Soliton dynamics and stability of the time-fractional higher-order nonlinear Schrödinger equation: Analytical solutions and modulational instability analysis

Kanza Noor; Jamshad Ahmad; Kanza Noor; Jamshad Ahmad

doi:10.3934/math.2025981

AIMS Mathematics

2025, Volume 10, Issue 9: 22053-22074. doi: 10.3934/math.2025981

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Soliton dynamics and stability of the time-fractional higher-order nonlinear Schrödinger equation: Analytical solutions and modulational instability analysis

Kanza Noor ,
Jamshad Ahmad ^,

Department of Mathematics, Faculty of Science, University of Gujrat, Gujrat-50700, Pakistan

Received: 06 July 2025 Revised: 28 August 2025 Accepted: 05 September 2025 Published: 22 September 2025
MSC : 35Q55, 35B35, 35C08, 35A20, 34A34

The paper investigates the soliton dynamics and stability analysis of the time-fractional higher-order nonlinear Schrödinger equation. A Caputo time-fractional derivative is included in the time fractional higher-order nonlinear Schrödinger equation, along with dispersive higher-order and nonlinear terms, which allow us to describe wave propagation in arbitrarily complex nonlinear and dispersive media in greater detail. Through the use of the $ \phi^{6} $-model expansion method, a vast range of precise analytical soliton solutions is obtained, comprising nonlinear, regular, and singular periodic solitons. The effects of the fractional higher-order physical parameters on the amplitude, width, and nature of these solitons are systematically examined. Moreover, the modulational instability is studied through the use of linear stability analysis. Plots are given in order to explain the change in the form and stability behavior, and the evolution of the soliton solutions as the parameters vary.
- Schrödinger equation,
- $ \phi^{6} $-model expansion method,
- soliton solutions,
- stability analysis
Citation: Kanza Noor, Jamshad Ahmad. Soliton dynamics and stability of the time-fractional higher-order nonlinear Schrödinger equation: Analytical solutions and modulational instability analysis[J]. AIMS Mathematics, 2025, 10(9): 22053-22074. doi: 10.3934/math.2025981

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Abstract

The paper investigates the soliton dynamics and stability analysis of the time-fractional higher-order nonlinear Schrödinger equation. A Caputo time-fractional derivative is included in the time fractional higher-order nonlinear Schrödinger equation, along with dispersive higher-order and nonlinear terms, which allow us to describe wave propagation in arbitrarily complex nonlinear and dispersive media in greater detail. Through the use of the $ \phi^{6} $-model expansion method, a vast range of precise analytical soliton solutions is obtained, comprising nonlinear, regular, and singular periodic solitons. The effects of the fractional higher-order physical parameters on the amplitude, width, and nature of these solitons are systematically examined. Moreover, the modulational instability is studied through the use of linear stability analysis. Plots are given in order to explain the change in the form and stability behavior, and the evolution of the soliton solutions as the parameters vary.

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