Localized modes in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="bold-script">PT</mml:mi></mml:math>-symmetric sine-Gordon couplers with phase shift

Khan, Wajahat Ali; Ali, Amir; Gul, Zamin; Ahmad, Saeed; Ullah, Arif

doi:10.1016/j.chaos.2020.110290

Cited by 7 publications

(2 citation statements)

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“…As future work, it will be interesting to investigate the hybrid planar waveguide arrays by a variety of fractional coupled sine-Gordon equations with different phase shifts reported in [38] for the integer order. The consideration of such systems is very useful to investigate different physical phenomena for the applications of the parity time symmetry in optics, Bose-Einstein condensates, and nonlinear physical phenomena, where the coupling of nonlinearity fundamentally advances the problem and generates completely novel characteristics.…”

Section: Discussionmentioning

confidence: 99%

Ulam Stability of Fractional Hybrid Sequential Integro-Differential Equations with Existence and Uniqueness Theory

Algahtani

2022

Symmetry

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In this paper, a variety of boundary value problems (BVPs) known as hybrid fractional sequential integro-differential equations (HFSIDs) with two point orders (p,q) are investigated. The uniqueness and existence of the solution are discussed via Banach fixed-point theorems. Certain particular theorems associated with Hyers–Ulam and Hyers–Ulam–Rassias stability to the solution, as well as the uniqueness and existence of the solution of the BVPs are studied. The results are illustrated with some particular examples, and the numerical data are analyzed for confirmation of the results. The results obtained in this work are simple and can easily be applicable to physical systems. Furthermore, symmetry analysis of fractional differential equations and HFSIDs are also presented. This is due to the fact that the aforementioned analysis plays a significant role in both the optimization and qualitative theory of fractional differential equations.

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Section: Discussionmentioning

confidence: 99%

Ulam Stability of Fractional Hybrid Sequential Integro-Differential Equations with Existence and Uniqueness Theory

Algahtani

2022

Symmetry

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“…By considering u≃ sin u, the considered system becomes the Sine-Gordon equation. In future work, it will be interesting to investigate the Sine-Gordon model with nonlinear AC/DC drives with different fractional operators to study the solitonic behavior, localized modes in single and in stacked long Josephson junctions with a variety of potentials, parity time symmetry, the nonlinearity/dispersion effects, and evolution of the localized monotonic shocks [67][68][69][70][71][72][73].…”

Section: Discussionmentioning

confidence: 99%

Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions

Saifullah

Ali

Irfan

et al. 2021

Mathematical Problems in Engineering

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In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient α significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.

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