2 – 0 and f(x) has the same form (3.11). In this case, the exact solutions of Eq. (1.1) have the forms:
uðnÞ ¼
Z qffiffiffiffiffiffiffiffiffi 1 f ðxÞ secðnÞ exp if ðxÞ t 2 þ c1 t þ ðc21 þ c22 Þ þ cðxÞdx ; 4
ð5:9Þ
where n is given by (3.18). Remark 3 (i) When k(x) is a constant we deduce that the solution (5.9) of case 2 reduces to the solution (3.16) of case 3 in Section 3. R To show this, we see that if k(x) is a constant then, we deduce from the conditions (5.8) that f ðxÞ / e2 cðxÞdx ; aðxÞ / bðxÞf 2 ðxÞ and c(x) = b(x)f(x). Thus the conditions (5.8) of case 2 agree with the conditions (3.15) of case 3 in Section 3. (ii) By the use of the transformation c2 ? ic2 we get sec (n) ? sech (n). 6. Description of the exp-function method and its use to solve Eq. (1.1) The exp-function method was first proposed by He and Wu [14]. Using the generalized wave transformation (2.2), Eqs. (3.2) and (3.3) become
1 1 0 tðnÞft2 ½f 0 ðxÞ þ 2bðxÞf ðxÞ2 þ t½g 0 ðxÞ þ 2bðxÞf ðxÞgðxÞ þ bðxÞgðxÞ2 þ h ðxÞg þ bðxÞpðxÞ2 t00 ðnÞ þ aðxÞt3 ðnÞ ¼ 0 2 2
ð6:1Þ
t0 ðnÞftp0 ðxÞ þ q0 ðxÞ þ ½2tf ðxÞ þ gðxÞbðxÞpðxÞg þ tðnÞ½bðxÞf ðxÞ cðxÞ ¼ 0:
ð6:2Þ
and
In this method, we admit the use of the relation (3.1), where t (n) has the form
Pd
tðnÞ ¼ Pqn¼c
an expðnnÞ ; expðmnÞ
m¼p bm
ð6:3Þ
where c, d, p and q are positive integers which are unknown to be determine later, an and bm are unknown constants. Eq. (6.3) can be written in the form
tðnÞ ¼
ac expðcnÞ þ . . . þ ad expðdnÞ : ap expðpnÞ þ . . . þ aq expðqnÞ
ð6:4Þ
2265
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
In order to determine values of c and p, we balance the linear term of the highest order derivatives t00 with the highest order nonlinear term t3 in Eq. (6.1). By simple calculations, we have
c1 exp½ðc þ 3pÞn þ . . . ; c2 exp½4pn þ c3 exp½ð3c þ pÞn þ . . . ¼ ; c4 exp½4pn þ
t00 ¼
ð6:5Þ
t3
ð6:6Þ
where ci are coefficients for simplicity. By balancing the highest order of exp-function in Eqs. (6.5) and (6.6), we have c + 3p = 3c + p which leads to
p ¼ c:
ð6:7Þ
Similarly, from the ansatz (6.4), we have
. . . þ d1 exp½ðd þ 3qÞn ; þ d2 exp½4qn þ d3 exp½ð3d þ qÞn ¼ ; þ d4 exp½4qn
t00 ¼
ð6:8Þ
t3
ð6:9Þ
where di are coefficients for simplicity. By balancing the lowest order of exp-function in Eqs. (6.8) and (6.9), we have (d + 3q) = (3d + q) which leads to
q ¼ d:
ð6:10Þ
Choosing p = c = 1 and q = d = 1, Eq. (6.4) becomes
tðnÞ ¼
a1 expðnÞ þ a0 þ a1 expðnÞ : b1 expðnÞ þ b0 þ b1 expðnÞ
ð6:11Þ
Substituting Eq. (6.11) into Eqs. (6.1) and (6.2), we get 3 1 X ðri þ t 2 qi þ t ei Þ ¼ 0; A i¼3
2 1X ðli þ t xi Þ ¼ 0; B i¼2
ð6:12Þ
where
A ¼ ðb1 expðnÞ þ b0 þ b1 expðnÞÞ3 ;
B ¼ ðb1 expðnÞ þ b0 þ b1 expðnÞÞ2 ;
dhðxÞ 1 2 a1 b1 bðxÞgðxÞ2 þ a31 aðxÞ ¼ 0; dx 2 dhðxÞ 1 1 1 2 dhðxÞ 2 2 ¼ a0 b1 2a1 b1 b0 a1 b1 b0 bðxÞgðxÞ2 a0 b1 bðxÞgðxÞ2 þ 3a21 a0 aðxÞ a1 b1 b0 bðxÞpðxÞ2 þ a0 b1 bðxÞpðxÞ2 ¼ 0; dx dx 2 2 2 dhðxÞ dhðxÞ 1 2 dhðxÞ 2 dhðxÞ 2 ¼ a1 b0 a1 b1 2a0 b1 b0 2a1 b1 b1 a1 b0 bðxÞgðxÞ2 dx dx dx dx 2 1 1 1 2 2 a1 b1 bðxÞgðxÞ2 a1 b1 b1 bðxÞgðxÞ2 a0 b1 b0 bðxÞgðxÞ2 a0 b1 b0 bðxÞpðxÞ2 þ a1 b0 bðxÞpðxÞ2 2a1 b1 b1 bðxÞpðxÞ2 2 2 2
r3 ¼ a1 b21 r2 r1
2
þ 2a1 b1 bðxÞpðxÞ2 þ 3a21 a1 aðxÞ þ 3a1 a20 aðxÞ ¼ 0;
r0 ¼ 2a0 b1 b1
dhðxÞ dhðxÞ dhðxÞ 1 2 dhðxÞ 2 2 2a1 b1 b0 2a1 b1 b0 a0 b0 a0 b0 bðxÞgðxÞ dx dx dx dx 2 2
2
2
a1 b1 b0 bðxÞgðxÞ a0 b1 b1 bðxÞgðxÞ a1 b1 b0 bðxÞgðxÞ þ a30 aðxÞ þ 6a0 a1 a1 aðxÞ 3 3 2 2 2 3a0 b1 b1 bðxÞpðxÞ þ a1 b1 b0 bðxÞpðxÞ þ a1 b1 b0 bðxÞpðxÞ ¼ 0; 2 2 dhðxÞ dhðxÞ 1 2 dhðxÞ 2 dhðxÞ 2 2 r1 ¼ 2a1 b1 b1 2a0 b1 b0 a1 b1 a1 b0 a1 b1 bðxÞgðxÞ dx dx dx dx 2 1 1 2 2 a1 b0 bðxÞgðxÞ2 a1 b1 b1 bðxÞgðxÞ2 a0 b1 b0 bðxÞgðxÞ2 þ 3a1 a21 aðxÞ þ 3a1 a20 aðxÞ þ a1 b0 bðxÞpðxÞ2 2 2 1 2 a0 b0 b1 bðxÞpðxÞ2 þ 2a1 b1 bðxÞpðxÞ2 2a1 b1 b1 bðxÞpðxÞ2 ¼ 0; 2 dhðxÞ dhðxÞ 1 2 r2 ¼ a0 b21 2a1 b1 b0 a1 b1 b0 bðxÞgðxÞ2 a0 b1 bðxÞgðxÞ2 þ 3a0 a21 aðxÞ dx dx 2 1 1 1 2 2 dhðxÞ 2 a1 b1 b0 bðxÞpðxÞ2 þ a0 b1 bðxÞpðxÞ2 ¼ 0; r3 ¼ a1 b1 a1 b1 bðxÞgðxÞ2 þ a31 aðxÞ ¼ 0; 2 2 dx 2 df ðxÞ 2 df ðxÞ 2 2 df ðxÞ 2 2 2 2 2a1 b1 bðxÞf ðxÞ ¼ 0; t 2 q2 ¼ a0 b1 2a1 b1 b0 2a0 b1 bðxÞf ðxÞ 4a1 b1 b0 bðxÞf ðxÞ ¼ 0; t 2 q3 ¼ a1 b1 dx dx dx
2266
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
df ðxÞ df ðxÞ df ðxÞ 2 df ðxÞ 2 2 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞ 4a0 b1 b0 bðxÞf ðxÞ dx dx dx dx df ðxÞ 2 2 2 df ðxÞ 2a1 b1 bðxÞf ðxÞ2 2a1 b0 bðxÞf ðxÞ2 ¼ 0; t 2 q0 ¼ a0 b0 2a1 b1 b0 dx dx df ðxÞ df ðxÞ 2 2 2 2a1 b1 b0 2a0 b0 bðxÞf ðxÞ 4a1 b1 b0 bðxÞf ðxÞ 4a1 b1 b0 bðxÞf ðxÞ2 4a0 b1 b1 bðxÞf ðxÞ2 ¼ 0; 2a0 b1 b1 dx dx df ðxÞ df ðxÞ 2 df ðxÞ 2 df ðxÞ 2 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞ ¼ a1 b1 dx dx dx dx 2 2 4a0 b1 b0 bðxÞf ðxÞ2 2a1 b1 bðxÞf ðxÞ2 2a1 b0 bðxÞf ðxÞ2 ¼ 0; 2
t2 q1 ¼ a1 b1
t2 q1
df ðxÞ df ðxÞ 2 2 2 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞ 2a0 b1 bðxÞf ðxÞ ¼ 0; dx dx 2 df ðxÞ 2 2 dgðxÞ 2 t2 q3 ¼ a1 b1 2a1 b1 bðxÞf ðxÞ2 ¼ 0; te3 ¼ a1 b1 2a1 b1 bðxÞf ðxÞgðxÞ ¼ 0; dx dx dgðxÞ 2 dgðxÞ 2 te2 ¼ a0 b1 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞgðxÞ 2a0 b1 bðxÞf ðxÞgðxÞ ¼ 0; dx dx dgðxÞ dgðxÞ 2 dgðxÞ 2 dgðxÞ te1 ¼ a1 b1 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞgðxÞ 4a0 b1 b0 bðxÞf ðxÞgðxÞ dx dx dx dx dgðxÞ dgðxÞ 2 2 2 dgðxÞ 2a1 b1 bðxÞf ððxÞgðxÞ 2a1 b0 bðxÞf ððxÞgðxÞ ¼ 0; t e0 ¼ a0 b0 2a1 b1 b0 2a0 b1 b1 dx dx dx dgðxÞ 2 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞgðxÞ 4a1 b1 b0 bðxÞf ðxÞgðxÞ 4a0 b1 b1 bðxÞf ðxÞgðxÞ 2a0 b0 bðxÞf ðxÞgðxÞ ¼ 0; dx dgðxÞ dgðxÞ 2 dgðxÞ 2 dgðxÞ te1 ¼ a1 b1 a1 b0 2a1 b1 b1 2a0 b1 b0 4a1 b1 b1 bðxÞf ðxÞgðxÞ 4a0 b1 b0 bðxÞf ðxÞgðxÞ dx dx dx dx dgðxÞ 2 2 2 dgðxÞ 2a1 b1 bðxÞf ðxÞgðxÞ 2a1 b0 bðxÞf ðxÞgðxÞ ¼ 0; t e2 ¼ a0 b1 2a1 b1 b0 4a1 b1 b0 bðxÞf ðxÞgðxÞ dx dx 2 2 dgðxÞ 2 2a0 b1 bðxÞf ðxÞgðxÞ ¼ 0; t e3 ¼ a1 b1 2a1 b1 bðxÞf ðxÞgðxÞ ¼ 0; l2 ¼ a1 b1 bðxÞf ðxÞ a1 b1 cðxÞ ¼ 0; dx dqðxÞ dqðxÞ l1 ¼ a1 b0 a0 b1 þ a1 b0 bðxÞgðxÞpðxÞ a0 b1 bðxÞgðxÞpðxÞ þ a0 b1 bðxÞf ðxÞ þ a1 b0 bðxÞf ðxÞ a1 b0 cðxÞ a0 b1 cðxÞ ¼ 0; dx dx dqðxÞ dqðxÞ l0 ¼ 2a1 b1 2a1 b1 þ a1 b1 bðxÞf ðxÞ þ a1 b1 bðxÞf ðxÞ þ a0 b0 bðxÞf ðxÞ þ 2a1 b1 bðxÞgðxÞpðxÞ dx dx dqðxÞ dqðxÞ a1 b0 þ a0 b1 bðxÞgðxÞpðxÞ 2a1 b1 bðxÞgðxÞpðxÞ a1 b1 cðxÞ a1 b1 cðxÞ a0 b0 cðxÞ ¼ 0; l1 ¼ a0 b1 dx dx a1 b0 bðxÞgðxÞpðxÞ þ a0 b1 bðxÞf ðxÞ þ a1 b0 bðxÞf ðxÞ a0 b1 bðxÞ a1 b0 cðxÞ ¼ 0; 2
t2 q2 ¼ a0 b1
l2 ¼ a1 b1 bðxÞf ðxÞ a1 b1 cðxÞ ¼ 0;
dpðxÞ dpðxÞ a0 b1 þ 2a1 b0 bðxÞf ðxÞpðxÞ 2a0 b1 bðxÞf ðxÞpðxÞ ¼ 0; dx dx dpðxÞ dpðxÞ 2a1 b1 þ 4a1 b1 bðxÞf ðxÞpðxÞ 4a1 b1 bðxÞf ðxÞpðxÞ ¼ 0; t x0 ¼ 2a1 b1 dx dx dpðxÞ dpðxÞ t x1 ¼ a0 b1 a1 b0 þ 2a0 b1 bðxÞf ðxÞpðxÞ 2a1 b0 bðxÞf ðxÞpðxÞ ¼ 0: dx dx
t x1 ¼ a1 b0
ð6:13Þ
Solving the system (6.13) by Maple, we have the following cases of solutions: Case1.
a0 ¼ a0 ;
a1 ¼ a1 ;
a1 ¼ 0;
b0 ¼
a0 b1 ; a1
1 c1 c2 f ðxÞ þ c3 ; gðxÞ ¼ c1 f ðxÞ; 2 2 2 0 c b f ðxÞ aðxÞ ¼ 2 1 2 ; cðxÞ ¼ bðxÞf ðxÞ; 8a1 qðxÞ ¼
b1 ¼ b1 ;
hðxÞ ¼
b1 ¼ 0;
pðxÞ ¼ c2 f ðxÞ;
1 2 c þ 2c21 f ðxÞ þ c4 ; 8 2
ð6:14Þ
where ci(i = 14) are arbitrary constants, such that c1, c2 – 0 and f(x) has the same form (3.11). In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a1 ða1 en þ a0 Þ 1 2 2 2 ; c þ c t þ þ 2c f ðxÞ þ c exp i t 1 4 1 b1 ða1 en a0 Þ 8 2
which can be written in the form
ð6:15Þ
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
uðnÞ ¼
a1 nþ/ 1 exp i t 2 þ c1 t þ c22 þ 2c21 f ðxÞ þ c4 ; coth 2 8 b1
2267
ð6:16Þ
where e/ ¼ aa10 . Case2.
a0 ¼ 0;
a1 ¼ a1 ;
a1 ¼ a1 ;
b0 ¼ 0;
1 c1 c2 f ðxÞ þ c3 ; gðxÞ ¼ c1 f ðxÞ; 2 2 c2 b f 0 ðxÞ aðxÞ ¼ 2 1 2 ; cðxÞ ¼ bðxÞf ðxÞ: 2a1 qðxÞ ¼
b1 ¼ b1 ; hðxÞ ¼
b1 ¼
a1 b1 ; a1
pðxÞ ¼ c2 f ðxÞ;
1 2 c þ 2c22 f ðxÞ þ c4 ; 4 1
ð6:17Þ
In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a1 ða1 en þ a1 en Þ 1 2 2 2 þ c t þ þ 2c ; c f ðxÞ þ c exp i t 1 4 2 b1 ða1 en a1 en Þ 4 1
ð6:18Þ
which can be written in the form
uðnÞ ¼
a1 1 cothðn þ /Þ exp i t2 þ c1 t þ c21 þ 2c22 f ðxÞ þ c4 ; 4 b1
ð6:19Þ
where e2/ ¼ aa11 . Case3.
a0 ¼ a0 ; a1 ¼ 0; a1 ¼ 0; b0 ¼ 0; b1 ¼ b1 ; b1 ¼ b1 ; pðxÞ ¼ c2 f ðxÞ; 1 1 qðxÞ ¼ c1 c2 f ðxÞ þ c3 ; gðxÞ ¼ c1 f ðxÞ; hðxÞ ¼ c21 c22 f ðxÞ þ c4 ; 2 4 2b b c2 f 0 ðxÞ aðxÞ ¼ 1 12 2 ; cðxÞ ¼ bðxÞf ðxÞ: a0
ð6:20Þ
In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a0 1 2 2 2 ; exp i t þ c t þ 2c c f ðxÞ þ c 1 4 1 2 4 b1 en þ b1 en
ð6:21Þ
which can be written in the form
a0 1 uðnÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi sechðn þ /Þ exp i t 2 þ c1 t þ c21 2c22 f ðxÞ þ c4 ; 4 2 b1 b1
ð6:22Þ
where e2/ ¼ bb11 . Case4.
a0 ¼ a0 ;
a1 ¼ a1 ;
pðxÞ ¼ c2 f ðxÞ; hðxÞ ¼
a1 ¼ a1 ;
qðxÞ ¼
b0 ¼
1 c1 c2 f ðxÞ þ c3 ; 2
1 2 c2 þ 2c21 f ðxÞ þ c4 ; 8
b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a20 4a1 a1 a1
;
b1 ¼ b1 ;
b1 ¼
a1 b1 ; a1
2
gðxÞ ¼ c1 f ðxÞ;
aðxÞ ¼
c22 b1 f 0 ðxÞ ; 8a21
ð6:23Þ
cðxÞ ¼ bðxÞf ðxÞ:
In this case, the exact solution of Eq. (1.1) has the from:
uðnÞ ¼
a1 ða1 en þ a0 þ a1 en Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
exp i t2 þ c1 t þ c22 þ 2c21 f ðxÞ þ c4 ; 8 b1 a1 en a20 4a1 a1 a1 en
ð6:24Þ
which can be written in the form
uðnÞ ¼ where e/ ¼
a1 nþ/ 1 exp i t 2 þ c1 t þ c22 þ 2c21 f ðxÞ þ c4 ; coth 2 8 b1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2
a0
a0 4a1 a1 2a1
and n is given by (3.18).
ð6:25Þ
2268
E.M.E. Zayed, M.A.M. Abdelaziz / Applied Mathematics and Computation 218 (2011) 2259–2268
Remark 4. From cases 1–4, we deduce that a(x) / f0 (x). But c(x) = b(x)f(x) where f(x) is given by (3.11). Differentiating (3.11), we get f0 (x) / b (x)f(x)2. Consequently we have
aðxÞbðxÞ / cðxÞ2 ;
ð6:26Þ
which is equivalent to (3.19). Remark 5. Each of these solutions is just a disguised version of one or other of the solutions of Section 3. Remark 6. All solutions of this article have been checked with Maple by putting them back into the original Eq. (1.1).
7. Conclusions In this article, the generalized extended tanh-function method, the sine–cosine method and the exp-function method have been applied to find the exact traveling wave solutions of the generalized nonlinear Schrödinger (GNLS) equation with variable coefficients. We have shown in remarks 2 and 4 that if the coefficients a(x), b (x) and c(x) in Eq. (1.1) satisfying the condition a (x)b(x) / c(x)2 then Eq. (1.1) has the solutions given in this article. On comparing the solutions constructed in Sections 3, 5 and 6 using the above three methods respectively, we have noted that, when k(x) is not a constant, the solution of Eq. (1.1) in Section 5 are different from those obtained in Sections 3 and 6. But, when k(x) is a constant we have shown in remark 3 that, the solution of case 2 in Section 5 reduces to the solution of case 3 in Section 3. Finally, we have noted under certain transformations that the solutions obtained by the exp-function method are just a disguised version of one or other of the solutions obtained by using the tanh-function method. Acknowledgment The authors thank the referee for his comments on this paper. References [1] S.A. El-Wakil, M.A. Abdou, New exact traveling wave solutions using modified extended tanh-function method, Chaos Solitons Fract. 31 (2007) 840– 852. [2] E. Fan, Extended tanh-function method, and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [3] A.M. Wazwaz, The tanh-method for traveling wave solutions of nonlinear wave equations, Appl. Math. Comput. 187 (2007) 1131–1142. [4] E.M.E. Zayed, H.M. AbdelRahman, The extended tanh-method for finding traveling wave solutions of nonlinear PDEs, Nonlinear Sci. Lett. A 1 (2) (2010) 193–200. [5] E.M.E. Zayed, M.A.M. Abdelaziz, The tanh-function method using a generalized wave transformation for nonlinear equations, Int. J. Nonlinear Sci. Numer. Simul. 11 (8) (2010) 595–601. [6] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) 650–654. [7] W. Malfliet, W. Hereman, The tanh-method part I. Exact solutions of nonlinear evolution and wave equations, Phys. Script. 54 (1996) 563–568. [8] A.M. Wazwaz, The tanh-method for traveling wave solutions of nonlinear equations, Appl. Math. Comput. 154 (2004) 713–723. [9] M. Yaghobi Moghaddam, A. Asgari, H. Yazdani, Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by extended tanh–coth, sine–cosine and Exp-function methods, Appl. Math. Comput. 210 (2009) 422–435. [10] A.M. Wazwaz, A sine–cosine method for handling nonlinear wave equations, Math. Comput. Model. 40 (2004) 499–508. [11] Y. Yang, Z.L. Tao, F.R. Austin, Solutions of the generalized KdV equation with time-dependent damping and dispersion, Appl. Math. Comput. 216 (2010) 1029–1035. [12] E. Fan, H. Zhang, A note on the homogeneous balance method, Phys. Lett. A 246 (1998) 403–406. [13] M.L. Wang, Exact solutions of a compound Kdv-Burgers equations, Phys. Lett. A 213 (1996) 279–287. [14] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fract. 30 (2006) 700–708. [15] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos Solitons Fract. 34 (2007) 1421–1429. [16] S. Zhang, Application of exp-function method to high dimensional evolution equation, Chaos Solitons Fract. 38 (2008) 270–276. [17] S.D. Zhu, Exp-function method for the discrete mKdV lattice, Int. J. Nonlinear Sci. Numer. Simul. 8 (2007) 465–469. [18] C.Q. Dai, J.F. Zhang, Jacobian elliptic function method for nonlinear differential-difference equations, Chaos Solitons Fract. 27 (2006) 1042–1047. [19] E. Fan, J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A 305 (2002) 383–392. [20] S. Liu, Z.Fu. S. Liu, Q. Zhao, Jacobi elliptic function method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001) 69–74. [21] X.Q. Zhao, H.Y. Zhi, H.Q. Zhang, Improved Jacobi-function method with symbolic computation to construct new double-periodic solutions for the generalized Ito system, Chaos Solitons Fract. 28 (2006) 112–116. [22] J.L. Zhang, M.L. Wang, Y.M. Wang, Z.D. Fang, The improved F-expansion method and its applications, Phys. Lett. A 350 (2006) 103–109. [23] J.H. He, New interpretation of homotopy perturbation method, Int. J. Mod. Phys. B 20 (2006) 2561–2568. [24] J.H. He, Asymptology by homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul. 6 (2005) 207–208. [25] J.H. He, X.H. Wu, Construction of solitary solutions and compact on like solution by variational iteration method, Chaos Solitons Fract. 29 (2006) 108– 113. [26] M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981. [27] R. Hirota, Direct method of finding exact solutions of nonlinear evolution equations, in: R. Bullough, P. Caudrey (Eds.), Backlund Transformation, Springer, Berlin, 1980. [28] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004. [29] A.M. Wazwaz, On multiple soliton solutions for coupled KdV-mKdV equation, Nonlinear Sci. Lett. A 1 (3) (2010) 289–296. [30] J.L. Zhang, B.A. Li, M.L. Wang, The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients, Chaos Solitons Fract. 39 (2009) 858–865. [31] E.J. Parkes, Observation on the tanh-coth expansion method for finding solutions to nonlinear evolution equations, Appl. Math. Comput. 217 (2010) 1749–1754.