Exact solutions and conservation laws for the time-fractional nonlinear dirac system: A study of classical and nonclassical lie symmetries

Farzaneh Alizadeh; Samad Kheybari; Kamyar Hosseini; Farzaneh Alizadeh; Samad Kheybari; Kamyar Hosseini

doi:10.3934/math.2025532

AIMS Mathematics

2025, Volume 10, Issue 5: 11757-11782. doi: 10.3934/math.2025532

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Exact solutions and conservation laws for the time-fractional nonlinear dirac system: A study of classical and nonclassical lie symmetries

1.
Department of Mathematics, Near East University TRNC, Mersin10, Nicosia 99138, Turkey
2.
Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey
3.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
4.
Faculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, Turkey
5.
Faculty of Engineering and Natural Sciences, Istanbul Okan University, Istanbul, Turkey

Received: 31 March 2025 Revised: 08 May 2025 Accepted: 13 May 2025 Published: 21 May 2025
MSC : 35R11, 70H33, 76M60

Time-fractional Dirac-type systems arise in quantum field theory, plasma physics, and condensed matter systems where fractional calculus captures nonlocal interactions. In this study, we employ classical and nonclassical Lie symmetry methods to analyze the underlying symmetry structure of the system. By deriving infinitesimal generators and performing similarity reductions, we transform the governing fractional partial differential equations (FPDEs) into fractional ordinary differential equations (FODEs). Exact solutions are constructed using the power series method. Furthermore, we establish conservation laws in the fractional setting, ensuring the physical consistency of the system. Our findings offer new insights into the interplay among symmetry, conservation principles, and exact solutions in fractional quantum field models, expanding the analytical toolkit for studying nonlinear relativistic wave equations.
- classical Lie symmetries,
- nonclassical lie symmetries,
- time-fractional nonlinear Dirac system,
- exact solutions,
- conservation laws
Citation: Farzaneh Alizadeh, Samad Kheybari, Kamyar Hosseini. Exact solutions and conservation laws for the time-fractional nonlinear dirac system: A study of classical and nonclassical lie symmetries[J]. AIMS Mathematics, 2025, 10(5): 11757-11782. doi: 10.3934/math.2025532

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Abstract

Time-fractional Dirac-type systems arise in quantum field theory, plasma physics, and condensed matter systems where fractional calculus captures nonlocal interactions. In this study, we employ classical and nonclassical Lie symmetry methods to analyze the underlying symmetry structure of the system. By deriving infinitesimal generators and performing similarity reductions, we transform the governing fractional partial differential equations (FPDEs) into fractional ordinary differential equations (FODEs). Exact solutions are constructed using the power series method. Furthermore, we establish conservation laws in the fractional setting, ensuring the physical consistency of the system. Our findings offer new insights into the interplay among symmetry, conservation principles, and exact solutions in fractional quantum field models, expanding the analytical toolkit for studying nonlinear relativistic wave equations.

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