Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example

In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with...

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Bibliographic Details
Published in:Complexity (New York, N.Y.) N.Y.), 2019, Vol.2019 (2019), p.1-9
Main Authors: Zhang, Sheng, Xu, Bo, Wei, Yuanyuan
Format: Article
Language:English
Subjects:
Applied mathematics
Calculus
Exact solutions
Fourier transforms
Fractals
Integral equations
Investigations
Mathematical analysis
Nonlinear systems
Partial differential equations
Physics
Solitary waves
Spectra
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Summary:In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with local time-fractional derivative. Based on the spectral problem with the equipped local time-fractional derivative, the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transform. As a result, a formula of exact solution with Mittag-Leffler functions is obtained. Finally, in the case of reflectionless potential the obtained exact solution is reduced to fractional n-soliton solution. In order to gain more insights into the fractional n-soliton dynamics, the dynamical evolutions of the reduced fractional one-, two-, and three-soliton solutions are simulated. It is shown that the velocities of the reduced fractional one-, two-, and three-soliton solutions change with the fractional order.
ISSN:1076-2787
1099-0526
DOI:10.1155/2019/7952871