Int. J. Anal. Appl. (2022), 20:22 Application of the F-Expansion Method for Solving the Fokas-Lenells Equation Ohoud A. Alshahrani∗ Department of Mathematics, Faculty of Sciences, University of Tabuk, P.O.Box 741, Tabuk 71491, Saudi Arabia ∗Corresponding author: ohoud1972@yahoo.com, ahoud.ksa@hotmail.com Abstract. By the aid of traveling wave hypothesis, the F-expansion method has been implemented in this paper to obtain Jacobian-Elliptic function solutions for the optical Fokas-Lenells model. The hyperbolic-function solutions are derived as special cases from the Jacobian-Elliptic function solutions. The present approach is straightforward to determine the exact solutions for the Fokas-Lenells equa- tion. The existence criteria of the obtained solutions are also reported. 1. Introduction In the field of telecommunications engineering, the optical soliton perturbation and the PDEs are the most active areas of research [1-15]. The dynamics of soliton molecules are administered by a variety of nonlinear evolution equations. The nonlinear Schrödinger’s equation is the most studied model in this context. Although the nonlinear Schrödinger’s model has been extensively studied by many authors with different forms of nonlinearity, the present work analyzes the pulse propagation engineering through optical fibers and PCF with a newly established model known as the Fokas– Lenells equation (FLE) [12, 13]. Such model has been studied to obtain various kinds of soliton solutions by using the complex–amplitude ansatz and other approaches [12, 13]. The objective of this paper is to apply the F-expansion method by the aid of the traveling wave hypothesis to deduct more general solution as well as soliton solutions. The paper is organized as follows. In the next subsection, the FLE is presented. In section 2, the traveling wave hypothesis is introduced. Section 3 is devoted to the application of the F-expansion method on the current model. The exact solutions in terms of Jacobian-Elliptic functions are displayed in section 4. The hyperbolic-function solutions are derived Received: Feb. 4, 2022. 2010 Mathematics Subject Classification. 78A60, 37K10, 35Q51, 35Q55. Key words and phrases. optics; Fokas–Lenells equation; solitons. https://doi.org/10.28924/2291-8639-20-2022-22 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-22 2 Int. J. Anal. Appl. (2022), 20:22 from the Jacobian-Elliptic function solutions in section 5. The paper is ended by a conclusion section 6 and two appendices. 1.1. Governing model. The dimensionless form of perturbed FLE that has been proposed takes the form [16, 17]: iqt +a1qxx +a2qxt + |q|2 (B1q + iσqx)= i [ αqx +λ ( |q|2m q ) x +µ ( |q|2m ) x q ] , (1) where the right hand side represents all the perturbation terms. In equation (1), the independent variables are x and t that represent spatial and temporal variables respectively while q(x,t) is the complex–valued wave profile representing the soliton profile. Here, a1 is the coefficient of group velocity dispersion while a2 is the coefficient of spatio–temporal dispersion that was proposed to be included a few years ago. Then, σ is the coefficient of nonlinear dispersion. On the right hand side, α is the coefficient of inter–modal dispersion while λ accounts for self–steepening effect and finally µ gives another form of nonlinear dispersion. The parameter B1 indicates the effect of self-phase modulation while the parameter m refers to the full nonlinearity. 2. Traveling wave hypothesis The solutions of (1) may be supposed as q(x,t)= eiθ(x,t) u(ω), (2) where ω = x −γt and the phase θ(x,t) = −kx +βt + θ0, u(ω) is the amplitude component of the wave and γ is its speed. k is the soliton frequency, β is its wave-number and θ0 is the phase constant. Equation (1) can be decomposing into real and imaginary parts yields a pair of relations. The real and imaginary parts of Eq. (1) are respectively (a1 −a2)u′′ +(a2ωk −a1k2 −αk −ω)u +(B1 +kσ)u3 −ku[(2m+1)λ+2mµ]u2m =0, (3) and γ +2ka1 −a2(γk +ω)−σu2 +α+[(2m+1)λ+2mµ]u2m =0. (4) We notice from (4) that [λ(2m+1)+2mµ]u2m =−γ −2ka1 +a2(γk +ω)+σu2 −α. (5) Setting (2m+1)λ+2mµ =0, σ =0, (6) in (5), then λ = −2mµ (2m+1) , (7) Int. J. Anal. Appl. (2022), 20:22 3 and γ = 2ka1 −a2ω +α (a2k −1) . (8) Accordingly, Eq. (3) reduces to (a1 −a2γ)u′′ +(a1k2 −ω +kγ(1−a2k))u +B1u3 =0. (9) Using n1 =(a1 −a2γ), n2 =(a1k2 −ω +kγ(1−a2k)), (10) hence (9) gives n1u ′′ +n2u +B1u 3 =0. (11) 3. Application the F-expansion method to the FLE Assume that the solution of (11) is in the form [18] u(ω)= n∑ i=0 aiF i(ω), (12) where ai, i = 0,1,2, . . . ,n, are constants to be determined, n is a positive integer which can be evaluated by balancing the highest-order linear term u ′′ and nonlinear term u3, this gives n = 1. Moreover, F(ω) satisfies the following auxiliary equation F ′(ω)=± √ PF4(ω)+QF2(ω)+R, (13) where P, Q, and R are constants. Eq. (13) for F(ω) leads to  F ′′ =2PF3 +QF, F ′′′ =(6PF2 +Q)F ′, F ′′′′ =24P2F5 +20PQF3 +(12PR+Q2)F . . . (14) In Appendix A, we present 46 forms of exact solutions for Eq. (13), (see ref. [18] for details). In fact, these exact solutions can be used to construct more exact solutions for Eq. (11). According to n =1, Eq. (12) becomes u(ω)= a0 +a1F(ω). (15) 4 Int. J. Anal. Appl. (2022), 20:22 Substituting (15) into (11), we obtain the following system of algebraic equations  B1a 3 0 +a0n2 =0, 3B1a 2 0a1 +Qa1n1 +a1n2 =0, 3B1a 2 1a0 =0, B1a 3 1 +2Pa1n1 =0. (16) Solving the last system (16), we have a0 =0, a1 =± √ −2Pn1 B1 , Q =− n2 n1 . (17) With these values of a0, a1 and Q, the exact solution of (11) can be obtained by 46 forms depending on the values of P, Q and R with the corresponding solution F(ω) in Appendix A. However, some selected cases are presented in the following sections. 4. Jacobian-elliptic function solutions Case: 1: Let us consider the inputs of case 1 in Appendix A with implementing (15) and (17), thus  P = m2, Q =−n2 n1 =−(1+m2), R =1, F(ω)= snω, u =± √ −2n1m2 B1 snω, n2 =(1+m2)n1, n1 < 0, q(x,t)=± √ −2n1m2 B1 ei (−kx+βt+θ0) sn(x −γt). (18) This general solution has not been reported in [16, 17]. Moreover, it will be demonstrated in section 5 that the solution (18) and other solutions of this section reduce to hyperbolic forms as special cases. Case: 2: The solution of this case (case 2 in Appendix A) is expressed in terms of another kind of Jacobian- Elliptic functions as   P = m2, Q =−n2 n1 =−(1+m2), R =1, F(ω)= cdω, u =± √ −2n1m2 B1 cdω, n2 =(1+m2)n1, n1 < 0, q(x,t)=± √ −2n1m2 B1 ei (−kx+βt+θ0) cd(x −γt). (19) which was not also reported in Refs. [16, 17]. Int. J. Anal. Appl. (2022), 20:22 5 Case: 3: The solution of this case (by using the inputs of case 26 in Appendix A) is expressed as  P > 0, Q =−n2 n1 < 0, R = m 2Q2 (1+m2)2P , F(ω)= √ −m2Q (1+m2)P sn (√ −Q 1+m2 ω ) , u =± √ −2m2n2 (1+m2)B1 sn (√ n2 (1+m2)n1 ω ) , n2n1 > 0, , n2B1 < 0, q(x,t)=± √ −2m2n2 (1+m2)B1 ei (−kx+βt+θ0) sn (√ n2 (1+m2)n1 (x −γt) ) . (20) Case: 4: The solution of this case (by using the inputs of case 27 in Appendix A) is expressed as  P < 0, Q =−n2 n1 > 0, R = (1−m2)Q2 (m2−2)2P , F(ω)= √ −Q (2−m2)P dn (√ Q 2−m2 ω ) , u =± √ 2n2 (2−m2)B1 dn (√ −n2 (2−m2)n1 ω ) , n2n1 < 0, , n2B1 > 0, q(x,t)=± √ 2n2 (2−m2)B1 ei (−kx+βt+θ0) dn (√ −n2 (2−m2)n1 (x −γt) ) . (21) which was not reported in Refs. [16, 17]. Case: 5: On using the inputs of case 28 in Appendix A, we have  P < 0, Q =−n2 n1 > 0, R = m2(m2−1)Q2 (2m2−1)2P , F(ω)= √ −m2Q (2m2−1)P cn (√ Q 2m2−1 ω ) , u =± √ 2m2n2 (2m2−1)B1 cn (√ −n2 (2m2−1)n1 ω ) , n2n1 < 0, , n2B1 > 0, q(x,t)=± √ 2m2n2 (2m2−1)B1 ei (−kx+βt+θ0) cn (√ −n2 (2m2−1)n1 (x −γt) ) . (22) 5. Hyperbolic-function solutions Some soliton−like solutions of Eq. (11) can be obtained in the limited case when the modulus m → 1 (see Appendix B), as: Case: 1:   P =1, Q =−n2 n1 =−2, R =1, F(ω)= tanhω, u =± √ −2n1 B1 tanhω, n2 =2n1, n1 < 0, q(x,t)=± √ −2n1 B1 ei (−kx+βt+θ0) tanh(x −γt). (23) 6 Int. J. Anal. Appl. (2022), 20:22 Case: 2:   P =1, Q =−n2 n1 =−2, R =1, F(ω)=1, u =± √ −2n1 B1 , n2 =2n1, n1 < 0, q(x,t)=± √ −2n1 B1 ei (−kx+βt+θ0). (24) Case: 3:   P > 0, Q =−n2 n1 < 0, R = Q 2 4P , F(ω)= √ −Q 2P tanh (√ −Q 2 ω ) , u =± √ −n2 B1 tanh (√ n2 2n1 ω ) , n2n1 > 0, , n2B1 < 0, q(x,t)=± √ −n2 B1 ei (−kx+βt+θ0) tanh (√ n2 2n1 (x −γt) ) . (25) Cases: 4:   P < 0, Q =−n2 n1 > 0, R =0, F(ω)= √ −Q P sech (√ Q ω ) , u =± √ 2n2 B1 sech (√ −n2 n1 ω ) , n2n1 < 0, , n2B1 > 0, q(x,t)=± √ 2n2 B1 ei (−kx+βt+θ0) sech (√ −n2 n1 (x −γt) ) . (26) Case: 5: This case leads to the same hyperbolic-function solution given in (26). Here, it should be noted that the solutions presented in the previous section in terms of the Jacobian- Elliptic function are more general than those previously obtained in the relevant literature. In addition, the obtained hyperbolic-function solutions were derived as special cases from our Jacobian-Elliptic function solutions. Moreover, some of the present solutions have not been reported in previous works [16, 17] which analyzed the same Fokas-Lenells equation. As a final observation is that all of the current solutions are obtained by using only one method, however, three different methods have been applied in [16, 17] to obtain only three solutions. Finally, several kinds of soliton solutions such as singular soliton solution and dark-singular combo soliton solution can be derived by considering more inputs of the 46 cases in Appendix A with the aid of Appendix B. 6. Conclusions This paper revealed new types of exact solutions for the perturbed FLE, where the perturbation terms are of Hamiltonian type and appeared with full nonlinearity. The F-expansion method was applied in this paper to obtain several kinds of Jacobian-Elliptic function solutions for the optical Fokas-Lenells model. In special cases, the solito-like solutions in terms of the hyperbolic-functions are Int. J. Anal. Appl. (2022), 20:22 7 derived from the Jacobian-Elliptic function solutions. The results have not been reported in previous works in relevant literatures. Several kinds of soliton solutions such as singular soliton solution and dark-singular combo soliton solution can be derived by further investigations of the suggested method. Appendix A Relations between values of (P, Q, R) and corresponding F(ω) in Eq. (13), where A, B and C are arbitrary constants and m1 = √ 1−m2. Case P Q R F(ω) 1 m2 −(1+m2) 1 snω 2 m2 −(1+m2) 1 cdω=cnω/dnω 3 −m2 2m2 −1 1−m2 cnω 4 −1 2−m2 m2 −1 dnω 5 1 −(1+m2) m2 nsω =(snω)−1 6 1 −(1+m2) m2 dcω = dnω/cnω 7 1−m2 2m2 −1 −m2 ncω =(cnω)−1 8 m2 −1 2−m2 −1 ndω =(dnω)−1 9 1−m2 2−m2 1 scω = snω/cnω 10 −m2(1−m2) 2m2 −1 1 sdω = snω/dnω 11 1 2−m2 1−m2 csω = cnω/snω 12 1 2m2 −1 −m2(1−m2) dsω = dnω/snω 13 1/4 (1−2m2)/2 1/4 nsω ±csω 14 (1−m2)/4 (1+m2)/2 (1−m2)/4 ncω ± scω 15 1/4 (m2 −2)/2 m2/4 nsω ±dsω 16 m2/4 (m2 −2)/2 m2/4 snω ± icnω 17 m2/4 (m2 −2)/2 m2/4 √ 1−m2sdω ±cdω 18 1/4 (1−m2)/2 1/4 m cdω ± i √ 1−m2ndω 19 1/4 (1−2m2)/2 1/4 m snω ± idnω 20 1/4 (1−m2)/2 1/4 √ 1−m2scω ±dcω 21 (m2 −1)/4 (m2 +1)/2 (m2 −1)/4 m sdω ±ndω 22 m2/4 (m2 −2)/2 1/4 snω 1±dnω 23 −1/4 (m2 +1)/2 (1−m2)2/4 m cnω ±dnω 24 (1−m2)2/4 (m2 +1)/2 1/4 dsω ±csω 25 m 4(1−m2) 2(2−m2) 2(1−m2) m2−2 1−m2 2(2−m2) dcω ± √ 1−m2ncω 26 P > 0 Q < 0 m 2Q2 (m2+1)2P √ −m2Q (m2+1)P sn (√ −Q m2+1 ω ) 27 P < 0 Q > 0 (1−m 2)Q2 (m2−2)2P √ −Q (2−m2)P dn (√ Q 2−m2ω ) 8 Int. J. Anal. Appl. (2022), 20:22 28 P < 0 Q > 0 m 2(m2−1)Q2 (2m2−1)2P √ − m2Q (2m2−1)P cn (√ Q 2m2−1ω ) 29 1 2−4m2 1 snωdnωcnω 30 m4 2 1 snωcnωdnω 31 1 m2 +2 1−2m2 +m4 dnωcnωsnω 32 A 2(m−1)2 4 m2+1 2 +3m (m−1)2 4A2 dnωcnω A(1+snω)(1+msnω) 33 A 2(m+1)2 4 m2+1 2 −3m (m+1) 2 4A2 dnωcnω A(1+snω)(1−msnω) 34 − 4 m 6m−m2 −1 −2m3 +m4 +m2 mcnωdnω msn2ω+1 35 4 m −6m−m2 −1 2m3 +m4 +m2 mcnωdnω msn2ω−1 36 1/4 1−2m 2 2 1/4 sn 1±cnω 37 1−m 2 4 1+m2 2 1−m2 4 cnω 1±snω 38 4m1 2+6m1 −m2 2+2m1 −m2 m 2snωcnω m1−dn2ω 39 −4m1 2−6m1 −m2 2−2m1 −m2 −m 2snωcnω m1+dn2ω Case P Q R F(ω) 40 2−m 2−2m1 4 m2 2 −1−3m1 2−m 2−2m1 4 m2snωcnω sn2ω+(1+m1)dnω−1−m1 41 2−m 2+2m1 4 m2 2 −1+3m1 2−m 2+2m1 4 m2snωcnω sn2ω+(−1+m1)dnω−1+m1 42 C 2m4−(B2+C2)m2+B2 4 m2+1 2 m2−1 4(C2m2−B2) √ (B2−C2) (B2−C2m2) +snω B cnω+C dnω 43 B 2+C2m2 4 1 2 −m2 1 4(C2m2+B2) √ (C2m2+B2−C2) (B2+C2m2) +cnω B snω+C dnω 44 B 2+C2 4 m2 2 −1 m 4 4(C2+B2) √ (B2+C2−C2m2) (B2+C2) +dnω B snω+C cnω 45 −(m2 +2m+1)B2 2m2 +2 2m−m 2−1 B2 m sn2ω−1 B(m sn2ω+1) 46 −(m2 −2m+1)B2 2m2 +2 −2m+m 2+1 B2 m sn2ω+1 B(m sn2ω−1) Appendix B The Jacobi−elliptic functions degenerate into hyperbolic functions when m → 1 as: snω → tanhω, {cnω, dnω}→ sechω, {scω, sdω}→ sinhω, {dsω, csω}→ cschω, {ncω, ndω}→ coshω, nsω → cothω, {cdω, dcω}→ 1. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] A. Biswas, 1-Soliton solution of 1+2 dimensional nonlinear Schrödinger’s equation in power law media, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 1830–1833. https://doi.org/10.1016/j.cnsns.2008.08.003. [2] A. Biswas, D. Milovic, Travelling wave solutions of the non-linear Schrödinger’s equation in non-Kerr law media, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 1993–1998. https://doi.org/10.1016/j.cnsns.2008.04. 017. https://doi.org/10.1016/j.cnsns.2008.08.003 https://doi.org/10.1016/j.cnsns.2008.04.017 https://doi.org/10.1016/j.cnsns.2008.04.017 Int. J. Anal. Appl. (2022), 20:22 9 [3] S.H. Crutcher, A.J. Osei, A. Biswas, Wobbling phenomena with logarithmic law nonlinear Schrödinger equations for incoherent spatial Gaussons, Optik. 124 (2013), 4793–4797. https://doi.org/10.1016/j.ijleo.2013.01.081. [4] M. Eslami, M. Mirzazadeh, B. Fathi Vajargah, A. Biswas, Optical solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients by the first integral method, Optik. 125 (2014), 3107–3116. https: //doi.org/10.1016/j.ijleo.2014.01.013. [5] A.J. Mohamad Jawad, M.D. Petković, A. Biswas, Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation. 217 (2010), 869–877. https://doi.org/10.1016/j.amc.2010.06.030. [6] A.A. Gaber, A.F. Aljohani, A. Ebaid, J.T. Machado, The generalized Kudryashov method for nonlinear space–time fractional partial differential equations of Burgers type, Nonlinear Dyn. 95 (2019), 361–368. https://doi.org/ 10.1007/s11071-018-4568-4. [7] A.A. AlQarni, A. Ebaid, A.A. Alshaery, H.O. Bakodah, A. Biswas, S. Khan, M. Ekici, Q. Zhou, S.P. Moshokoa, M.R. Belic, Optical solitons for Lakshmanan–Porsezian–Daniel model by Riccati equation approach, Optik. 182 (2019), 922–929. https://doi.org/10.1016/j.ijleo.2019.01.057. [8] Y.M. Mahrous, S.M. Khaled, A. Ebaid, An internet traffic flow model via a conformable derivative: The exact soliton solutions, Adv. Differ. Equ. Control Processes. 21 (2019), 227–237. https://doi.org/10.17654/DE021020227. [9] D.A. Lott, A. Henriquez, B.J.M. Sturdevant, A. Biswas, A numerical study of optical soliton-like structures resulting from the nonlinear Schrödinger’s equation with square-root law nonlinearity, Appl. Math. Comput. 207 (2009), 319–326. https://doi.org/10.1016/j.amc.2008.10.038. [10] M. Mirzazadeh, M. Eslami, B.F. Vajargah, A. Biswas, Optical solitons and optical rogons of generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity, Optik. 125 (2014), 4246–4256. https: //doi.org/10.1016/j.ijleo.2014.04.014. [11] H.O. Bakodah, M.A. Banaja, B.A. Alrigi, A. Ebaid, R. Rach, An efficient modification of the decomposition method with a convergence parameter for solving Korteweg de Vries equations, J. King Saud Univ. - Sci. 31 (2019), 1424–1430. https://doi.org/10.1016/j.jksus.2018.11.010. [12] H. Triki, A.-M. Wazwaz, Combined optical solitary waves of the Fokas—Lenells equation, Waves Rand. Complex Media. 27 (2017), 587–593. https://doi.org/10.1080/17455030.2017.1285449. [13] H. Triki, A.-M. Wazwaz, New types of chirped soliton solutions for the Fokas–Lenells equation, Int. J. Numer. Methods Heat Fluid Flow. 27 (2017), 1596–1601. https://doi.org/10.1108/HFF-06-2016-0252. [14] B. Salah, E.R. El-Zahar, A.F. Aljohani, A. Ebaid, M. Krid, Optical soliton solutions of the time-fractional perturbed Fokas-Lenells equation: Riemann-Liouville fractional derivative, Optik. 183 (2019), 1114–1119. https://doi.org/ 10.1016/j.ijleo.2019.02.016. [15] A. Ebaid, E.R. El-Zahar, A.F. Aljohani, B. Salah, M. Krid, J.T. Machado, Exact solutions of the generalized nonlinear Fokas-Lennells equation, Results Phys. 14 (2019), 102472. https://doi.org/10.1016/j.rinp.2019.102472. [16] A.J. Mohamad Jawad, A. Biswas, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation of Fokas–Lenells equation with two integration schemes, Optik. 165 (2018), 111–116. https://doi.org/10.1016/j.ijleo.2018. 03.104. [17] A.F. Aljohani, E.R. El-Zahar, A. Ebaid, M. Ekici, A. Biswas, Optical soliton perturbation with Fokas-Lenells model by Riccati equation approach, Optik. 172 (2018), 741–745. https://doi.org/10.1016/j.ijleo.2018.07.072. [18] A. Ebaid, E.H. Aly, Exact solutions for the transformed reduced Ostrovsky equation via the -expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions, Wave Motion. 49 (2012), 296–308. https://doi. org/10.1016/j.wavemoti.2011.11.003. https://doi.org/10.1016/j.ijleo.2013.01.081 https://doi.org/10.1016/j.ijleo.2014.01.013 https://doi.org/10.1016/j.ijleo.2014.01.013 https://doi.org/10.1016/j.amc.2010.06.030 https://doi.org/10.1007/s11071-018-4568-4 https://doi.org/10.1007/s11071-018-4568-4 https://doi.org/10.1016/j.ijleo.2019.01.057 https://doi.org/10.17654/DE021020227 https://doi.org/10.1016/j.amc.2008.10.038 https://doi.org/10.1016/j.ijleo.2014.04.014 https://doi.org/10.1016/j.ijleo.2014.04.014 https://doi.org/10.1016/j.jksus.2018.11.010 https://doi.org/10.1080/17455030.2017.1285449 https://doi.org/10.1108/HFF-06-2016-0252 https://doi.org/10.1016/j.ijleo.2019.02.016 https://doi.org/10.1016/j.ijleo.2019.02.016 https://doi.org/10.1016/j.rinp.2019.102472 https://doi.org/10.1016/j.ijleo.2018.03.104 https://doi.org/10.1016/j.ijleo.2018.03.104 https://doi.org/10.1016/j.ijleo.2018.07.072 https://doi.org/10.1016/j.wavemoti.2011.11.003 https://doi.org/10.1016/j.wavemoti.2011.11.003 1. Introduction 1.1. Governing model 2. Traveling wave hypothesis 3. Application the F-expansion method to the FLE 4. Jacobian-elliptic function solutions 5. Hyperbolic-function solutions 6. Conclusions Appendix A Appendix B References