New optical solitons of space-time conformable fractional perturbed Gerdjikov-Ivanov equation by sine-Gordon equation method

Yaşar, Elif; Yıldırım, Yakup

doi:10.1016/j.rinp.2018.04.058

Cited by 79 publications

(11 citation statements)

References 14 publications

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“…The governing NLSE shows up in distinctive fields, including fluid dynamics, nonlinear optics and plasma physics. A lot of work has been done to find soliton solutions for various forms of NLSEs [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Eslami and Neirameh studied the exact soliton solutions for higher order NLSE [13].…”

Section: Introductionmentioning

confidence: 99%

Stationary Solutions for Nonlinear Schrödinger Equations by Lie Group Analysis

et al. 2019

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Section: Introductionmentioning

confidence: 99%

Stationary Solutions for Nonlinear Schrödinger Equations by Lie Group Analysis

et al. 2019

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“…The solitary wave solutions of the complex perturbed GIE were investigated by employing two methods (Khater 2021). Exact solutions of the space-time conformable fractional perturbed GIE were derived in Yaşar et al (2018). The GIE was decomposed into two systems of solvable ordinary differential equations and the solutions were derived in terms of the Riemann theta functions (Dai and Fan 2004).…”

Section: Introductionmentioning

confidence: 99%

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Abdel‐Gawad

2023

Opt Quant Electron

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The Gerdjikov–Ivanov equation (GIE) occupied a remarkable area of research in the literature. In the present work, a modified GIE (MGIE) is considered which is new and was not studied in the literature. Also, the modified-unified method (MUM) is used to obtain approximate analytic solutions (AASs) of MGIE. Up to our knowledge, no AASs for non-integrable complex field equation were found up to now. Thus the AASs found, here, are novel. The UM addresses finding the exact solutions to integrable equations. In this sense as no exact solution for MGIE exists, consequently, it is not integrable. So, here, approximate analytic optical soliton solutions are invoked. The UM stands for expressing the solution of nonlinear evolution equations in polynomial and rational forms in an auxiliary function (AF) with an appropriate auxiliary equation. For finding exact solutions by the UM, the coefficients of the AF, with all powers, are set equal to zero, For a non-integrable equation, only approximate solutions are affordable. In this case, we are led to utilizing the MUM. Herein, non-zero coefficients (residue terms (RTs)) are considered as errors, which are space and time-independent. It is worth mentioning that, this is in contrast to the errors found by the different numerical methods, where they are space and time-dependent. Further, in the present case, the maximum error is controlled via an adequate choice of the parameters in the RTs. These solutions are displayed in graphs. Breather soliton, chirped soliton and M-shape soliton, among others, are observed. Furthermore, modulation instability (MI) is studied and it is found MI triggers when the coefficient of the nonlinear dispersion exceeds a critical value.

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“…However, the derivation of the exact solution of FDEs is not an easy task since the properties of a fractional derivative are harder than the classical derivative. Recently, several research groups have been developed to derive the exact and numerical solutions of FDEs such as invariant subspace method [11,21,39,[51][52][53]57,58,71], variational iteration method [44], homotopy perturbation method [45], operational matrix method [55], collocation method [40] and so on [14,23,68,69]. Among those methods, the Lie symmetry analysis method is an algorithmic approach that provides an efficient tool to construct an exact solution of FDEs in a systematic way.…”

Section: Introductionmentioning

confidence: 99%

On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations

Sethukumarasamy¹,

Vijayaraju²,

Prakash³

2021

JNMP

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In this article, we explain how to extend the Lie symmetry analysis method for n-coupled system of fractional ordinary differential equations in the sense of Riemann-Liouville fractional derivative. Also, we systematically investigated how to derive Lie point symmetries of scalar and coupled fractional ordinary differential equations namely (i) fractional Thomas-Fermi equation, (ii) Bagley-Torvik equation, (iii) two-coupled system of fractional quartic oscillator, (iv) fractional type coupled equation of motion and (v) fractional Lotka-Volterra ABC system.The dimensions of the symmetry algebras for the Bagley-Torvik equation and its various cases are greater than 2 and for this reason we construct optimal system of one-dimensional subalgebras. In addition, the exact solutions of the above mentioned fractional ordinary differential equations are explicitly derived wherever possible using the obtained symmetries.

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New optical solitons of space-time conformable fractional perturbed Gerdjikov-Ivanov equation by sine-Gordon equation method

Cited by 79 publications

References 14 publications

Stationary Solutions for Nonlinear Schrödinger Equations by Lie Group Analysis

Stationary Solutions for Nonlinear Schrödinger Equations by Lie Group Analysis

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations

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