Ablowitz Mark J, Been Joel B, Carr Lincoln D
Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA.
Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, Colorado 80401, USA.
Phys Rev Lett. 2022 May 6;128(18):184101. doi: 10.1103/PhysRevLett.128.184101.
Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg-deVries and nonlinear Schrödinger equations. These equations are shown to predict superdispersive transport of nondissipative solitons in fractional media.
非线性可积方程是非线性动力学的基础，而分数阶方程在反常扩散中广为人知。我们通过展示一类新的可积分数阶非线性演化方程的发现，将这两个领域联系起来，这类方程描述了分数阶介质中的色散输运。这些方程可以通过一个广泛通用的数学过程，利用完备性关系、色散关系和逆散射变换技术，从非线性可积方程构造出来。作为例子，这种通用方法被用于刻画两个与物理相关的、普遍存在的可积非线性方程的分数阶扩展：科特韦格 - 德弗里斯方程和非线性薛定谔方程。结果表明，这些方程能够预测分数阶介质中非耗散孤子的超色散输运。
