Optical solitons and other solutions to the conformable space time fractional complex Ginzburg Landau equation under Kerr law nonlinearity

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1 Pramana J. Phys. (2018) 91:58 Indian Academy of Sciences Optical solitons other solutions to the conformale space time fractional complex Ginzurg Lau equation under Kerr law nonlinearity TUKUR ABDULKADIR SULAIMAN 12 HACI MEHMET BASKONUS 3 HASAN BULUT 14 1 Department of Mathematics Firat University Elazig Turkey 2 Department of Mathematics Federal University Dutse Jigawa State Nigeria 3 Department of Computer Engineering Munzur University Tunceli Turkey 4 Department of Mathematics Education Final International University Kyrenia Cyprus Corresponding author. sulaiman.tukur@fud.edu.ng MS received 15 March 2018; revised 28 March 2018; accepted 3 April 2018; pulished online 24 August 2018 Astract. This study reveals the dark right comined dark right singular comined singular optical solitons singular periodic solutions to the conformale space time fractional complex Ginzurg Lau equation. We reach such solutions via the powerful extended sinh-gordon equation expansion method (ShGEEM). Constraint conditions that guarantee the existence of valid solitary wave solutions are given. Under suitale choice of the parameter values interesting three-dimensional graphs of some of the otained solutions are plotted. Keywords. PACS Nos Sinh-Gordon equation; Ginzurg Lau equation; optical soliton; periodic wave Hq; J; Rz; e 1. Introduction The investigation of soliton propagation through nonlinear optical fires as a theory of optical solitons has ecome one of the interesting research areas [12. It is well known that optical solitons represent a treasure trove of research in the field of nonlinear optics. Optical solitons play vital roles in the field of fire optics photonics quantum electronics optoelectronics so on [3. Various integral schemes have een used to investigate different kinds of nonlinear Schrödinger equations (NLSE) [4 32. For the past two decades the study of fractional differential equations has attracted the attention of many scientists [3334. Several definitions of fractional differential equations have een sumitted to the literature such as Riemann Liouville Grunwald Letnikov the Caputo Atangana Baleanu derivative in Caputo sense Atangana Baleanu fractional derivative in Riemann Liouville sense modified Riemann Liouville [ Recently Khalil et al [39 developed the conformale fractional derivative which can simply e used to transform fractional nonlinear partial differential equations to nonlinear ordinary equations. This paper is devoted for otaining optical solitons singular periodic wave solutions to the conformale space time fractional complex Ginzurg Lau (GL) equation [23 y utilising the powerful extended sinh-gordon equation expansion method (ShGEEM) [40 44: id t + ad2 x + F( 2 ) ( 2 ) 1 [ν 2 Dx 2 ( 2 ) ϱ(dx ( 2 )) 2 γ = 0 (1) where x t are the non-dimensional distance along the fire time in dimensionless form respectively. The unknown function is the complex-valued function of x t the symol sts for the complex conjugate of the unknown function a are the coefficients of the group velocity dispersion nonlinearity respectively. The real-valued constants ν ϱ γ are coefficients from the perturation effects [4546. F is a real-valued algeraic function which must have the smoothness of the function F( 2 ) :C C. When the complex plane C is considered as 2D linear space R 2 the function F( 2 ) is k times continuously differentiale [47.

2 58 Page 2 of 8 Pramana J. Phys. (2018) 91:58 F( 2 ) σδ=1 C k( ( δ δ) (σ σ ); R 2). (2) 2. The conformale fractional derivative In this section asic definition properties theorem aout conformale fractional derivative are discussed [39. DEFINITION 1 Let g : (0 ) R then the conformale fraction derivative of g of order is defined as g(t + ɛt 1 ) g(t) T (g)(t) = lim ɛ 0 ɛ t > 0 0 <<1. (3) Here some asic properties of the conformale fractional derivative [39 are presented: 1. T (g + ch) = T (g) + ct (h) c R. 2. T (t λ ) = λt λ λ R. 3. T (gh) = gt (h) + ht (g). 4. T (g/h) = ((ht (g) gt (h))/h 2 ). 5. If g is differentiale then T (g)(t)=t 1 (dg/dt). Theorem 1. Let g h : (0 ) R e differentiale also differentiale functions. Then the following rule holds: T (g h)(t) = t 1 h (t)g (h(t)). (4) 3. The extended ShGEEM Consider the following space time fractional sinh-gordon equation [40: Dx (D t ) = λ sinh( ) 0 < 1 (5) where λ is a non-zero constant. Sustituting the fractional travelling wave transformations (x t) = ψ(η) η = x υ t (6) into eq. (5) we otain the following nonlinear ordinary differential equation (NODE): ψ = λ sinh(ψ) (7) υ where υ is the velocity of the travelling wave. Integrating eq. (7) once we otain ( ψ ) 2 = λ ( ) ψ 2 υ sinh2 + c (8) 2 where c is the constant of integration. Setting (λ/υ) = ψ/2 = eq.(8) ecomes = sinh 2 ( ) + c. (9) For different values of parameters a eq.(9) possesses the following sets of solutions: Set I: When c = 0 = 1 eq. (9) ecomes = sinh( ). (10) Simplifying eq. (10) the following solutions are secured: sinh( ) =±csch(η) or sinh( ) =±i sech(η) (11) cosh( ) =±coth(η) or cosh( ) =±tanh(η) (12) where i = 1. Set II: When c = 1 = 1 eq. (9) ecomes = cosh( ). (13) Simplifying eq. (13) the following solutions are otained: sinh(θ) = tan(η) or sinh( ) = cot(η) (14) cosh( ) =±sec(η) or cosh( ) =± csc(η). (15) For a given nonlinear space time fractional partial differential equation say P( Dx D x (D t ) 2 Dx t...)= 0 0 < 1 (16) the following steps are followed to secure various wave solutions to (16): Step 1: We first transform eq. (16) into the following NODE y using eq. (6): Q(ψ ψ ψ ψ 2 ψ...)= 0. (17) Step 2: We assume that eq. (17) has a new ansatz solution of the form ψ( ) = Bk sinh( ) + A k cosh( ) k + A0 (18) where is a function of η it satisfies eq. (9) A 0 A k B k (k = m) areconstantstoedetermined later. To otain the value of m the homogeneous

3 Pramana J. Phys. (2018) 91:58 Page 3 of 8 58 alance principle is used on the highest derivatives highest power nonlinear term in eq. (17). Step 3: We sustitute eq. (28) its derivatives with a fixed value of m along with eq. (10) or (13) into eq. (17) to otain a polynomial equation in s sinh j ( ) cosh k ( ) (s = 0 1 j k = ). We collect a set of overdetermined nonlinear algeraic equations in A 0 A k B k υy summing the coefficients of s sinh j ( ) cosh k ( ) with the same power setting each summation to zero. Step 4: The otained set of overdetermined nonlinear algeraic equations is then solved with the aid of symolic software to determine the values of the parameters A 0 A k B k υ. Step 5: Using the results otained in eqs (11) (12) (14) (15) we have the wave solutions to the given nonlinear partial differential equation in the following forms: ψ(η) = ± Bk i sech(η)± A k tanh(η) k + A0 ψ(η) = ψ(η) = ψ(η) = (19) ± Bk csch(η) ± A k coth(η) k + A0 (20) ± Bk sec(η) + A k tan(η) k + A0 (21) ± Bk csc(η) A k cot(η) k + A0. (22) 4. Application In this section we otain various solutions to the complex GL equation [3 y using the extended ShGEEM. Consider the following complex fractional travelling wave transformation: (x t) = ψ(η)e iφ φ = ϑ x + ω t + η = x υ t (23) where ψ(η) sts for the pulse shape υ is the velocity of the soliton φ is the phase component ϑ is the soliton frequency ω is the soliton wave numer is the phase constant. Sustituting eq. (23) into eq. (1) yields the following NODE: ωψ + a(ψ ϑ 2 ψ) + F(ψ 2 )ψ 2(ν 2ϱ) (ψ ) 2 ψ 2νψ γψ = 0 (24) from the real part the relation υ = 2aϑ from the imaginary part. Setting ν = 2ϱ eqs(1) (24) ecome id t + ad2 x + F( 2 ) ϱ( 2 ) 1 [2 2 Dx 2 ( 2 ) (Dx ( 2 )) 2 γ = 0 (25) (a 4ϱ)ψ (ω + aϑ 2 + γ)ψ+ F(ψ 2 )ψ = 0 (26) respectively. For the Kerr law nonlinearity F(ψ) = ψ [3. Therefore eq. (26) ecomes (a 4ϱ)ψ (ω + aϑ 2 + γ)ψ + ψ 3 = 0. (27) The Kerr law nonlinearity arises from the fact that light wave in an optical fire encounters nonlinear responses from the non-harmonic motion of electrons with an external electric field [2. Balancing the terms ψ ψ 3 in eq. (27) yields m = 1. With m = 1 eqs (28) (32) take the following forms: ( ) = B 1 sinh( ) + A 1 cosh( ) + A 0 (28) ψ(η) =±B 1 i sech(η) ± A 1 tanh(η) + A 0 (29) ψ(η) =±B 1 csch(η) ± A 1 coth(η) + A 0 (30) ψ(η) =±B 1 sec(η) + A 1 tan(η) + A 0 (31) ψ(η) =±B 1 csc(η) A 1 cot(η) + A 0 (32) respectively (figure 1). Putting eq. (28) its second derivative into eq. (27) give a polynomial in the power hyperolic functions. We get a set of algeraic equations y equating each summation of the coefficients of the hyperolic functions of the same power to zero. We solve the set of algeraic equations to get the values of the parameters involved. To explicitly secure the solutions of eq. (1) we insert the values of the parameters into each of eqs (29) (32). Set 1: When A 0 A 1 = B 1 = A 1 (a + 2(γ 2ϱ + ω)) ϑ = 2a eq. (1) possesses the following comined dark right comined singular optical solitons:

4 58 Page 4 of 8 Pramana J. Phys. (2018) 91:58 Figure 1. The 3D graphics of eq. (35) when (a) = = 0.7 () = = (x t) =± + tanh +2aϑ t 2 (x t) =± + csch + 2aϑ t ( i sech + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (33) ( coth + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (34) respectively where () >0a(a + 2(γ 2ϱ + ω)) < 0 for the existence of valid solitons (figure 2). Set 2: When 2() A 0 = 0 A 1 = B 1 = 0 Figure 2. The 3D graphics of eq. (43) when (a) = = 0.7 () = = ϱ 2a γ ω ϑ = a eq. (1) possesses the following dark singular optical solitons: 2() 3 (x t) =± tanh + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (35) 2() 4 (x t) =± coth + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (36)

5 Pramana J. Phys. (2018) 91:58 Page 5 of 8 58 csch + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (38) respectively. Valid solitons exist for oth () <0or(4ϱ a) >0a(a γ 4ϱ ω) > 0. Set 4: When A 0 = 0 A 1 = B 1 = A 1 ω = 1 2 a(1 + 2ϑ2 ) γ + 2ϱ eq. (1) possesses the following comined dark right comined singular optical solitons: ( [ x 7 (x t) =± i sech + 2aϑ t + tanh + 2aϑ t e i( ϑ(x /) ( 1/2a(1+2ϑ 2 )+γ 2ϱ ) (t /)+ ) (39) Figure 3. The 3D graphics of eq. (44) when (a) = = 0.7 () = = 0.6. respectively where () >0a(8ϱ 2a γ ω) > 0 for the existence of valid solitons (figure 3). Set 3: When 2() A 0 = 0 A 1 = 0 B 1 = a γ 4ϱ ω ϑ = a eq. (1) possesses the following right singular optical solitons: 2() 5 (x t) =± sech + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (37) 2() 6 (x t) =± ( [ x 8 (x t) =± coth + 2aϑ t + csch + 2aϑ t e i( ϑ(x /) ((1/2)a(1+2ϑ 2 )+γ 2ϱ)(t /)+ ) (40) respectively where () >0 for valid solitons. Set 5: When 2() A 0 = 0 A 1 = B 1 = 0 ω = a(2 + ϑ 2 ) γ + 8ϱ eq. (1) possesses the following dark singular optical solitons: [ 2() x 9 (x t) =± tanh + 2aϑ t e i( ϑ(x /) (a(2+ϑ 2 )+γ 8ϱ)(t /)+ ) (41) 2() 10 (x t) =± coth + 2aϑ t e i( ϑ(x /) (a(2+ϑ 2 )+γ 8ϱ)(t /)+ ) (42)

6 58 Page 6 of 8 Pramana J. Phys. (2018) 91:58 respectively where () >0 for valid solitons. Set 6: When 2() A 0 = 0 A 1 = 0 B 1 = ω = a(1 ϑ 2 ) γ 4ϱ eq. (1) possesses the following right singular optical solitons: [ 2(a 4ϱ) x 11 (x t) =± sech + 2aϑ t e i( ϑ(x /)+(a(1 ϑ 2 ) γ 4ϱ)(t /)+ ) (43) [ 2() x 12 (x t) =± csch + 2aϑ t e i( ϑ(x /)+(a(1 ϑ 2 ) γ 4ϱ)(t /)+ ) (44) respectively where (a 4ϱ) > 0or(a 4ϱ) < 0 for valid solitons. Set 7: When 2(γ + 4ϑ A 0 = 0 A 1 = 2 ϱ + ω) (2 + ϑ 2 ) (γ 8ϱ + ω) B 1 = 0 a = 2 + ϑ 2 eq. (1) possesses the following dark singular optical solitons: 13 (x t)=± tanh 2(γ + 4ϑ 2 ϱ + ω) (2 + ϑ 2 ) 2ϑ(γ 8ϱ + ω) 2 + ϑ 2 t ( ) e i ϑ (x ) () +ω (t ) () + 14 (x t)=± 2(γ + 4ϑ 2 ϱ + ω) (2 + ϑ 2 ) 2ϑ(γ 8ϱ + ω) t coth 2 + ϑ 2 (45) e i( ϑ(x /)+ω(t /)+ ) (46) respectively where (2 + ϑ 2 )(γ + 4ϑ 2 ϱ + ω) > 0for valid solitons. Set 8: When A 0 = 0 A 1 = 0 B 1 = 2(γ + 4ϑ 2 ϱ + ω) (ϑ 2 a = γ + 4ϱ + ω 1) 1 ϑ 2 eq. (1) possesses the following right singular optical solitons: 2(γ + 4ϑ 15 (x t) =± 2 ϱ + ω) (ϑ 2 1) (γ + 4ϱ + ω) t sech + 1 ϑ 2 e i( ϑ(x /)+ω(t /)+ ) (47) 2(γ + 4ϑ 16 (x t) =± 2 ϱ + ω) (ϑ 2 1) (γ + 4ϱ + ω) t csch + 1 ϑ 2 e i( ϑ(x /)+ω(t /)+ ) (48) respectively where (ϑ 2 1)(γ + 4ϑ 2 ϱ + ω) > 0or (ϑ 2 1)(γ + 4ϑ 2 ϱ + ω) < 0 for valid solitons. Set 9: When A 0 = 0 A 1 = a 2(γ + 2ϱ + ω) B 1 = A 1 ϑ = 2a eq. (1) possesses the following singular periodic solutions: ( [ x 17 (x t) =± sec + 2aϑ t + tan + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (49) 18 (x t) =± + csc + 2aϑ t ( cot + 2aϑ t e i( ϑ(x /)+ω(t /)+ ) (50) where (4ϱ a) >0or(4ϱ a) <0a(a 2(γ + 2ϱ + ω)) > 0 for valid solitons.

7 Pramana J. Phys. (2018) 91:58 Page 7 of 8 58 Set 10: When A 0 = 0 A 1 = B 1 = A 1 ω = 1 2 (a 2aϑ2 2(γ + 2ϱ)) eq. (1) possesses the following singular periodic solutions: ( [ x 20 (x t) =± sec + 2aϑ t + tan + 2aϑ t e i( ϑ(x /)+(1/2)(a 2aϑ 2 2(γ +2ϱ))(t /)+ ) ( [ x 21 (x t) =± cot + 2aϑ t + csc + 2aϑ t (51) e i( ϑ(x /)+(1/2)(a 2aϑ 2 2(γ +2ϱ))(t /)+ ) (52) where (4ϱ a) >0 or(4ϱ a) <0 forvalidsolitons. 5. Conclusions This study used the powerful extended ShGEEM in constructing the dark right comined dark right singular comined singular optical solitons singular periodic wave solutions to the conformale space time fractional complex GL equation. The constraint conditions for the existence of a valid soliton to each of the otained solutions are stated. Under suitale choice of parameters the 3D graphics of some of the otained solutions are also plotted. The results reported may e useful in explaining the physical meaning of the studied equation in this paper other complex nonlinear models arising in various fields of nonlinear science. The computations in this paper provide a lot of encouragement to the future findings. It can e seen that the integral scheme used in this study is a powerful efficient mathematical tool which produces a family of solutions when applied to a complex nonlinear model. In our future research we are going to consider investigating the conformale space time fractional complex GL with a different type of nonlinearity. Its numerical solutions will also e computed. References [1 M Mirzazadeh M Eslami Q Zhou J. Nonlinear Opt. Phys. Mater. 24(2) (2015) [2 A H Arnous A R Seadawy R T Alqahtani A Biswas Optik (2017) [3 M Mirzazadeh M Ekici A Sonmezoglu M Eslami Q Zhou A H Kara D Milovic F B Majid A Biswas MBelić Nonlinear Dyn. 85(3) 1979 (2016) [4 A Ali A R Seadawy D Lu Optik (2017) [5 A M Wazwaz Chaos Solitons Fractals (2008) [6 H Bulut T A Sulaiman B Demirdag Nonlinear Dyn. 91(3) 1985 (2018) [7 M Mirzazadeh A H Arnous M F Mahmood E Zerrad A Biswas Nonlinear Dyn. 81(8) 277 (2015) [8 C Cattani Int. J. Fluid Mech. Res. 30(5) 23 (2003) [9 H Triki A M Wazwaz J. Electromagn. Waves Appl. 30(6) 788 (2016) [10 T A Sulaiman T Akturk H Bulut H M Baskonus J. Electromagn. Waves Appl. 32(9) 1093 (2017) [11 H Bulut T A Sulaiman H M Baskonus Superlatt. Microstruct. (2018) j.spmi [12 S T R Rizvi I Ali K Ali M Younis Optik 127(13) 5328 (2016) [13 A Sardar K Ali S T R Rizvi M Younis Q Zhou E Zerrad A Biswas A Bhrawy J. Nanoelectron. Optoelectron. 11(3) 382 (2016) [14 H M Baskonus T A Sulaiman H Bulut T Akturk Superlatt. Microstruct (2018) [15 M Eslami A Neirameh Opt. Quantum Electron (2018) [16 M Arshad A R Seadawy D Lu Eur.Phys.J.Plus (2017) [17 J Manafian M F Aghdaei Eur.Phys.J.Plus13197 (2016) [18 M T Darvishi S Ahmadian S B Arai M Najafi Opt. Quantum Electron (2018) [19 O Uruc A Esen Int. J. Mod. Phys. C 27(9) (2016) [20 H Bulut T A Sulaiman H M Baskonus Eur. Phys. J. Plus (2017) [21 A Esen O Tasozan Ann. Math. Sil (2017) [22 M Eslami M Mirzazadeh A Neirameh Pramana J. Phys. 84(1) 3 (2015) [23 M Matinfar M Eslami M Kordy Pramana J. Phys. 85(4) 583 (2015) [24 J Manafian M Lakestani Pramana J. Phys. 85(1) 31 (2015) [25 A R Seadawy Pramana J. Phys. 89: 49 (2017) [26 K Roy S K Ghosh P Chatterjee Pramana J. Phys. 86(4) 873 (2016) [27 H Bulut T A Sulaiman H M Baskonus Opt. Quantum Electron. 50(2) 87 (2018) [28 O A Ilhan T A Sulaiman H Bulut H M Baskonus Eur.Phys.J.Plus133(1) 27 (2018) [29 E Bas B Kilics World Appl. Sci. J. 11(3) 256 (2010)

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