Fractional Curve Flows and Solitonic Hierarchies in Gravity and Geometric Mechanics

TL;DR

This paper develops a geometric framework using fractional calculus and nonholonomic manifolds to model fractional gravity and mechanics, revealing solitonic hierarchies and fractional integrable equations.

Contribution

It introduces a novel geometric approach combining fractional calculus with bi-Hamiltonian structures to derive fractional solitonic equations in gravity and mechanics.

Findings

Derivation of fractional vector sine-Gordon equations

Construction of fractional vector mKdV equations

Identification of solitonic hierarchies in fractional models

Abstract

Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and associated solitonic hierarchies. The constructions yield horizontal/vertical pairs of fractional vector sine-Gordon equations and fractional vector mKdV equations when the hierarchies for corresponding curve fractional flows are described in explicit forms by fractional wave maps and analogs of Schrodinger maps.