Opt Quant Electron (2018)50:164 https://doi.org/10.1007/s11082-018-1426-z
The /6 -model expansion method for solving the nonlinear conformable time-fractional Schro¨dinger equation with fourth-order dispersion and parabolic law nonlinearity Elsayed M. E. Zayed1 • Abdul-Ghani Al-Nowehy2
Received: 20 December 2017 / Accepted: 7 March 2018 Ó Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract The /6 -model expansion method combined with the conformable time-fractional derivative is applied in this paper for finding many new exact solutions including Jacobi elliptic function solutions, solitary wave solutions, trigonometric function solutions and other solutions to the nonlinear conformable time-fractional Schro¨dinger equation with fourth-order dispersion and parabolic law nonlinearity. This method presents a wider applicability for handling the nonlinear partial differential equations. Comparing our results with the well-known results are given. Keywords The /6 -model expansion method Conformable fractional derivative Jacobi elliptic function solutions Exact solutions Solitary wave solutions Other solutions Nonlinear Schro¨dinger equation
1 Introduction Fractional differential equations have attracted much attention during recent decades since these equations have been proved to be valuable tools in the modeling of many phenomena in areas of biology, chemistry, economic, engineering, physics and other areas of applications (Biswas et al. 2014; El-Sayed et al. 2007; Ahmed et al. 2007; Laskin 2000; Miller and Ross 1993), conformable fractional derivative (Khalil et al. 2014; Abu Hammad and Khalil 2014; Abdeljawad 2015; Hosseini and Ansari 2017). Many powerful methods for solving nonlinear fractional or nonfractional differential equations were appeared in open literature, such as the generalized Riccati & Abdul-Ghani Al-Nowehy
[email protected] Elsayed M. E. Zayed
[email protected] 1
Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt
2
Mathematics Department, Faculty of Education and Science, Taiz University, Taiz, Yemen
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equation mapping method (Zayed and Al-Nowehy 2017a), the Jacobi elliptic function method (Zayed et al. 2014a; Zayed and Al-Nowehy 2017b), the improved ðG0 =GÞ-expansion method (Zayed et al. 2014b; Gepreel and Omran 2012), the local fractional differential equations (Yang 2012), the multiple exp-function method (Zayed et al. 2016e), the modified simple equation method (Zayed and Al-Nowehy 2016a; Arnous et al. 2017; Jawad et al. 2017), the generalized Kudryashov method (Zayed and Al-Nowehy 2016b, c), the soliton ansatz method (Triki et al. 2016; Zayed and Al-Nowehy 2016d), the /6 -model expansion method (Zhou et al. 2013, 2015; Zayed and Al-Nowehy 2017c), the extended trial function method (Biswas et al. 2018), the Riccati–Bernoulli’s sub-ODE method (Mirzazadeh et al. 2018, 2017), the first integral method with conformable fractional derivative (Ekici et al. 2016), the sinecosine function method (Mirzazadeh et al. 2015), and so on. The objective of this paper is to apply the /6 -model expansion method combined with the conformable fractional derivative of the nonlinear conformable time-fractional Schro¨dinger equation with fourth-order dispersion and parabolic law nonlinearity (Biswas and Milovic 2009; Douvagai et al. 2016; Xu 2011; Zayed et al. 2017e; Kohl et al. 2008; Biswas et al. 2008): i
oa u o2 u o4 u 2 4 þ a b þ c u þk u u ¼ 0; j j j j 1 ota ox2 ox4
i¼
pffiffiffiffiffiffiffi 1;
ð1:1Þ
which describes the propagation of optical pulse in a medium, and uðx; tÞ is the slowly a varying envelope of the electromagnetic field, where 0\a 1 and ootau is the conformable time-fractional derivative of the function u(x, t). The coefficients of a represents the group velocity dispersion (GVD), the coefficients of b represents the fourth-order dispersion terms, the coefficients of c represents the self-phase modulation (SPM) with parabolic law nonlinearity and k1 is a nonzero constant. When a ¼ 1, Eq. (1.1) has been discussed in Xu (2011) using two direct algebraic methods and in Zayed et al. (2017e) using the extended auxiliary equation method and the new mapping method. When a ¼ 1 and b ¼ 0, Eq. (1.1) has been discussed in Kohl et al. (2008). Also, when a ¼ 1 and k1 ¼ 0; Eq. (1.1) has been solved in Zayed et al. (2017e) using the Jacobi elliptic function method and in Biswas et al. (2008) using the soliton ansatz method. To our knowledge, Eq. (1.1) has not been discussed elsewhere using the proposed method. This paper is organized as follows: In Sect. 2, the description of the /6 -model expansion method combined with the conformable time-fractional derivative is given. In Sect. 3, we apply this method to solve Eq. (1.1). In Sect. 4, we present some graphical representations for some solutions of Eq. (1.1). In Sect. 5, conclusions are obtained.
2 Description of the /6 -model expansion method combined with the conformable time-fractional derivative Khalil et al. (2014), introduced the definition of the conformable fractional derivative as follows: Definition 1 Let f : ½0; 1Þ ! R be a function. Then, the conformable fractional derivative of f of order a is defined as f ðt þ t1a Þ f ðtÞ ; !0
Ta ð f ÞðtÞ ¼ lim
123
ð2:1Þ
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for all t [ 0; a 2 ð0; 1: The conformable fractional derivative satisfies some properties which are given in the following theorem (Khalil et al. 2014; Abu Hammad and Khalil 2014; Abdeljawad 2015; Hosseini and Ansari 2017): Theorem 1 Let a 2 ð0; 1, and f, g be a -conformable differentiable at a point t, then: ðiÞTa ðaf þ bgÞ ¼ aTa ð f Þ þ bTa ðgÞ; for all a; b 2 R: ðiiÞTa ðtl Þ ¼ ltla ; for all l 2 R: ð2:2Þ
ðiiiÞTa ð fgÞ ¼ fTa ðgÞ þ gTa ð f Þ: f gTa ð f Þ fTa ðgÞ ðivÞTa ¼ : g g2 If, in addition, f is differentiable, then Ta ð f ÞðtÞ ¼ t1a df dt ðt Þ:
Theorem 2 Let f : ½0; 1Þ ! R, be a differentiable function and also a -conformable differentiable. Let g be a function defined in the range of f and also differentiable; then, one has the following rule: ð2:3Þ Ta ð f gÞðtÞ ¼ t1a g0 ðtÞf 0 ðgðtÞÞ: Suppose we have the following nonlinear conformable time-fractional PDE: a o u ou o2a u o2 u P u; a ; ; 2a ; 2 ; . . . ¼ 0; ot ox ot ox
ð2:4Þ
where P is a polynomial in u(x, t) and its partial derivatives with respect to x and conformable time-fractional derivative in which the highest order derivatives and nonlinear highest terms are involved. In the following, we give the main steps of the /6 -model expansion method combined with the conformable time-fractional derivative: Step 1 We use the conformable time-fractional derivative wave transformation a t ; ð2:5Þ uðx; tÞ ¼ uðgÞ; g ¼ kx þ w a where k and w are nonzero constants, to reduce Eq. (2.4) to the following integer order nonlinear ordinary differential equation (ODE): Gðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð2:6Þ
where G is polynomial in uðgÞ and its total derivatives u0 ðgÞ; u00 ðgÞ and so on. Step 2 We assume that Eq. (2.6) has the formal solution : uðgÞ ¼ F ½/ðgÞ;
ð2:7Þ
where F is a suitable variable transformation, and /ðgÞ satisfies /6 model in the form 8 < /02 ðgÞ ¼ c þ c /2 ðgÞ þ 1 c /4 ðgÞ þ 1 c /6 ðgÞ; 0 1 3 5 2 3 ð2:8Þ : /00 ðgÞ ¼ c1 /ðgÞ þ c3 /3 ðgÞ þ c5 /5 ðgÞ;
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where ci for i ¼ 0; 1; 3; 5 being real constants. Step 3 It is well known (Zhou et al. 2013, 2015; Zayed and Al-Nowehy 2017c) that Eq. (2.8) has the solution: UðgÞ /ðgÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; f U2 ðgÞ þ g
ð2:9Þ
where f U 2 ðgÞ þ g [ 0 and UðgÞ is the solution of the Jacobi elliptic equation: U 02 ðgÞ ¼ l0 þ l2 U 2 ðgÞ þ l4 U 4 ðgÞ;
ð2:10Þ
and lj ðj ¼ 0; 2; 4Þ are constants to be determined later, while f and g are constants given by f ¼
c3 ðl2 c1 Þ 2ðl2 c1 Þ2 þ 6l0 l4 4l2 ðl2 c1 Þ
;
ð2:11Þ
;
ð2:12Þ
and g¼
3l0 c3 2
2ðl2 c1 Þ þ 6l0 l4 4l2 ðl2 c1 Þ
under the constraint condition 2 9l0 l4 3l0 l4 ð2l2 þ c1 Þ þ 4c5 ðl2 þ c1 Þ ¼ 0: c23 ðl2 c1 Þ ðl2 c1 Þ
ð2:13Þ
Step 4 It is well-known (Zayed et al. 2014a), Zayed and Al-Nowehy (2017b, d) that Eq. (2.10) has the following Jacobi elliptic function solutions: No.
l0
1
1
2
l2 ð1 þ m2 Þ 2
2
1m
2m 1
2
U ðgÞ
l4
2
m2
sn ðgÞ or cd ðgÞ
m
2
cn ðgÞ
1
dn ðgÞ
3
m 1
2m
4
m2
ð1 þ m2 Þ
5
m2
2m2 1
1 m2
nc ðgÞ
6
-1
2 m2
ð1 m2 Þ
nd ðgÞ
7
1
2 m2
1 m2
8
1
2
2m 1 2
2
ns ðgÞ or dc ðgÞ
1
2
sc ðgÞ 2
m ð1 m Þ
sd ðgÞ
9
1m
2m
1
cs ðgÞ
10
m2 ð1 m2 Þ
2m2 1
1
ds ðgÞ
11
1m2 4
1þm2 2
1m2 4
nc ðgÞ sc ðgÞ or
12
ð1m2 Þ 4 1 4
1þm2 2
1 4
m cn ðgÞ dn ðgÞ
12m2 2
1 4
sn ðgÞ 1cn ðgÞ
1 4
1þm2 2
ð1m2 Þ
13 14
123
2
2
4
sn ðgÞ cn ðgÞdn ðgÞ
cn ðgÞ 1sn ðgÞ
The /6 -model expansion method for solving the nonlinear...
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In this table, sn ðgÞ ¼ sn ðg; mÞ; cd ðgÞ ¼ cd ðg; mÞ; cn ðgÞ ¼ cn ðg; mÞ; dn ðgÞ ¼ dn ðg; mÞ; ns ðgÞ ¼ ns ðg; mÞ; cs ðgÞ ¼ cs ðg; mÞ; ds ðgÞ ¼ ds ðg; mÞ; sc ðgÞ ¼ sc ðg; mÞ; sd ðgÞ ¼ sd ðg; mÞ are the Jacobi elliptic functions with the modulus 0 m 1. These functions degenerate into hyperbolic functions when m ! 1 as follows: sn ðg; 1Þ ¼ tanhðgÞ; cn ðg; 1Þ ¼ sech ðgÞ; dn ðg; 1Þ ¼ sech ðgÞ; ns ðg; 1Þ ¼ cothðgÞ; cs ðg; 1Þ ¼ csch ðgÞ; ds ðg; 1Þ ¼ csch ðgÞ; sc ðg; 1Þ ¼ sinhðgÞ; sd ðg; 1Þ ¼ sinhðgÞ; nc ðg; 1Þ ¼ coshðgÞ; cd ðg; 1Þ ¼ 1 and into trigonometric function when m ! 0 as follows: snðg; 0Þ ¼ sinðgÞ;cd ðg; 0Þ ¼ cosðgÞ;cn ðg; 0Þ ¼ cosðgÞ; ns ðg; 0Þ ¼ cscðgÞ; cs ðg;0Þ ¼ cotðgÞ; dsðg; 0Þ ¼ cscðgÞ; sc ðg; 0Þ ¼ tanðgÞ; sdðg; 0Þ ¼ sinðgÞ;nc ðg; 0Þ ¼ secðgÞ; dnðg; 0Þ ¼ 1: Step 5 Substituting Eq. (2.9) into Eq. (2.7) along with (2.5), we have the Jacobi elliptic function solutions of Eq. (2.4).
3 Application In this section, we solve Eq. (1.1) using the /6 -model expansion method combined with the conformable time-fractional derivative. To this aim, we assume that Eq. (1.1) has the following solution in the form of soliton ansatz uðx; tÞ ¼ A/½gðx; tÞ exp½iQðx; tÞ;
ð3:1Þ
where a t ; gðx; tÞ ¼ B x v a
a t Qðx; tÞ ¼ kx þ w þ h0 ; a
0\a 1:
ð3:2Þ
Here /ðgÞ is the soliton amplitude component that represents the pulse shape, Q(x, t) is the soliton phase, while the constants A, B, v, k, w and h0 are the soliton amplitude, soliton width, soliton velocity, soliton frequency, wave number and phase constant, respectively. Substituting (3.1) along with (3.2) into Eq. (1.1) and separating the real and imaginary parts, we obtain the two ODEs: Im : 4bkB2 /000 ðgÞ ðv þ 2ak þ 4bk3 Þ/0 ðgÞ ¼ 0;
ð3:3Þ
Re : B2 ða þ 6bk2 Þ/00 ðgÞ bB4 /0000 ðgÞ ðw þ ak2 þ bk4 Þ/ðgÞ þ cA2 /3 ðgÞ þ ck1 A4 /5 ðgÞ ¼ 0:
ð3:4Þ Differentiating Eq. (3.3) with respect to g and substituting into Eq. (3.4) we get: 2 B ð2ak þ 20bk3 vÞ 00 / ðgÞ ðw þ ak2 þ bk4 Þ/ðgÞ þ cA2 /3 ðgÞ þ k1 cA4 /5 ðgÞ ¼ 0: 4k ð3:5Þ Substituting (2.8) into Eq. (3.5), collecting all the coefficients of Qi ði ¼ 1; 3; 5Þ and setting them to zero, we have the following algebraic equations:
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c1 B2 ð2ak þ 20bk3 vÞ ¼ 0; 4k c3 B2 ð2ak þ 20bk3 vÞ cA2 þ ¼ 0; 4k c5 B2 ð2ak þ 20bk3 vÞ ¼ 0: k1 cA4 þ 4k
ðw þ ak2 þ bk4 Þ þ
ð3:6Þ
Solving the system of algebraic equations (3.6), we obtain the following results sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c3 ðw þ ak2 þ bk4 Þ kðw þ ak2 þ bk4 Þ cc1 c5 ; k1 ¼ 2 ; B¼2 ; A¼ 3 cc1 c1 ð2ak þ 20bk vÞ c3 ðw þ ak2 þ bk4 Þ ð3:7Þ provided that cc1 c3 ðw þ ak2 þ bk4 Þ\0 and c1 kðw þ ak2 þ bk4 Þð2ak þ 20bk3 vÞ [ 0: With reference to Sect. 2, we have the following solutions of Eq. (1.1) as follows: (1) If l0 ¼ 1; l2 ¼ ð1 þ m2 Þ; l4 ¼ m2 ; then UðgÞ ¼ sn ðgÞ or UðgÞ ¼ cd ðgÞ; and we have the Jacobi elliptic function solutions: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ sn ðgÞ ð3:8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ cc1 f sn 2 ðgÞ þ g or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ cd ðgÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f cd ðgÞ þ g where g ¼ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 4
kðwþak þbk Þ c1 ð2akþ20bk3 vÞðx
f ¼ g¼
ð3:9Þ
a
v ta Þ; while f and g are given by c3 ð1 þ m2 þ c1 Þ
2ð1 þ m2 þ c1 Þ2 þ6m2 4ð1 þ m2 Þð1 þ m2 þ c1 Þ 3c3 2ð1 þ m2 þ c1 Þ2 þ6m2 4ð1 þ m2 Þð1 þ m2 þ c1 Þ
; ;
under the constraint condition
9m2 2 c c23 þ 2 1 þ m 1 ð 1 þ m2 þ c 1 Þ
2 3m2 2 þ 4c5 þ 1 þ m c1 ¼ 0: ð1 þ m2 þ c1 Þ In particular, if m ! 1; then we have the solitary wave solutions: 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c3 ðw þ ak2 þ bk4 Þ6 2ð2 þ c1 Þ 2ð5 þ 4c1 Þ tanhðgÞ7 ½iðkxþwðtaa Þþh0 Þ ; uðx; tÞ ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5e cc1 c3 ð2 þ c1 Þ tanh2 ðgÞ þ 3c3 ð3:10Þ
123
The /6 -model expansion method for solving the nonlinear...
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or 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2 þ c1 Þ2 2ð5 þ 4c1 Þ 7 i kxþw ta þh c3 ðw þ ak2 þ bk4 Þ6 ð a Þ 0 Þ ; uðx; tÞ ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½ ð cc1 2 c ð2 þ c Þ þ 3c coth ðgÞ 3
1
3
ð3:11Þ under the constraint condition 2 9 3 2 þ ð4 c1 Þ þ 4c5 þ ð2 c1 Þ ¼ 0: c3 ð 2 þ c1 Þ ð2 þ c1 Þ While, if m ! 0; then we have the periodic wave solutions: 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
ffi 2 c21 1 sinðgÞ ta c3 ðw þ ak2 þ bk4 Þ6 7 uðx; tÞ ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½iðkxþwð a Þþh0 Þ ; cc1 c3 ð1 þ c1 Þ sin2 ðgÞ þ 3c3 ð3:12Þ or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
ffi 2 c21 1 cosðgÞ ta c3 ðw þ ak2 þ bk4 Þ4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 c3 ð1 þ c1 Þ cos ðgÞ þ 3c3 ð3:13Þ under the constraint condition c23 ð2 c1 Þ þ 4c5 ð1 c1 Þ2 ¼ 0: (2) If l0 ¼ 1 m2 ; l2 ¼ 2m2 1; l4 ¼ m2 ; then UðgÞ ¼ cn ðgÞ; and we have the Jacobi elliptic function solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ cn ðgÞ ð3:14Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f cn ðgÞ þ g where f and g are constants given by f ¼ g¼
c3 ð2m2 1 c1 Þ 2
2ð2m2 1 c1 Þ 6m2 ð1 m2 Þ 4ð2m2 1Þð2m2 1 c1 Þ 3c3 ð1 m2 Þ 2ð2m2 1 c1 Þ2 6m2 ð1 m2 Þ 4ð2m2 1Þð2m2 1 c1 Þ
; ;
under the constraint condition
2 9m2 ð1 m2 Þ 2 3m2 ð1 m2 Þ 2 c23 þ 4c 1 þ c 1 þ c ¼ 0: 2 2m 2m 1 5 1 ð2m2 1 c1 Þ ð2m2 1 c1 Þ (3) If l0 ¼ m2 1; l2 ¼ 2 m2 ; l4 ¼ 1; then UðgÞ ¼ dn ðgÞ; and we have the Jacobi elliptic function solution
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ dn ðgÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f dn ðgÞ þ g
ð3:15Þ
where f and g are constants given by f ¼ g¼
c3 ð2 m2 c1 Þ 2ð2
m2
2
c1 Þ 6ðm2 1Þ 4ð2 m2 Þð2 m2 c1 Þ 3c3 ðm2 1Þ
2ð2 m2 c1 Þ2 6ðm2 1Þ 4ð2 m2 Þð2 m2 c1 Þ
; ;
under the constraint condition 2
2 3ðm2 1Þ 2 9ðm 1Þ 2 2 c3 2 2 m þ c1 þ 4c5 2 m þ c1 ¼ 0: ð2 m2 c1 Þ ð2 m2 c1 Þ (4) If l0 ¼ m2 ; l2 ¼ ð1 þ m2 Þ; l4 ¼ 1; then UðgÞ ¼ ns g or UðgÞ ¼ dc ðgÞ; and we have the Jacobi elliptic function solutions sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ ns ðgÞ ð3:16Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f ns ðgÞ þ g or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ dc ðgÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ cc1 f dc 2 ðgÞ þ g
ð3:17Þ
where f and g are constants given by f ¼ g¼
c3 ð1 þ m2 þ c1 Þ 2ð1 þ m2 þ c1 Þ2 þ6m2 4ð1 þ m2 Þð1 þ m2 þ c1 Þ 3c3 m2 2ð1 þ m2 þ c1 Þ2 þ6m2 4ð1 þ m2 Þð1 þ m2 þ c1 Þ
; ;
under the constraint condition
2 9m2 3m2 2 2 c23 þ 2 1 þ m þ 1 þ m c þ 4c c ¼ 0: 1 5 1 ð1 þ m2 þ c1 Þ ð1 þ m2 þ c1 Þ (5) If l0 ¼ m2 ; l2 ¼ 2m2 1; l4 ¼ 1 m2 ; then UðgÞ ¼ nc ðgÞ; and we have the Jacobi elliptic function solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ nc ðgÞ ð3:18Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f nc ðgÞ þ g where f and g are constants given by
123
The /6 -model expansion method for solving the nonlinear...
f ¼ g¼
Page 9 of 19 164
c3 ð2m2 1 c1 Þ 2ð2m2 1 c1 Þ2 6m2 ð1 m2 Þ 4ð2m2 1Þð2m2 1 c1 Þ 3c3 m2 2ð2m2 1 c1 Þ2 6m2 ð1 m2 Þ 4ð2m2 1Þð2m2 1 c1 Þ
; ;
under the constraint condition c23
2 9m2 ð1 m2 Þ 2 3m2 ð1 m2 Þ 2 1 þ c 1 þ c ¼ 0: 2 2m 2m þ 4c 1 5 1 ð2m2 1 c1 Þ ð2m2 1 c1 Þ
In particular, if m ! 1; then we have the solitary wave solution: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
ffi 2 c21 1 ta c3 ðw þ ak2 þ bk4 Þ4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½iðkxþwð a Þþh0 Þ ; ð3:19Þ uðx; tÞ ¼ 2 cc1 c3 ð1 c1 Þ 3c3 sech ðgÞ under the constraint condition c23 ð2 þ c1 Þ 4c5 ð1 þ c1 Þ2 ¼ 0: (6) If l0 ¼ 1; l2 ¼ 2 m2 ; l4 ¼ ð1 m2 Þ; then UðgÞ ¼ nd ðgÞ; and we have the Jacobi elliptic function solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ nd ðgÞ ð3:20Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f nd ðgÞ þ g where f and g are constants given by f ¼ g¼
c3 ð2 m2 c1 Þ 2ð2
m2
2
c1 Þ þ6ð1 m2 Þ 4ð2 m2 Þð2 m2 c1 Þ 3c3
2ð2 m2 c1 Þ2 þ6ð1 m2 Þ 4ð2 m2 Þð2 m2 c1 Þ
; ;
under the constraint condition
2 9ð1 m2 Þ 3ð1 m2 Þ 2 2 c23 þ c þ 4c 2 2 m 2 m þ c ¼ 0: 1 5 1 ð2 m2 c1 Þ ð2 m2 c1 Þ (7) If l0 ¼ 1; l2 ¼ 2 m2 ; l4 ¼ 1 m2 ; then UðgÞ ¼ sc ðgÞ; and we have the Jacobi elliptic function solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ sc ðgÞ ð3:21Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f sc ðgÞ þ g where f and g are constants given by
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c3 ð2 m2 c1 Þ 2ð2 m2 c1 Þ2 þ6ð1 m2 Þ 4ð2 m2 Þð2 m2 c1 Þ 3c3 2ð2 m2 c1 Þ2 þ6ð1 m2 Þ 4ð2 m2 Þð2 m2 c1 Þ
; ;
under the constraint condition
2 9ð1 m2 Þ 3ð1 m2 Þ 2 2 c23 2 2 m 2 m þ c þ c ¼ 0: þ 4c 1 5 1 ð2 m2 c1 Þ ð2 m2 c1 Þ In particular, if m ! 1; then we have the solitary wave solution qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2
ffi 1 2 c 2 4 1 ta c3 ðw þ ak þ bk Þ4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½iðkxþwð a Þþh0 Þ ; ð3:22Þ uðx; tÞ ¼ 2 cc1 c3 ð1 c1 Þ þ 3c3 csch ðgÞ under the constraint condition c23 ð2 þ c1 Þ 4c5 ð1 þ c1 Þ2 ¼ 0: While, if m ! 0; then we have the periodic wave solutions: 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c3 ðw þ ak2 þ bk4 Þ4 2ð2 c1 Þ 2ð5 4c1 Þ tanðgÞ5 ½iðkxþwðtaa Þþh0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ ; e cc1 c3 ð2 c1 Þ tan2 ðgÞ þ 3c3 ð3:23Þ or 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c3 ðw þ ak2 þ bk4 Þ4 2ð2 c1 Þ 2ð5 4c1 Þ 5 ½iðkxþwðtaa Þþh0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ; ð3:24Þ uðx; tÞ ¼ cc1 c3 ð2 c1 Þ þ 3c3 cot2 ðgÞ under the constraint condition c23 ðc1 2Þ 4c5 ðc1 1Þ2 ¼ 0: (8) If l0 ¼ 1; l2 ¼ 2m2 1; l4 ¼ m2 ð1 m2 Þ; then UðgÞ ¼ sd ðgÞ; and we have the Jacobi elliptic function solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ sd ðgÞ ð3:25Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ cc1 f sd 2 ðgÞ þ g where f and g are constants given by
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c3 ð 2m2 1 c1 Þ 2ð 2m2 1 c1 Þ2 6m2 ð1 m2 Þ 4ð 2m2 1Þð 2m2 1 c1 Þ 3c3 2ð 2m2 1 c1 Þ2 6m2 ð1 m2 Þ 4ð 2m2 1Þð 2m2 1 c1 Þ
; ;
under the constraint condition
9m2 ð1 m2 Þ 2 2 2m 1 þ c c23 1 ð 2m2 1 c1 Þ
2 3m2 ð1 m2 Þ 2 2m 1 þ c1 ¼ 0: þ 4c5 ð 2m2 1 c1 Þ (9) If l0 ¼ 1 m2 ; l2 ¼ 2 m2 ; l4 ¼ 1; then UðgÞ ¼ cs ðgÞ; and we have the Jacobi elliptic function solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ cs ðgÞ ð3:26Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ cc1 f cs 2 ðgÞ þ g where f and g are constants given by f ¼ g¼
c3 ð2 m2 c1 Þ 2
2ð2 m2 c1 Þ þ6ð1 m2 Þ 4ð2 m2 Þð2 m2 c1 Þ 3c3 ð1 m2 Þ 2ð2 m2 c1 Þ2 þ6ð1 m2 Þ 4ð2 m2 Þð2 m2 c1 Þ
; ;
under the constraint condition
2 9ð1 m2 Þ 3ð1 m2 Þ 2 2 c23 2 2 m 2 m þ 4c þ c ¼ 0: þ c 1 5 1 ð2 m2 c1 Þ ð2 m2 c1 Þ (10) If l0 ¼ m2 ð1 m2 Þ; l2 ¼ 2m2 1; l4 ¼ 1; then UðgÞ ¼ ds ðgÞ; and we have the Jacobi elliptic function solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ta c3 ðw þ ak2 þ bk4 Þ ds ðgÞ ð3:27Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e½iðkxþwð a Þþh0 Þ ; uðx; tÞ ¼ 2 cc1 f ds ðgÞ þ g where f and g are constants given by f ¼ g¼
c3 ð2m2 1 c1 Þ 2ð2m2 1 c1 Þ2 6m2 ð1 m2 Þ 4ð2m2 1Þð2m2 1 c1 Þ 3c3 m2 ð1 m2 Þ 2ð2m2 1 c1 Þ2 6m2 ð1 m2 Þ 4ð2m2 1Þð2m2 1 c1 Þ
; ;
under the constraint condition
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E. M. E. Zayed, A.-G. Al-Nowehy
2 9m2 ð1 m2 Þ 2 3m2 ð1 m2 Þ 2 1 þ c 1 þ c ¼ 0: þ 4c 2 2m 2m 1 5 1 ð2m2 1 c1 Þ ð2m2 1 c1 Þ
cn ðgÞ 1þm 1m (11) If l0 ¼ 1m 4 ; l2 ¼ 2 ; l4 ¼ 4 ; then UðgÞ ¼ nc ðgÞ sc ðgÞ or UðgÞ ¼ 1sn ðgÞ ; and we have the Jacobi elliptic function solutions 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2
uðx; tÞ ¼
2
2
ta c3 ðw þ ak2 þ bk4 Þ6 nc ðgÞ sc ðgÞ 7 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½iðkxþwð a Þþh0 Þ ; cc1 f ½nc ðgÞ sc ðgÞ2 þg
ð3:28Þ
or 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ½iðkxþwðta Þþh0 Þ c3 ðw þ ak2 þ bk4 Þ6 cn ðgÞ 6 a ffi7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ ; ð3:29Þ 4 5e h i 2 cc1 cn ðgÞ ð1 sn ðgÞÞ f 1sn ðgÞ þg where f and g are constants given by f ¼ 2 2 1þm 2 g¼ 2 2 1þm 2
1þm2 2 c1
2 1m2 2 c1 þ6 4 ð1 þ m2 Þð1 3 2 4 c3 ð1 m Þ
2 1m2 2 c1 þ6 4 ð1 þ m2 Þð1
c3
; þ m2 2c1 Þ ; þ m2 2c1 Þ
under the constraint condition " # " #2 2 2
9ð1 m2 Þ 3ð1 m2 Þ 2 2 2 c3 1 þ m þ c1 þ c5 1 þ m þ 2c1 ¼ 0: 8ð1 þ m2 2c1 Þ 4ð1 þ m2 2c1 Þ In particular, if m ! 0; then we have the periodic wave solutions: 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
2 5 2 4 2 ½ sec ð g Þ tan ð g Þ c þ2c 1 1 c3 ðw þ ak þ bk Þ6 2 8 7 ½iðkxþwðtaa Þþh0 Þ uðx; tÞ ¼ ; 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5e
cc1 c 1 c ½secðgÞ tanðgÞ2 þ 3 c 3 2
1
4 3
ð3:30Þ or 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 c3 ðw þ ak þ bk Þ6 6 uðx; tÞ ¼ 4 cc1
3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 12 c1 þ2c1 58 cosðgÞ 7 ½iðkxþwðta Þþh0 Þ 7e a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; h i 1
cosðgÞ 2 3 5 ð1 sinðgÞÞ c3 2 c1 1sinðgÞ þ 4 c3
ð3:31Þ under the constraint condition 2c23 ð2c1 1Þ c5 ð4c1 1Þ2 ¼ 0:
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2
ð1m2 Þ 2 1 (12) If l0 ¼ ; l2 ¼ 1þm 4 2 ; l4 ¼ 4 ; then UðgÞ ¼ m cn ðgÞ dn ðgÞ; and we have the Jacobi elliptic function solutions 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
uðx; tÞ ¼
ta c3 ðw þ ak2 þ bk4 Þ6 m cn ðgÞ dn ðgÞ 7 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½iðkxþwð a Þþh0 Þ ; cc1 f ½m cn ðgÞ dn ðgÞ2 þg
ð3:32Þ
where f and g are constants given by f ¼ 2 2 1þm 2
1þm2 2 c1
2 1m2 2 c1 þ6 4 ð1 þ
c3
; m2 Þð1 þ m2 2c1 Þ 2
3 2 4 c3 ð 1 m Þ ; g¼ 2
2 2 2 1m 2 Þð1 þ m2 2c Þ 2 1þm c þ6 ð 1 þ m 1 1 2 4
under the constraint condition " # " #2 2 2
9ð1 m2 Þ 3ð1 m2 Þ 2 2 2 1 þ m þ c1 þ c5 1 þ m þ 2c1 ¼ 0: c3 8ð1 þ m2 2c1 Þ 4ð1 þ m2 2c1 Þ sn ðgÞ 1 (13) If l0 ¼ 14 ; l2 ¼ 12m 2 ; l4 ¼ 4 ; then UðgÞ ¼ 1cn ðgÞ ; and we have the Jacobi elliptic function solutions 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ½iðkxþwðta Þþh0 Þ c3 ðw þ ak2 þ bk4 Þ6 sn ðgÞ 6 7e a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ ; ð3:33Þ 4 5 h i 2 cc1 sn ðgÞ ð1 cn ðgÞÞ f 1cn ðgÞ þg 2
where f and g are constants given by c3
12m2 2
c1
f ¼ ;
2 2 2 Þð1 2m2 2c Þ þ 3 c ð 1 2m 2 12m 1 1 2 8 3 4 c3 g¼ ;
2 2 12m 2 2 c1 ð1 2m2 Þð1 2m2 2c1 Þ þ 38
under the constraint condition c23
2 9 3 2 2 þ c þ 2c ¼ 0: 1 2m 1 2m þ c 1 5 1 8ð1 2m2 2c1 Þ 4ð1 2m2 2c1 Þ
In particular, if m ! 1; then we have the solitary wave solutions:
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2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c3 ðw þ ak2 þ bk4 Þ6 6 uðx; tÞ ¼ 4 cc1
3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
2 5 2 2 þ c1 2c1 8 tanhðgÞ 7 ta ffi7e½iðkxþwð a Þþh0 Þ ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
h tanhðgÞ i2 3 5 ð1 sech ðgÞÞ c3 2 þ c1 1sech ðgÞ þ 4 c3
ð3:34Þ under the constraint condition 2c23 ð2c1 þ 1Þ c5 ð4c1 þ 1Þ2 ¼ 0: While, if m ! 0; then we have the periodic wave solutions: 2 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
2 5 2 4 2 2 c1 þ2c1 8 sinðgÞ 7 ½iðkxþwðta Þþh0 Þ c3 ðw þ ak þ bk Þ6 6 7e a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ ; 4 cc1 1
h sinðgÞ i2 3 5 ð1 cosðgÞÞ c3 2 c1 1cosðgÞ þ 4 c3
ð3:35Þ under the constraint condition 2c23 ð2c1 1Þ c5 ð4c1 1Þ2 ¼ 0: 2
2
(14) If l0 ¼ 14 ; l2 ¼ 1þm 2 ; l4 ¼ elliptic function solutions
ð1m2 Þ 4
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c3 ðw þ ak2 þ bk4 Þ6 6 uðx; tÞ ¼ 4 cc1
ðg Þ ; then UðgÞ ¼ cn ðgsnÞdn ðgÞ ; and we have the Jacobi
3 7 ½iðkxþwðta Þþh0 Þ sn ðgÞ 7 a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; h i ffi5e
ðcn ðgÞ dn ðgÞÞ
f
sn ðgÞ cn ðgÞdn ðgÞ
2
þg
ð3:36Þ where f and g are constants given by c3 ð1 þ m2 2c1 Þ ; f ¼ 2
2 3 2 2 2 2 4 1þm 2 c1 þ 4 ð1 m Þ 2ð1 þ m Þð1 þ m 2c1 Þ 3 2 c3 g¼ 2 ;
2 3 1þm 4 2 c1 þ 4 ð1 m2 Þ2 2ð1 þ m2 Þð1 þ m2 2c1 Þ
under the constraint condition " # " #2 2 2 2 2
9 ð 1 m Þ 3 ð 1 m Þ 1 þ m2 þ c 1 þ c 5 1 þ m2 þ 2c1 ¼ 0: c23 8ð1 þ m2 2c1 Þ 4ð1 þ m2 2c1 Þ In particular, if m ! 1; then we have the solitary wave solution:
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3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2
ffi 1 tanh ð g Þ 2 c 2 4 1 ta c3 ðw þ ak þ bk Þ6 7 uðx; tÞ ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5e½iðkxþwð a Þþh0 Þ ; cc1 c ð1 c Þ tanh2 ðgÞ þ 3c sech 2 ðgÞ 3
1
3
ð3:37Þ under the constraint condition c23 ð2 þ c1 Þ 4c5 ð1 þ c1 Þ2 ¼ 0:
4 Graphical representations of some solutions In this section, we have presented some graphs of the exact solutions including Jacobi elliptic function solutions. Exact solutions of the results describe different nonlinear waves. These solutions have a remarkable property that keeps its identity upon interacting with other. Let us now examine Figs. 1, 2, 3 and 4 as it illustrates some of our solutions obtained in this article. To this aim, we select some special values of the parameters obtained, for example, in some of the Jacobi elliptic function solutions (3.8), (3.18), (3.20) and (3.26) of the nonlinear conformable time-fractional Schro¨dinger equation with fourth-order dispersion and parabolic law nonlinearity (1.1), with a ¼ b ¼ k ¼ c1 ¼ w ¼ v ¼ 1; 10\x; t\10. From these Figures, one can see that the obtained solutions possess the Jacobi elliptic function solutions. Also, these Figures expressing the behavior of these solutions which give some perspective readers how the behavior solutions are produced. For more convenience the graphical representations of these solutions are shown in the following Figures:
Fig. 1 Plot solution |u(x, t)| of (3.8) with c3 ¼ 1; c ¼ 1; m ¼ a ¼ 1=2
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Fig. 2 Plot solution |u(x, t)| of (3.18) with c ¼ 1; c3 ¼ 1; m ¼ a ¼ 1=3
Fig. 3 Plot solution |u(x, t)| of (3.20) with c ¼ 1; c3 ¼ 1; m ¼ a ¼ 1=4
5 Conclusions In this paper, based on the /6 -model expansion method combined with the conformable time-fractional derivative, we have obtained many new exact solutions including Jacobi elliptic function solutions, solitary wave solutions, trigonometric function solutions and other solutions to the nonlinear conformable time-fractional Schro¨dinger equation with
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Fig. 4 Plot solution |u(x, t)| of (3.26) with c3 ¼ 1; c ¼ 1; m ¼ a ¼ 1=5
fourth-order dispersion and parabolic law nonlinearity (1.1). Comparing our results obtained in this paper with the results obtained in Biswas and Milovic (2009), Douvagai et al. (2016), Xu (2011), Zayed et al. (2017e), Kohl et al. (2008), and Biswas et al. (2008), we conclude that our results are new and not found elsewhere. Finally, our results in this paper have been checked with the aid of the Maple by putting them back into the original Eq. (1.1).
References Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015) Abu Hammad, M., Khalil, R.: Conformable fractional heat differential equation. Int. J. Pure. Appl. Math. 94, 215–221 (2014) Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325, 542–553 (2007) Arnous, A.H., Ullah, M.Z., Asma, M., Moshokoa, S.P., Zhou, Q., Mirzazadeh, M., Biswas, A., Belic, M.: Dark and singular dispersive optical solitons of Schro¨dinger–Hirota equation by modified simple equation method. Optik 136, 445–450 (2017) Biswas, A., Milovic, D.: Optical solitons with fourth order dispersion and dual-power law nonlinearity. Int. J. Nonlinear Sci. 7, 443–447 (2009) Biswas, A., Milovic, D., Zerrad, E., Majid, F.: Optical solitons in a Kerr law media with fourth order dispersion. Adv. Studies Theor. Phys. 20, 1007–1012 (2008) Biswas, A., Bhrawy, A.H., Abdelkawy, M.A., Alshaery, A.A., Hilal, E.M.: Symbolic computation of some nonlinear fractional differential equations. Rom. J. Phys. 59, 0433–0442 (2014) Biswas, A., Ekici, M., Triki, H., Sonmezoglu, A., Mirzazadeh, M., Zhou, Q., Mahmood, M.F., Ullah, M.Z., Moshokoa, S.P., Belic, M.: Resonant optical soliton perturbation with anti-cubic nonlinearity by extended trial function method. Optik 156, 784–790 (2018) Douvagai, Y.salathiel, Betchewe, G., Doka, S.Y., Creptin, K.T.: Exact traveling wave solutions to the fourth-order dispersive nonlinear Schr o¨dinger equation with dual-power law nonlinearity. Math. Methods Appl. Sci. 39, 1135–1143 (2016)
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