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

Sheng Zhang,Bo Xu,Yuanyuan Wei

doi:10.1155/2019/7952871

Sheng Zhang, Bo Xu + Show 1 more

Open Access

https://doi.org/10.1155/2019/7952871

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Abstract

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.

Highlights

Since the increasing interest on fractional calculus and its applications, dynamical processes and dynamical systems of fractional orders have attracted much attention
We have derived and solved the local time-fractional Korteweg-de vries (tfKdV) eq (1) in the fractional framework of the spectral transform method. This is due to the linear spectral problem (3) equipped with the local time-fractional evolution (4)
Whether or not the spectral transform can be extended to some other nonlinear evolution equations with another type of fractional derivative depends on whether fractional derivative has the good properties required by the spectral transform method

Summary

Introduction

Since the increasing interest on fractional calculus and its applications, dynamical processes and dynamical systems of fractional orders have attracted much attention. With the close attentions of fractional calculus and its applications [27–53], some of the natural questions are whether the existing methods like those in [54–70] in soliton theory can be extended to nonlinear. This paper is motivated by the desire to extend the spectral transform to nonlinear fractional PDEs and gain more insights into the fractional soliton dynamics of the obtained solutions. For such a purpose, we consider the following local tfKdV equation: Dt(α)u + 6uux + uxxx = 0, 0 < α ≤ 1.

Derivation of the Local tfKdV Equation

Local Fractional Spectral Transform

Fractional Soliton Dynamics

Conclusion