Study of Optical Soliton Perturbation with Quadratic-Cubic Nonlinearity by Lie Symmetry and Group Invariance

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1 ISSN X, Physics of Wave Phenomena, 018, Vol. 6, No. 4, pp c Allerton Press, Inc., 018. THEORY OF NONLINEAR WAVES Study of Optical Soliton Perturbation with Quadratic-Cubic Nonlinearity by Lie Symmetry Group Invariance M. Asma 1*, A.Bansal **, W.A.M.Othman 1***, B.R.Wong 1**** 3, 4*****, A.Biswas 1 Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur, Malaysia Department of Mathematics, D.A.V. College for Women, Ferozepur, India 3 Department of Physics, Chemistry Mathematics, Alabama A&M University, Normal, AL-3576 USA 4 Department of Mathematics Statistics, Tshwane University of Technology, Pretoria, 0008 South Africa Received September 4, 018 Abstract Here, optical soliton perturbation with quadratic-cubic nonlinearity has been discussed by applying Lie symmetry group invariants. The perturbation terms include third fourth order dispersions in addition to self-steepening, intermodal dispersion higher order dispersion effects. Using presented algorithms, Bright dark soliton solutions are revealed. DOI: /S X INTRODUCTION In the field of mathematical physics, optical soliton perturbation has attracted researchers due to predominant applications to telecommunications industry. While several forms of non-kerr laws of nonlinearity are visible studied across the globe, this paper will study a particular form of such law. It is the quadratic-cubic (QC) nonlinearity that first appeared in the literature a few years ago. This form of optical fibers gained popularity ever since several papers were published with such nonlinearity with the inclusion of Hamiltonian type perturbation terms 1 4. Besides these popular integration schemes, there is a powerful analytical approach that is centuries old is yet popular still flourishing 5 8. This is the Lie symmetry analysis. Several forms of nonlinear evolution equations have been fruitfully addressed using this method during the past This paper will address perturbed nonlinear Schrödinger s equation (NLSE) with QC nonlinearity using this technique. The perturbation terms include third order dispersion (3OD) fourth order dispersion (4OD) as well as self-steepening effect along with intermodal dispersion nonlinear dispersion. The nonlinear * miraszaka@gmail.com ** anupma51@yahoo.co.in *** wanainun@um.edu.my **** bernardr@um.edu.my ***** biswas.anjan@gmail.com perturbative effects due to self-steepening nonlinear dispersion appear with full nonlinearity in order to maintain the model on a generalized flavor. The detailed analysis for the extraction of bright dark solitons are explored Governing Model Governing model is perturbed NLSE with QC nonlinearity, in dimensionless form is given as 1 3 ( iq t + aq xx + b 1 q + b q ) q = i αq x γq xxx iσq xxxx + λ ( q m q ) x + θ( q m) x q, (1) where x t are spatial temporal variables; q(x, t) the complex-valued wave profile; a, b 1,b represent the group velocity dispersion, quadratic cubic nonlinearities, respectively; α the intermodal dispersion; γ σ are 3OD 4OD, respectively; θ m>0denote the nonlinear dispersion nonlinearity parameter, respectively. To avoid the formation of shock waves, the coefficient λ due to selfsteepening is included... LIE GROUP ANALYSIS The concept of continuous symmetry is formalized by Lie groups, thus, become a part of the foundation of mathematics. The main motivation for investigating symmetries of differential equations is 31

2 STUDY OF OPTICAL SOLITON PERTURBATION 313 mapping of known solutions to other solutions, construction of invariant solutions, detection of linearizing transformation, etc. The key idea of Lie theory of symmetry analysis of differential equations relies on the invariance under a transformation of independent dependent variables. In this section, we will perform Lie symmetry analysis 4 10 for Eq. (1). Consider q(x, t) =u(x, t)+iv(x, t). () From (1) after using (), the real imaginary parts are v t + au xx + b 1 u u + v + b u(u + v )= αv x + γv xxx + σu xxxx m(λ + θ)(u + v ) m 1 (uvu x +v v x ) λ(u + v ) m v x, (3) u t + av xx + b 1 v u + v + b v(u + v )=αu x γu xxx + σv xxxx + m(λ + θ)(u + v ) m 1 (u u x +uvv x )+λ(u + v ) m u x. (4) One-parameter Lie group of transformations: u = u + ɛη(x, t, u, v), x = x + ɛξ(x, t, u, v), v = v + ɛφ(x, t, u, v), t = t + ɛτ(x, t, u, v), (5) where ɛ is an infinitesimal parameter. The associated vector field with the above group of transformations can be written as V = ξ(x, t, u, v) x + τ(x, t, u, v) + η(x, t, u, v) t u + φ(x, t, u, v) v. (6) Applying the second prolongation pr () V of V to Eqs. (3) (4), we find that the coefficient functions ξ,τ,η,φ must satisfy the invariance condition φ t + aη xx + b 1 η u + v u(uη + vφ) + + b (3u η +uvφ + ηv ) u + v = λ (u + v ) m φ x λ mv x (u + v ) m 1 (uη + vφ) m(λ + θ) (u + v ) m 1 (uvη x +uu x φ +u x vη +v φ x +4vv x φ) m(m 1)(λ + θ)(u + v ) m (uη + vφ)(uvu x +v v x )+γφ xxx + ση xxxx αφ x, (7) η t + aφ xx + b 1 (φ u + v v(uη + vφ) + + b (3v φ +uvη + φu ) u + v = λ (u + v ) m η x +mλu x (u + v ) m 1 (uη + vφ)+m(λ + θ)(u + v ) m 1 (u η x +4uu x η +uvφ x +uφv x +vv x η)+m(m 1)(λ + θ)(u + v ) m (uη + vφ)(u u x +uvv x ) γη xxx + σφ xxxx + αη x, (8) where η t, φ t, η x, φ x, η xx, φ xx, η xxx, φ xxx, η xxxx, φ xxxx are extended (prolonged) infinitesimals acting on an enlarged space that includes all derivatives of the dependent variables u t, v t, u x, v x, u xx, v xx, u xxx, v xxx, u xxxx, v xxxx. From (7) (8), by replacing differential coefficients to zero, the infinitesimals are determined. The following obtained differential equations in η, φ, ξ, τ that need to be satisfied: PHYSICS OF WAVE PHENOMENA Vol. 6 No

3 314 ASMA et al. (i) ξ u =0, ξ v =0, (ii) τ x =0, τ u =0, τ v =0, (iii) φ u =0, φ vv =0, (iv) η x =0, η v =0, η uu =0, (v) η u +4ξ x =0, (vi) 3γφ xv +3γξ xx =0, (vii) τ t φ v =0, (viii) φ v + ξ x =0, (ix) v λξ x v ξ x θ + v φ v λ + v φ v θ + λuη +3λvφ +θvφ, =0, (x) 4m(m 1)(λ + θ)v (u + v ) m (uη + vφ) =0, (xi) αφ v + γξ xxx αξ x + ξ t =0, (xii) 4m(m 1)(λ + θ)uv(u + v ) m (uη + vφ) =0, (xiii) m(λ + θ)(u + v ) m 1 (uvξ x uvη u uφ vη) =0, (xiv) σξ xxxx aξ xx =0, (xv) φ v +3γξ x =0, b 1 uvφ (xvi) u + v + b 1u η u + v +b uvφ φ t + b 1 η u + v + b ηv +3b u η =0. (9) The general solution of this large system provides following forms for the infinitesimal elements η, φ, ξ, τ: ξ = C 1, τ = C, η =0, φ =0, (10) where C 1, C are arbitrary constants. Obtained symmetries reduce by applying characteristic equations: dx ξ = dt τ = du η = dv φ. (11) For (1), solving (11), after applying symmetries, we get ζ = x wt, q(x, t) =F (ζ) exp ig(ζ). (1) 3. REDUCED ODEs AND SOLITON SOLUTIONS TO NLSE The suitable similarities obtained in (1) reduce (1) to the following ordinary differential equations: wf(ζ) G (ζ)+af (ζ) af (ζ) G (ζ) + b1 F (ζ) + b F (ζ) 3 = λf (ζ) m+1 G (ζ)+3γf (ζ) G (ζ) +3γF (ζ) G (ζ)+γf(ζ) G (ζ) γf(ζ) G (ζ) 3 αf (ζ) G (ζ)+σ F (ζ) 6F (ζ) G (ζ) } + σ 1F (ζ) G (ζ) G (ζ) 3F (ζ) G (ζ) 4F (ζ) G (ζ) G (ζ)+f(ζ) G (ζ) } 4 (13) wf (ζ)+af (ζ) G (ζ)+af (ζ) G (ζ) =λ(m +1)F (ζ) m F (ζ)+mθf (ζ) m F (ζ) + αf (ζ) γf (ζ)+3γf (ζ) G (ζ) +3γF(ζ) G (ζ) G (ζ)+σ 4F (ζ) G (ζ)+6f (ζ) G (ζ) + σ 4F (ζ) G (ζ)+f(ζ)g (ζ) 4F (ζ) G (ζ) 3 6F (ζ) G (ζ) } G (ζ). (14) Now, Eqs. (13) (14) will be used to retrieve soliton-like solutions of NLSE in the following subsections. PHYSICS OF WAVE PHENOMENA Vol. 6 No

4 STUDY OF OPTICAL SOLITON PERTURBATION Bright Soliton Solution to NLSE By taking the following forms of F (ζ) G(ζ), we get bright soliton solutions to NLSE (1): F (ζ) =A sech(ζ) p, G(ζ) =Bζ, (15) where A, B are the amplitude width of the soliton. Balancing principle is applied to find p. By equating the exponents (m+1)p p+ using (15) in (13), (14), we get p = 1/m. Also the solution of (1) becomes q(x, t) =A sech t(b γ α + γ) x exp ib t(b γ α + γ)+x }, (16) where a = Bγ, b 1 =0,b = B(A λ +γ) A, σ =0,θ = 3(A λ +γ) A. 3.. Dark Solitons Other Solutions On substituting G(ζ) =Bζ in (13), (14) equating the coefficients equal to zero of linearly independent functions, we have σ = γ 4B, λ = θ 3, α = B γ +ab w, (17) Then reduced ODEs (13) (14) provides the following solutions to NLSE for m =1: q(x, t) = tanh 1 } B( wt + x)+c 1 C 4 C 4 exp ib( wt + x), (18) with a = 3γB, b 1 = 15B3 γ,b = B(45B γ +16C4 θ) 8C 4 4C4 ; with b = θb 3 ; with b = θb 3 q(x, t)= q(x, t) = 1 B (5γB 4a) exp i( wt + x)b, (19) 4 b 1 35 tanh (i/6)( wt + x)b + C1 } 4γB 3 16b 1 35 tanh (i/6)( wt + x)b + C 1 } γb 3 108b 1 a = 41γB 36 ; + 35γB3 exp i( wt + x)b, (0) 16b 1 q(x, t) = tanh 1 } 10B( tw + x)+c1 C 4 + C 3 exp ib( tw + x), (1) 0 with a = 5γB 4, b 1 = 3B3 γ,b = B(3B γ C3 θ) 800C 3 400C3 ; q(x, t) =C 4 tanh 1 B( wt + x)+c1 exp ib( wt + x), () with a = C 4 (θb 3b ) 3B, b 1 =0,γ =0. PHYSICS OF WAVE PHENOMENA Vol. 6 No

5 316 ASMA et al. 4. CONCLUSIONS The perturbed NLSE with QC nonlinearity was studied in this paper with Lie symmetry group invariants. Bright dark soliton solutions are retrieved. The perturbation terms included 3OD 4OD. It must be noted that traveling wave hypothesis the method of undetermined coefficients fail to retrieve soliton solutions to the perturbed NLSE with these two perturbation terms included. Thus, this integration scheme has a true edge over the other two. Although semi-inverse variational principle retrieved bright soliton solution to the model, it must be noted that this method, although analytical, does not yield an exact soliton solution 1. Thus Lie symmetry analysis still sts strong even today. The results of this paper pave way to further research activities in future. REFERENCES 1. M. Asma, W.A.M. Othman, B.R. Wong, A. Biswas, Cubic Nonlinearity by Semi-Inverse Variational Principle, Proc. Roman. Acad. Ser. A. 18(4), 331 (017).. M. Asma, W.A.M. Othman, B.R. Wong, A. Biswas, Cubic Nonlinearity by the Method of Undetermined Coefficients, J. Opt. Adv. Mater. 19(11 1), 699 (017). 3. M. Asma, W.A.M. Othman, B.R. Wong, A. Biswas, Cubic Nonlinearity by Traveling Wave Hypothesis, Opt. Adv. Mater. Rapid Commun. 11(9 10), 517- (017). 4. Hai-Qin Jin, Self-Similar Asymptotic Optical Waves in Quintic Nonlinear Media with Distributed Coefficients, Brazil. J. Phys. 45(4), 439 (015). 5. P.J. Olver, Applications of Lie Groups to Differential Equations (Springer Verlag, N.Y., 1993), Vol G.W. Bluman J.D. Cole, Similarity Methods for Differential Equations (Springer Verlag, N.Y., 1974). 7. L.V. Ovsiannikov, Group Analysis of Differential Equations (Academic Press, N.Y., 198). 8. S. Lie, On Integration of a Class of Linear Partial Differential Equations by Means of Definite Integrals, Archiv der Mathematik. 6,38 (1881). 9. R.K. Gupta A. Bansal. Similarity Reductions Exact Solutions of Generalized Bretherton Equation with Time-Dependent Coefficients, Nonlin. Dynam. 71, 1 (013). 10. A. Bansal R.K. Gupta, Lie Point Symmetries Similarity Solutions of the Time-Dependent Coefficients Calogero Degasperis Equation, Phys. Scripta. 86, (01). 11. A. Bansal R.K. Gupta, Painlevé Analysis, Lie Symmetries Invariant Solutions of Potential Kadomstev Petviashvili Equation with Time-Dependent Coefficients, Appl. Math. Comp. 19, 590 (013). PHYSICS OF WAVE PHENOMENA Vol. 6 No