Research Article New Exact Traveling Wave Solutions of the Time Fractional Complex GinzburgLandau Equation via the Conformable Fractional Derivative


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1 indawi Advances in Mahemaical Physics Volume 1, Aricle ID , 1 ages hs://doi.org/1.1155/1/ Research Aricle New Eac Traveling Wave Soluions of he Time Fracional Comle GinzburgLandau Equaion via he Conformae Fracional Derivaive Zhao Li and Tianyong an College of Comuer Science, Chengdu Universiy, Chengdu 6116, China Corresondence should be addressed o Zhao Li; Received 3 Seember ; Revised 17 November ; Acceed 8 December ; Puished 8 January 1 Academic Edior: Ming Mei Coyrigh 1 Zhao Li and Tianyong an. This is an oen access aricle disribued under he Creaive Commons Aribuion License, which ermis unresriced use, disribuion, and reroducion in any medium, rovided he original work is roerly cied. In his sudy, he eac raveling wave soluions of he ime fracional comle GinzburgLandau equaion wih he Kerr law and dualower law nonlineariy are sudied. The nonlinear fracional arial differenial equaions are convered o a nonlinear ordinary differenial equaion via a raveling wave ransformaion in he sense of conformae fracional derivaives. A range of soluions, which include hyerbolic funcion soluions, rigonomeric funcion soluions, and raional funcion soluions, is derived by uilizing he new eended ðg /GÞeansion mehod. y selecing aroriae arameers of he soluions, numerical simulaions are resened o elain furher he roagaion of oical ulses in oic fibers. 1. Inroducion The GinzburgLandau (GL) equaion [1 7] is one of he mos imoran arial differenial equaions in he field of mahemaics and hysics, which was inroduced ino he sudy of suerconduciviy henomenology heory in he h cenury by Ginzburg and Landau. The GL equaion usually describes he oical solion [8 15] roagaion hrough oical fibers over longer disances. Therefore, i is very imoran o analyze he dynamic behavior of he GL equaion. In aricular, searching for he aroimae analyical soluions or eac raveling soluions of he GL equaion has been more and more widely followed wih ineres because hey can hel o elain he comleiy of nonlinear oics. So, in recen years, many owerful mehods were used for finding he eac raveling soluions of he GL equaion, which include he firs inegral mehod [16], dynamical sysem mehod [17], ðg /GÞeansion mehod [18], iroa s mehod [19], and harmonic balance mehod []. In recen years, he fracional GL equaion [1 4] has been more and more widely followed wih ineres because i can be used o accuraely model some nonlinear oical henomena. In aricular, consrucing he eac raveling soluions of he fracional GL equaion is very imoran work because i can be beer elained he dynamics of solion roagaion hrough oical fibers over longer disances in nonlinear oics. So far, many imoran mehods have been roosed o find he eac soluions of he fracional GL equaion. In [5], Arshed eraced he hyerbolic, rigonomeric, and raional funcion soluions of he fracional GL equaion by he e ð ϕðξþþeansion mehod. In [6], Raza obained he eriodic and hyerbolic solion soluions of he conformae ime fracional GL equaion by using he soliary wave ansaz mehod and e ð φðχþþeansion mehod. In [7], Abdou e al. emloyed Jacobi s elliic funcion eansion aroach o rerieve douy eriodic funcion soluions for he imesace fracional GL equaion. In [8], Sirisubawee e al. invesigaed he cubicquinic GL equaion by he modified Kudryashov mehod and ðg /G,1 /GÞeansion mehod. In [9], Ghanbari and Gómez Aguilar obained he eriodic and hyerbolic solion soluions of he conformae GL equaion by using he generalized eonenial raional funcion mehod. Alhough eac raveling wave soluions of he fracional GL equaion have
2 Advances in Mahemaical Physics been successfully obained by uilizing he above mehods, i is far from enough on searching for he eac soluion of a more general fracional comle GL equaion. The main urose of his aer is o consruc eac raveling wave soluions of he ime fracional comle GL equaion wih he Kerr law and dualower law nonlineariy by using he new eended ðg /GÞeansion mehod, and a range of soluions which include hyerbolic funcion soluions, rigonomeric funcion soluions, raional funcion soluions, and negaive ower soluions is derived. In his aer, we will consider he following ime fracional comle GL equaion [6, 9]: id δ u + au + bf juj u = αju j juj β u j j juj + γu, <δ 1, u ð1þ where reresens disance across he fiber in dimensionless form. a and b denoe he coefficien of he grou velociy disersion and he coefficien of nonlineariy. The erms wih consans α and β arise from he erurbaion effecs. In aricular, γ comes from he deuning effec. a, b, α, β, and γ are realvalued consans. u is he comle conjugae numbers of u. Equaion (1) is used o model various hysical henomena, such as nonlinear waves, oseeinsein condensaion, secondorder hase ransiions, suerconduciviy, suerfluidiy, liquid crysals, and srings. In Equaion (1), Fðjuj Þu : C C is a realvalued algebraic funcion. Le Fðjuj Þu be k imes coninuously differeniae. Then, F juj [ u C k m,n=1 ð n, nþ ð m, mþ; R, ðþ where C is a comle lane of wodimensional linear sace in R. The res of his aer is organized as follows. In Secion, we will give he definiion of he conformae fracional derivaive and is roeries. In Secion 3, we will inroduce he eended ðg /GÞeansion mehod. In Secion 4, we will discuss he eac raveling soluions of he fracional comle GL equaion wih he Kerr law and dualower law nonlineariy by using he eended ðg /GÞeansion mehod and lo he rofiles of several reresenaive eac raveling soluions. In Secion 5, we will resen he concluding remarks.. Preliminaries Definiion 1 [8]. Le f : ½, Þ R. Then, he conformae fracional derivaive of f of order α is defined as D δ f ðþ= lim ε f + ε 1 δ fðþ ε, for all >,<δ 1: ð3þ If he limi in Equaion (3) eiss, hen we say ha f is δ conformae differeniae a a oin. Theorem [8]. Le α ð, 1 and f ðþ and gðþ be δconformae differeniae a a oin >. Then, D δ D δ D δ μ ð Þ= μ μ δ, for all μ R, ðafðþ+ bgðþ Þ= ad δ fðþ+ bdδ g ðþ, for all a, b R, D δ ðfðþg Þ= fðþd δ g ðþ+ g ðþdδ fðþ, fðþ = g ðþdδ fðþ fðþdδ g ðþ g ðþ g, ðþ D δ 1 δ df ðfðþ Þ= ðþ, rovided ha f ðþis differeniae: d ð4þ Theorem 3 [8]. Le f : ð, Þ R be a funcion such ha f is differeniae and δconformae differeniae. Furher, le g be a differeniae funcion defined in he range of f. Then, D α ðf gþðþ= 1 α g ðþ α 1 g ðþd α ðfðþ Þ =g where he rime noaion ð Þ denoes he classical derivaive. 3. Overview of he Eended ðg /GÞEansion Mehod ðþ, ð5þ Le us consider a general fracional arial differenial equaion in he form P u, u, u,,d δ u, Dδ u, Dδ Dδ, =, <δ 1, ð6þ where u is an unknown funcion and P is a olynomial of u and is arial fracional order derivaives, in which he highes order derivaives and nonlinear erms are involved. Se 1. y using he raveling wave ransformaion uð, Þ = UðξÞ, where ξ = lð δ /δþ, l is a nonzero arbirary consan, <δ 1, we can rewrie Equaion (6) as he following nonlinear ordinary differenial equaion (ODE) of he ineger order wih resec o ξ: Q U, U, U, =: ð7þ If i is ossie, we should inegrae Equaion (7) erm by erm one or more imes; he consans of inegraion are considered o be zero. Se. Assume ha he raveling wave soluion of Equaion (7) can be eressed as follows: N ðþ= U ξ i=! i! N i G G a i + b G i, ð8þ i=1 G
3 Advances in Mahemaical Physics 3 where eiher a N or b N may be zero, bu boh a N and b N canno be zero a a ime, and a i ði =,1,,,NÞ and b i ði = 1,,,NÞ are arbirary consans, where G = GðξÞ saisfies he following nonlinear ODE: GG = AG + GG + CG, where A,, and C are real arameers. ð9þ Se 3. The value of he osiive ineger N in Equaion (8) can be comued by using he homogeneous balance beween he highes order nonlinear erm and he highes order derivaive in UðξÞ aearing in Equaion (8). Se 4. Subsiuing Equaions (8) and (9) ino Equaion (7) wih he value of N obained in Se 3, we obain olynomials in ðg /GÞ N ðn =,1,, Þ and ðg /GÞ N ðn =,1,, Þ. Then, we collec all coefficiens of he resuled olynomial o zero, and we obain a se of algebraic equaions for a i ði =,1,, Þ and b i ði =1,,,NÞ. Se 5. y solving he algebraic equaions obained in Se 4, we can obain he values of he consans a i ði =,1,, Þ and b i ði =1,,,NÞ. Relacing he values ino Equaion (8), we consruc a more general ye and new eac raveling wave soluions of Equaion (3). The soluions of Equaion (9) can be caegorized ino he following hree cases. Case 1 (hyerbolic funcion soluions). When 4ðA 1Þ C >and A 1, hen 1 G ðþ ξ GðÞ ξ = +4C 4AC C 1 sinh +4C 4AC/ ξ + C cosh +4C 4AC/ A + C 1 cosh +4C 4AC/ ξ + C sinh +4C 4AC/ ξ : ð1þ Case (rigonomeric funcion soluions). When 4ðA 1ÞC >and A 1, hen 1 G ðþ ξ GðÞ ξ = 4AC 4C C 1 sin 4AC 4C / ξ + C cos 4AC 4C / A + C 1 cos 4AC 4C / ξ + C sin 4AC 4C / ξ : ð11þ Case 3 (raional funcion soluions). When 4ðA 1ÞC =and A 1, hen G ðþ ξ GðÞ ξ = 1 C 1 + : ð1þ 1 A C 1 ξ + C Remark 4. In fac, by aking A =, = λ, and C = μ in Equaion (9), he second nonlinear ODE reduces ino he following secondorder linear ODE: G + λg + μg =: 4. Eac Soluions of he Time Fracional Comle GinzburgLandau Equaion ð13þ In order o obain he raveling wave soluion of Equaion (1), we assume ha he eac soluion has he following form: u, ð Þ= UðÞe ξ iη, where ξ = vð δ /δþ and η = k + wð δ /δþ + θ. ð14þ Subsiuing (14) ino Equaion (1), hen collecing he real and imaginary ars, we ge he following equaions: Re : wu + a U k U +αu + γu, + bf U U U =ðα βþ Im : v = ak, U ð15þ ð16þ where U = d U/dξ. y aking α =β, Equaion (15) becomes wu + a U k U + bf U U =αu + γu: ð17þ 4.1. Kerr Law Nonlineariy. The Kerr law nonlineariy is he case when FðUÞ = U; hen, Equaion (17) reduces o ðα aþu bu 3 + γ + w + ak U =: ð18þ
4 4 Advances in Mahemaical Physics (a) δ = 1/ (b) δ = 1/ (c) δ = 4/5 Figure 1: The osiive real ar of u 11 ð, Þ in Equaion (3) for differenial values of he fracional arameer δ. y using he homogeneous balance rincile, we selec N =1. Then, we suose ha he soluion of Equaion (18) saisfies!! 1 G G UðÞ= ξ a 1 + a G + b 1, ð19þ G where a 1, a, and b 1 are consans o be deermined. Subsiuing (19) ogeher wih (9) ino (18) yields a olynomial in ðg /GÞ N (N =,1,, ) and ðg /GÞ N (N =,1,, ). Collecing he coefficien of he resuled olynomial and seing hem o zero, we obain a nonlinear algebraic equaion in a 1, a, and b 1. Solving he algebraic equaion using he comuer algebraic sofware of Male, we ge he following resuls: qffiffiffiffiffiffiffiffiffiffiffiffiffi Case 1. a 1 = ððα aþða 1Þ Þ/b, a = ffiffi ððα aþ Þ/b, b 1 =, k = k, and w = ððα aþð +4C 4ACÞÞ/ γ ak. Case. a 1 =, a = ffiffi ððα aþ Þ/b, b 1 = ffiffiffi ððα aþc Þ/b, k = k, and w = ðððα aþð +4C 4AC ÞÞ/Þ γ ak, where A,, C, a, b, α, and γ are free arameers. Subsiuing hese soluions ino (19), using (14), we obain he raveling wave soluion as follows: rffiffiffiffiffiffi!# ðα aþ u, ð Þ= α ð aþða 1Þ G ðþ ξ b b GðÞ ξ! 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ðα aþ α ð aþc G ðþ ξ u, ð Þ= b b GðÞ ξ e i k+w δ /δþ+θ e i k+w δ /δþ+θ ðþ where G ðξþ/gðξþ is defined in Secion 3, and ξ = +ak ð δ /δþ.
5 Advances in Mahemaical Physics 5 Family 1. The hyerbolic funcion soluions can be obained as 4ðA 1ÞC >and A 1. Then, rffiffiffiffiffiffi ðα aþ α ð aþða 1Þ u 1 ð, Þ= b b!# + +4C 4AC 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα aþ α ð aþc u ð, Þ= b b! C 4AC 1 e i k+w δ /δþ+θ e i k+w δ /δþ+θ ð1þ ðþ where 1 = ðc 1 sinh ðð +4C 4AC/Þð +akð δ /δþþþ + C cosh ðð +4C 4AC /Þ ð +akð δ /δþþþþ/ ðc 1 cosh ðð +4C 4AC/Þ ð +akð δ /δþþþ + C sinh ðð +4C 4AC/Þð +akð δ /δþþþþ, and C 1 and C are arbirary consans. In aricular, if we assume ha C =, A >1, >, and C 1 in (1), hen we have u 11 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα aþð +4C 4ACÞ ð, Þ= anh b!# +ak δ /δ e i ð k+w ð δ /δþ+θþ, +4C 4AC ð3þ and if C 1 =, A >1, >, and C in (1), hen we obain u 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα aþð +4C 4ACÞ ð, Þ= coh b!# +ak δ /δ e i ð k+w ð δ /δþ+θþ : +4C 4AC ð4þ The fied values are α =, a =1, b =1, A =, =4, C =3, and k =1, and he fracional arameer δ f1/5, 1/, 1g is used o lo he osiive real ar of he eac soluion u 11 ð, Þ as in Figures 1 and. Family. The rigonomeric funcion soluions can be obained as 4ðA 1ÞC >and A 1. Then, rffiffiffiffiffiffi u 3 ð, Þ= ðα aþ α ð aþða 1Þ b b!# 4AC 4C + e i ð k+w ð δ /δþ+θþ, ð5þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα aþ α ð aþc u 4 ð, Þ= b b! 1# + 4AC 4C e i k+w δ /δþ+θ ð6þ where = ð C 1 sin ðð 4AC 4C/Þð +akð δ /δþþþ + C cos ðð 4AC 4C /Þð + akð δ /δþþþþ/ ðc 1 cos ðð 4AC 4C/Þ ð + akð δ /δþþþ + C sin ðð 4AC 4C/Þð +akð δ /δþþþþ, and C 1 and C are arbirary consans. Secifically, if we suose ha C =, A >1, >, and C 1 in (5), hen we obain u 31 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα aþþð4ac 4C Þ ð, Þ= an b!# +ak δ e i ð k+w ð δ /δþ+θþ, δ 4AC 4C ð7þ and if C 1 =, A >1, >, and C in (5), hen we have u δ = 1 5 δ = 1 δ = 4 5 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα aþð4ac 4C Þ ð, Þ= co b!# +ak δ e i ð k+w ð δ /δþ+θþ : δ Figure : The osiive real ar of u 11 ð, Þ a =1: in Equaion (3) for differenial values of he fracional arameer δ: 4AC 4C ð8þ
6 6 Advances in Mahemaical Physics (a) δ = 1/ (b) δ = 1/ (c) δ = 4/5 Figure 3: The osiive real ar of u 31 ð, Þ in Equaion (7) for differenial values of he fracional arameer δ. The fied values are α =, a =1, b =1, A =4, =3, C =3, and k =1, and he fracional arameer δ f1/5, 1/, 4/5g is used o lo he osiive real ar of he eac soluion u 31 ð, Þ as in Figures 3 and 4. Family 3. The raional funcion soluions can be obained as 4ðA 1ÞC =and A 1. Then, rffiffiffiffiffiffi ðα aþ 1 α ð aþða 1Þ u 5 ð, Þ= b 1 A b C 1 +!# e i ð k+w ð δ /δþ+θþ, C 1 +ak δ /δ + C rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðα aþ 1 α ð aþc u 6 ð, Þ= b 1 A b C 1 +! 1 # e i ð k+w ð δ /δþ+θþ, C 1 +ak δ /δ + C where C 1 and C are arbirary consans. ð9þ 4.. DualPower Law. In his case, FðUÞ = U n + lu n ; hen, Equaion (17) akes he following form: ðα aþu + γ + w + ak U bu n+1 + lu 4n+1 =, ð3þ where l is a consan.
7 Advances in Mahemaical Physics δ = 1 5 δ = 1 δ = 4 5 Figure 4: The osiive real ar of u 31 ð, Þ a =1: in Equaion (7) for differenial values of he fracional arameer δ Using ransformaion U = V 1/n, ð31þ q ððn +1Þðα aþða 1Þ Þ/, b 1 =, k = k, and w = ðððα aþð +4C 4ACÞÞ/8n Þ ðð3bðn +1ÞÞ/ð8lðn +1Þ ÞÞ γ ak. Equaion (3) will be reduced ino he following ODE: h i Case. a ðα aþ ð1 nþv +nvv +4n γ + w + ak 1 =, a = ððn + 1Þ/ð4lðn + 1ÞÞÞ ð/ 4n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V CÞ ððn +1Þðα aþc Þ/, b 1 =ð1/nþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4bn V V + lv =: ðððn +1Þðα aþc Þ/Þ, k = k, and w = ðððα aþð + 4C 4ACÞÞ/8n Þ ðð3bðn +1ÞÞ/ð8lðn +1Þ ÞÞ γ ak, ð3þ where A,, C, a, b, α, and γ are free arameers. Now, alying he balancing rincile beween V 4 and VV in he above equaion, we ge N =1. Therefore, we have!! 1 G G VðÞ= ξ a 1 + a G + b 1, ð33þ G where a 1, a, and b 1 are consans o be deermined. Subsiuing (33) ogeher wih (9) ino (3) yields a olynomial in ðg /GÞ N (N =,1,, ) and ðg /GÞ N (N =,1,, ). Collecing he coefficien of he resuled olynomial and seing hem o zero, we obain a nonlinear algebraic equaion in a 1, a, and b 1. Solving he algebraic equaion using he comuer algebraic sofware of Male, we ge he following resuls: q Case 1. a 1 = ð1/nþ ððn +1Þðα aþða 1Þ Þ/, a = ððn + 1Þ/ð4lðn + 1ÞÞÞ ð/ð4nða 1ÞÞÞ Subsiuing hese soluions ino (33), using (14) and (31), we obain he raveling wave soluion as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n +1 u, ð Þ= 4lðn+1Þ ðn +1Þðα aþða 1Þ 4nðA 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# 1/n ðn +1Þðα aþða 1Þ G ðþ ξ e i ð k+w ð δ /δþ+θþ, n GðÞ ξ n +1 u, ð Þ= 4lðn+1Þ rffiffiffiffiffiffiffiffi ðn +1Þðα aþc 4nC 1 rffiffiffiffiffiffiffiffi! 1 # 1/n ðn +1Þðα aþc G ðþ ξ e i ð k+w ð δ /δþ+θþ, n GðÞ ξ ð34þ where G ðξþ/gðξþ is defined in Secion 3, and ξ = +akð δ /δþ.
8 8 Advances in Mahemaical Physics (a) δ = 1/5 (b) δ = 1/ (c) δ = 4/5 5 Figure 5: The osiive real ar of u 11 ð, Þ in Equaion (37) for differenial values of he fracional arameer δ. Family 1. The hyerbolic funcion soluions can be obained as 4ðA 1ÞC >and A 1. Then, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n +1 u 1 ð, Þ= 4lðn+1Þ ðn +1Þðα aþða 1Þ 4nðA 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn +1Þðα aþða 1Þ n!# 1/n + +4C 4AC 1 e i ð k+w ð δ /δþ+θþ, n +1 u ð, Þ= 4lðn+1Þ rffiffiffiffiffiffiffiffi ðn +1Þðα aþc 4nC 1 rffiffiffiffiffiffiffiffi ðn +1Þðα aþc n! 1# 1/n + +4C 4AC 1 ð35þ e i k+w δ /δþ+θ ð36þ where 1 = ðc 1 sinh ðð +4C 4AC/Þð + akð δ / δþþþ + C cosh ðð +4C 4AC/Þð +akð δ /δþþþþ/ðc 1 cosh ðð +4C 4AC/Þð + akð δ /δþþ + C sinh ðð +4C 4AC/Þð +akð δ /δþþþþ, and C 1 and C are arbirary consans. In aricular, if we assume ha C =, A >1, and C 1 in (35), hen we have u 11 n +1 ð, Þ= 4lðn+1Þ 1 rffiffiffiffiffiffiffi ðn +1Þðα aþð +4C 4ACÞ anh 4n +4C 4AC!# 1/n +ak δ e i k+w ð δ /δþ+θ δ ð Þ, ð37þ and if C 1 =, A >1, and C in (35), hen we obain u 1 n +1 ð, Þ= 4lðn+1Þ 1 rffiffiffiffiffiffiffi ðn +1Þðα aþð +4C 4ACÞ coh 4n +4C 4AC!# 1/n +ak δ e i k+w ð δ /δþ+θ δ ð Þ : ð38þ
9 Advances in Mahemaical Physics δ = 1 5 δ = 1 δ = 4 5 Figure 6: The osiive real ar of u 11 ð, Þ a =1: in Equaion (37) for differenial values of he fracional arameer δ The fied values are α =, a =1, b =1, A =, =4, C =3, k =1, l =3, and n =, and he fracional arameer δ f1/5, 1/, 1g is used o lo he real ar of he eac soluion u 11 ð, Þ as in Figures 5 and 6. Family. The rigonomeric funcion soluions can be obained as 4ðA 1ÞC >and A 1. Then, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n +1 u 3 ð, Þ= 4lðn+1Þ ðn +1Þðα aþða 1Þ 4nðA 1Þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn +1Þðα aþða 1Þ n!# 1/n + 4AC 4C e i ð k+w ð δ /δþ+θþ, n +1 u 4 ð, Þ= 4lðn+1Þ rffiffiffiffiffiffiffiffi ðn +1Þðα aþc 4nC 1 rffiffiffiffiffiffiffiffi ðn +1Þðα aþc n! 1# 1/n + 4AC 4C ð39þ e i k+w δ /δþ+θ ð4þ where = ð C 1 sin ðð 4AC 4C/Þð +ak ð δ / δþþþ + C cos ðð 4AC 4C/Þð +akð δ /δþþþþ/ ðc 1 cos ðð 4AC 4C/Þ ð + akð δ /δþþþ + C sin ðð 4AC 4C/Þð +akð δ /δþþþþ, and C 1 and C are arbirary consans. Secifically, if we suose ha C =, A >1, and C 1 in (39), hen we obain u 31 n +1 ð, Þ= 4lðn+1Þ 1 rffiffiffiffiffiffiffi ðn +1Þðα aþð4ac 4C Þ 4n 4AC 4C!# 1/n an +ak δ ð δ ð Þ, ð41þ e i k+w δ /δþ+θ and if C 1 =, A >1, and C in (39), hen we obain u 3 n +1 ð, Þ= 4lðn+1Þ 1 rffiffiffiffiffiffiffi ðn +1Þðα aþð4ac 4C Þ 4n 4AC 4C!# 1/n co +ak δ ð δ ð Þ : ð4þ e i k+w δ /δþ+θ The fied values are α =, a =1, b =1, A =4, =3, C =3, k =1, l =3,andn =, and he fracional arameer δ f1/5, 1/, 1g is used o lo he real ar of he eac soluion u 31 ð, Þ as in Figures 7 and 8. Family 3. The raional funcion soluions can be obained as 4ðA 1ÞC =and A 1. Then, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n +1 u 5 ð, Þ= 4lðn+1Þ ðn +1Þðα aþða 1Þ 4nðA 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðn +1Þðα aþða 1Þ nð1 AÞ C 1 + # 1/n e i ð k+w ð δ /δþ+θþ, C 1 ξ + C
10 1 Advances in Mahemaical Physics (a) δ = 1/ (b) δ = 1/ (c) δ = 4/5 Figure 7: The osiive real ar of u 31 ð, Þ in Equaion (41) for differenial values of he fracional arameer δ.
11 Advances in Mahemaical Physics δ = 1 5 δ = 1 δ = 4 5 Figure 8: The osiive real ar of u 31 ð, Þ a =1: in Equaion (41) for differenial values of he fracional arameer δ n +1 u 6 ð, Þ= 4lðn+1Þ rffiffiffiffiffiffiffiffi ðn +1Þðα aþc 4nC rffiffiffiffiffiffiffiffi 1 ðn +1Þðα aþc nð1 AÞ ð43þ solving more eensive and comlicaed fracional arial differenial equaions. Daa Availabiliy No daa were used o suor his sudy. 5. Conclusion C 1 + # 1/n 1 e i ð k+w ð δ /δþ+θþ : C 1 ξ + C The ime fracional comle GL equaion wih he Kerr law and dualower law nonlineariy, which deics he dynamics of solion roagaion hrough oical fibers over longer disances, is sudied by using he eended ðg /GÞeansion mehod. A series of new eac soluions are consruced, including hyerbolic funcion soluions, rigonomeric funcion soluions, and raional funcion soluions. Comaring our resuls wih he soluions in he revious lieraure, we ge many new eac soluions. In order o furher undersand he dynamic behaviors of hese resuls, wih he hel of sofware Male 18, we have rovided some grahical reresenaion of he eac soluions of he ime fracional comle GL equaion wih he Kerr law and dualower law nonlineariy by seing he aroriae arameers. We believe ha he obained soluions may be of cardinal significance in he field of nonlinear oics. Moreover, in he aer, we have confirmed ha he eended ðg /GÞeansion mehod is owerful, reliae, and sysemaic, and hey could be alied for Conflics of Ineres The auhors declare ha here is no conflic of ineres regarding he uicaion of his aer. Acknowledgmens This work was suored by he Scienific Research Funds of Chengdu Universiy under gran no References [1] A. Elmandouh, ifurcaion and new raveling wave soluions for he D GinzburgLandau equaion, The Euroean Physical Journal Plus, vol. 135, no. 8, ,. [] A. iswas and R. T. Alqahani, Oical solion erurbaion wih comle GinzburgLandau equaion by semiinverse variaional rincile, Oik, vol. 147, , 17. [3] A. iswas, Y. Yildirim, E. Yasar e al., Oical solion erurbaion wih comle GinzburgLandau equaion using rial soluion aroach, Oik, vol. 16,. 44 6, 18. [4] W. J. Corrêa and T. Özsari, Comle GinzburgLandau equaions wih dynamic boundary condiions, Nonlinear Anaysis: Real World Alicaions, vol. 41, , 18.
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