Fractional Integrable Nonlinear Soliton Equations

Abstract

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-de Vries and nonlinear Schrödinger equations. These equations are shown to predict super-dispersive transport of non-dissipative solitons in fractional media.

References in corpus (4)

• Spatiotemporal accessible solitons in fractional dimensions
• Propagation dynamics of the circular Airy Gaussian vortex beams in the fractional nonlinear Schrödinger equation
• Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities
• Bubbles and W-shaped solitons in Kerr media with fractional diffraction

Cited by in corpus (4)

• Integrable Fractional Modified Korteweg-de Vries, Sine-Gordon, and Sinh-Gordon Equations
• Dynamics of fractional N-soliton solutions with anomalous dispersions of integrable fractional higher-order nonlinear Schrödinger equations
• Interactions of fractional N-solitons with anomalous dispersions for the integrable combined fractional higher-order mKdV hierarchy
• Nondegenerate solitons in the integrable fractional coupled Hirota equation