In this paper, we study the different types of new soliton solutions to the Chiral nonlinear Schrodinger equation with the aid of the functional variable method. Then, we get some special soliton solutions for Chiral nonlinear Schrodinger equation. The parameters of the soliton envelope are obtained as a function of the dependent model coefficients.Publisher's Versio

Neirameh, Ahmad

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. and Eng. Math. V.13, N.1, 2023, pp. 225-231FUNCTIONAL VARIABLE METHOD TO THE CHIRAL NONLINEARSCHRODINGER EQUATIONA. NEIRAMEH1, §Abstract. In this paper, we study the different types of new soliton solutions to theChiral nonlinear Schrodinger equation with the aid of the functional variable method.Then, we get some special soliton solutions for Chiral nonlinear Schrodinger equation.The parameters of the soliton envelope are obtained as a function of the dependent modelcoefficients.Keywords: Chiral nonlinear Schrodinger equation, Functional variable method, Soliton.solutionAMS Subject Classification: 35A09.1. IntroductionThe nonlinear equations are prevalently used as models to identify numerous physicaloccurrences and have a very serious role in many natural sciences such as mathematics,mechanics and other fields. Nonlinear evolution equations (NEEs) which describe manyphysical phenomena are often illustrated by nonlinear partial differential equations. So,the exact solutions of NLPDE are explored in detail in order to understand the physi-cal structure of natural phenomena that are described by such equations. Searching forexplicit, exact solutions of NLPDE by many different methods is the main goal of thisactive research area. Some of these methods, the Riccati Equation method [1], Hirota’sbilinear operators [2], Hirota’s dependent variable transformation [3], the Jacobi ellipticfunction expansion [4], the homogeneous balance method [5], the tanh-function expansion[6], first integral method [7,8], the sub-equation method [9], the exp-function method [10],the Backlund transformation, and similarity reduction [11-17] are used to obtain the exactsolutions of NLPDE.1 Department of Mathematics, Faculty of sciences, Gonbad Kavous University, Gonbad, Iran.e-mail:a.neirameh@gonbad.ac.ir; ORCID: https://orcid.org/0000-0002-7677-4657.§ Manuscript received: October 24, 2020; accepted: January 01, 2021.TWMS Journal of Applied and Engineering Mathematics, Vol.13, No.1 © Işık University, Departmentof Mathematics, 2023; all rights reserved..225226 TWMS J. APP. AND ENG. MATH. V.13, N.1, 20232. Analysis of the methodIn order, we describe the functional variable method. Consider a given NLPDE for u (x,t) in the formH (u, ux, ut, uxx, .....) = 0, (1)here H is a polynomial of its arguments. Using the transformationu(x, t) = U (ξ) , ξ = x± ct (2)here c is the wave speed, then we get ordinary differential equation (ODE) likeQ (P, Pξ, Pξξ, .....) = 0, (3)Now make a transformation in which the unknown function P is considered as a functionalvariable in the form Pξ = G (P )and another derivatives of P arePξξ =12(G2)′, (4)Pξξξ =12(G2)′′ √G2, (5)Pξξξξ =12[(G2)′′G2 +(G2)′′ (G2)′]. (6)Eq. (1) can be written with respect to P, G and its derivatives upon using the statementof (2) into (1) givesR(U,G′, G′′, G′′′, .....)= 0, (7)by integrating of Eq. (7), Eq. (7) can be written with respect to G, and it is found theappropriate solutions by using Eq. (3) for the investigated problem.3. FVM to the Chiral nonlinear Schrödinger equationThe Chiral nonlinear Schrödinger equation is given by (see [1-3])i∂Ψ∂t+ ε(∂2Ψ∂x2+∂2Ψ∂x2) + i(c1(Ψ∂Ψ∗∂x−Ψ∗∂Ψ∂x) + c2(Ψ∂Ψ∗∂y−Ψ∗∂Ψ∂y))Ψ = 0, (8)here Ψ is the complex function of x, y and t, ε is the coefficient of the dispersion terms.Also, c1 and c2 are the coefficients of nonlinear coupling terms.In this case for solving Eq. (8), it is assumed that the soliton solution to Eq. (8) is givenbyΨ(x, y, t) = eiφ(x,y,t)P (x, y, t), (9)where P (x, y, t)is the amplitude portion of the soliton, while the phase portion of thesoliton is given byφ(x, y, t) = κ1x+ κ2y + ωt+ θ. (10)by using of conformable fractional derivatives [4-17]. Here in Eq. (10), κ1 and κ2 arethe frequencies in the x-and y-directions, ω is the soliton frequency while θ is the phaseconstant. Thus, from Eqs. (9) and (10),i∂Ψ∂t=(i∂P∂t− ωP)eiφ, (11)∂2Ψ∂x2=(∂2P∂x2+ 2iκ1∂P∂x− κ21P)eiφ, (12)∂2Ψ∂y2=(∂2P∂y2+ 2iκ2∂P∂y− κ22P)eiφ, (13)A. NEIRAMEH: FUNCTIONAL VARIABLE METHOD TO THE CHIRAL NONLINEAR ... 227(Ψ∂Ψ∗∂x−Ψ∗∂Ψ∂x)Ψ = −2iκ1P 3eiφ. (14)and(Ψ∂Ψ∗∂y−Ψ∗∂Ψ∂y)Ψ = −2iκ2P 3eiφ. (15)By substituting Eqs. (11)–(15) into Eq. (8) and decomposing into real and imaginaryparts yields, respectively,ωP − ε{(∂2P∂x2− κ21P)+(∂2P∂y2− κ22P)}− 2(κ1c1 + κ2c2)P 3 = 0, (16)∂P∂t+ 2a(κ1∂P∂x+ κ2∂P∂y) = 0. (17)This pair of equations will be analyzed further depending on the type of soliton solutionwhich is fetched. Under the traveling wave transformationP (x, y, t) = U(ξ), ξ = B1x+B2y − υt (18)we haveε(B21 +B22) d2Udξ2−(ε(κ21 + κ22) + ω)U + (2(κ1c1 + κ2c2))U3 = 0, (19)(−υ + 2ε(κ1B1 + κ2B2))dUdξ= 0. (20)from Eq. (20), we getυ = 2ε(κ1B1 + κ2B2).ε(B21 +B22) d2Udξ2−(ε(κ21 + κ22) + ω)U + (2(κ1c1 + κ2c2))U3 = 0, (21)Substituting (4) into Eq. (21) leads to the following equationG (P ) = P ′ =√(ε(κ21 + κ22) + ω)ε(B21 +B22) P 2 − (κ1c1 + κ2c2)ε(B21 +B22) P 4 + h0 (22)here h0 is an integration constant. We have solutions of Eq. (22) as following:Case 1: For h0 = 0,(ε(κ21+κ22)+ω)ε(B21+B22)< 0, and (κ1c1+κ2c2)ε(B21+B22)> 0, we know that Eq. (21) havetriangular soliton solutions as followingP (x, t) = ±√ε(κ21 + κ22) + ω(κ1c1 + κ2c2)sec[√−(ε(κ21 + κ22) + ω)ε(B21 +B22) (B1x+B2y − υt)]AndP (x, t) = ±√ε(κ21 + κ22) + ω(κ1c1 + κ2c2)csc[√−(ε(κ21 + κ22) + ω)ε(B21 +B22) (B1x+B2y − υt)]Case 2: For h0 = 0,(ε(κ21+κ22)+ω)ε(B21+B22)> 0, and (κ1c1+κ2c2)ε(B21+B22)< 0, we know that Eq. (21) havetriangular soliton solutions as followingP (x, t) = ±√ε(κ21 + κ22) + ω(κ1c1 + κ2c2)sech[√(ε(κ21 + κ22) + ω)ε(B21 +B22) (B1x+B2y − υt)]228 TWMS J. APP. AND ENG. MATH. V.13, N.1, 2023AndP (x, t) = ±√ε(κ21 + κ22) + ω(κ1c1 + κ2c2)csch[√(ε(κ21 + κ22) + ω)ε(B21 +B22) (B1x+B2y − υt)]Case 3: For h0 6= 0,(ε(κ21+κ22)+ω)ε(B21+B22)< 0, and (κ1c1+κ2c2)ε(B21+B22)> 0, we know that Eq. (21) havetriangular soliton solutions as followingP (x, t) = ±√ε(κ21 + κ22) + ω2(κ1c1 + κ2c2)tanh[√−(ε(κ21 + κ22) + ω)2ε(B21 +B22) (B1x+B2y − υt)]AndP (x, t) = ±√ε(κ21 + κ22) + ω2(κ1c1 + κ2c2)coth[√−(ε(κ21 + κ22) + ω)2ε(B21 +B22) (B1x+B2y − υt)]Case 4: For h0 6= 0,(ε(κ21+κ22)+ω)ε(B21+B22)> 0, and (κ1c1+κ2c2)ε(B21+B22)> 0we know that Eq. (21) havetriangular soliton solutions as followingP (x, t) = ±√−ε(κ21 + κ22) + ω2(κ1c1 + κ2c2)tan[√(ε(κ21 + κ22) + ω)2ε(B21 +B22) (B1x+B2y − υt)]AndP (x, t) = ±√−ε(κ21 + κ22) + ω2(κ1c1 + κ2c2)cot[√(ε(κ21 + κ22) + ω)2ε(B21 +B22) (B1x+B2y − υt)]4. Graphical behavior:The graphical behavior of the solutions for different values are represented below in thefollowing figures by using computation software Maple (see Figs. 1-3).A. NEIRAMEH: FUNCTIONAL VARIABLE METHOD TO THE CHIRAL NONLINEAR ... 229Figure 1:Figure 2:230 TWMS J. APP. AND ENG. MATH. V.13, N.1, 2023Figure 3:5. ConclusionsThis paper derived new exact soliton solutions of nonlinear Schrodinger equations,namely,the Chiral nonlinear Schrodinger equation which describe the propagation of ultrashort pulses in nonlinear optical fibers by using the Functional variable method. We boldlysay that the work here is valuable and may be beneficial for studying in other nonlinearscience. The exact solutions obtained from the model equations provide important insightinto the dynamics of solitary waves. The solutions obtained in this paper have not beenreported in the old research.References[1] Neirameh, A. and Eslami, M., (2019), New travelling wave solutions for plasma model of extendedK–dV equation, Afrika Matematika 30 (1-2), 335-344[2] Hirota, R., (1973), Exact N-soliton solutions of the wave equation of long waves in shallow-water andin nonlinear lattice, J. Math. Phys., 14 , 810-814.[3] Lei, Y., Fajiang, Y. and Yinghai, W., (2002), The homogeneous balance method, Lax pair, Hirotatransformation and a general fifth-order KdV equation, Chaos Soliton and Fractals, 13,337-340.[4] Liu, S. K., Fu, Z., Liu, S. and Zhao, Q., (2001), Jacobi elliptic function expansion method and periodicwave solutions of nonlinear wave equations, Phys. Lett. A, 289 , 69-74.[5] Wang, M. L., (1995), Solitary wave solutions for the variant Boussinesq equations, Phys. Lett. A, 199, 169-172.12[6] Parkes, E. J. and Duffy, B. R., (1996), An automated tanh-function method for finding solitary wavesolutions to non-linear evolution equations, Comput. Phys. Commun., 98 , 288-300.[7] He, Y., Li, S. and Long, Y., (2013), Exact solutions of the modified Benjamin-Bona-Mahoney (mBBM)equation by using the first integral method, Differential Equations and Dynamical Systems, 21 , 199-204.A. NEIRAMEH: FUNCTIONAL VARIABLE METHOD TO THE CHIRAL NONLINEAR ... 231[8] Hosseini, K. and Gholamin, P., (2015), Feng’s first integral method for analytic treatment of two higherdimensional nonlinear partial differential equations, Differential Equations and Dynamical Systems, 23, 317-325.[9] Zheng, B., (2011), A new Bernoulli sub-ODE method for constructing traveling wave solutions for twononlinear equations with any order, U.P.B. Sci. Bull., Series A, 73 , 85-94.[10] He, J. H., (2008),Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys., 22 ,3487–3578.[11] Eslami, M. and Neirameh, A., (2014), New solitary and double periodic wave solutions for a generalizedsinh-Gordon equation, Eur. Phys. J. Plus 129, 54.[12] Neirameh, A. and Eslami, M., (2017), An analytical method for finding exact solitary wave solutionsofthe coupled (2 1)-dimensional Painlevé Burgers equation, Scientia Iranica B , 24(2), 715-726.[13] Rezazadeh, H., Kumar, D., Neirameh, A., M. Eslami. and M. Mirzazadeh., (2019), Applications ofthree methods for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model withKerr law nonlinearity, Pramana 94(1), 39.[14] Rezazadeh, H., Neirameh, A., Eslami, M., Bekir, A. and A Korkmaz., (2019), A sub-equation methodfor solving the cubic–quartic NLSE with the Kerr law nonlinearity, Modern Physics Letters B 33 (18),195-197.[15] Rezazadeh, H., Neirameh, A., Raza, N., and Eslami, M., (2019), Exact Solutions of Nonlinear DiffusivePredator-Prey System by New Extension of Tanh Method, Journal of Computational and TheoreticalNanoscience 15, 3195-3200.[16] Fan, E. G., (2002), Auto-Backlund transformation and similarity reductions for general variable coef-ficient KdV equations, Phys. Lett. A, 294, 26-30.Ahmad Neirameh received his M.Sc. and Ph.D. in Mathematics from GuilanUniversity, 2006 and 2012, respectively. He has been a faculty member of Mathematicsin Gonbad university, Iran since 2013. His research interests focus mainly on PDEs.

Functional variable method to the Chiral nonlinear Schrodinger equation

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Functional variable method to the Chiral nonlinear Schrodinger equation

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