Petviashvili Method for the Fractional Schrödinger Equation

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Bayındır C., Farazande S., ALTINTAŞ A. A., Ozaydin F.

Fractal and Fractional, vol.7, no.1, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 7 Issue: 1
  • Publication Date: 2023
  • Doi Number: 10.3390/fractalfract7010009
  • Journal Name: Fractal and Fractional
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Social Sciences Citation Index (SSCI), Scopus, INSPEC, Directory of Open Access Journals
  • Keywords: fractional nonlinear Schrodinger equation, Petviashvili method, potential function, solitons, q-deformation
  • Istanbul Technical University Affiliated: Yes

Abstract

© 2022 by the authors.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, (Formula presented.), 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.