Petviashvili Method for the Fractional Schrödinger Equation
In this paper, we extend the Petviashvili method (PM) to the fractional nonlinear Schrödinger equation (fNLSE) for the construction and analysis of its soliton solutions. We also investigate the temporal dynamics and stabilities of the soliton solutions of the fNLSE by implementing a spectral method...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/1/9 |
| _version_ | 1797442359761305600 |
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| author | Cihan Bayındır Sofi Farazande Azmi Ali Altintas Fatih Ozaydin |
| author_facet | Cihan Bayındır Sofi Farazande Azmi Ali Altintas Fatih Ozaydin |
| author_sort | Cihan Bayındır |
| collection | DOAJ |
| description | In this paper, we extend the Petviashvili method (PM) to the fractional nonlinear Schrödinger equation (fNLSE) for the construction and analysis of its soliton solutions. We also investigate the temporal dynamics and stabilities of the soliton solutions of the fNLSE by implementing a spectral method, in which the fractional-order spectral derivatives are computed using FFT (Fast Fourier Transform) routines, and the time integration is performed by a 4th order Runge–Kutta time-stepping algorithm. We discuss the effects of the order of the fractional derivative, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, on the properties, shapes, and temporal dynamics of the soliton solutions of the fNLSE. We also examine the interaction of those soliton solutions with zero, photorefractive and q-deformed Rosen–Morse potentials. We show that for all of these potentials, the soliton solutions of the fNLSE exhibit a splitting and spreading behavior, yet their dynamics can be altered by the different forms of the potentials and noise considered. |
| first_indexed | 2024-03-09T12:40:45Z |
| format | Article |
| id | doaj.art-4db5e8311db34a5ba6264baa9f6d5f9c |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T12:40:45Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-4db5e8311db34a5ba6264baa9f6d5f9c2023-11-30T22:18:54ZengMDPI AGFractal and Fractional2504-31102022-12-0171910.3390/fractalfract7010009Petviashvili Method for the Fractional Schrödinger EquationCihan Bayındır0Sofi Farazande1Azmi Ali Altintas2Fatih Ozaydin3Engineering Faculty, İstanbul Technical University, Sarıyer, İstanbul 34469, TurkeyLaboratory of Environmental Hydraulics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, SwitzerlandDepartment of Physics, Faculty of Science, İstanbul University, Vezneciler, İstanbul 34116, TurkeyEuropean Organization for Nuclear Research CERN, CH-1211 Geneva, SwitzerlandIn this paper, we extend the Petviashvili method (PM) to the fractional nonlinear Schrödinger equation (fNLSE) for the construction and analysis of its soliton solutions. We also investigate the temporal dynamics and stabilities of the soliton solutions of the fNLSE by implementing a spectral method, in which the fractional-order spectral derivatives are computed using FFT (Fast Fourier Transform) routines, and the time integration is performed by a 4th order Runge–Kutta time-stepping algorithm. We discuss the effects of the order of the fractional derivative, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, on the properties, shapes, and temporal dynamics of the soliton solutions of the fNLSE. We also examine the interaction of those soliton solutions with zero, photorefractive and q-deformed Rosen–Morse potentials. We show that for all of these potentials, the soliton solutions of the fNLSE exhibit a splitting and spreading behavior, yet their dynamics can be altered by the different forms of the potentials and noise considered.https://www.mdpi.com/2504-3110/7/1/9fractional nonlinear Schrödinger equationPetviashvili methodpotential functionsolitonsq-deformation |
| spellingShingle | Cihan Bayındır Sofi Farazande Azmi Ali Altintas Fatih Ozaydin Petviashvili Method for the Fractional Schrödinger Equation Fractal and Fractional fractional nonlinear Schrödinger equation Petviashvili method potential function solitons q-deformation |
| title | Petviashvili Method for the Fractional Schrödinger Equation |
| title_full | Petviashvili Method for the Fractional Schrödinger Equation |
| title_fullStr | Petviashvili Method for the Fractional Schrödinger Equation |
| title_full_unstemmed | Petviashvili Method for the Fractional Schrödinger Equation |
| title_short | Petviashvili Method for the Fractional Schrödinger Equation |
| title_sort | petviashvili method for the fractional schrodinger equation |
| topic | fractional nonlinear Schrödinger equation Petviashvili method potential function solitons q-deformation |
| url | https://www.mdpi.com/2504-3110/7/1/9 |
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