Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons

TL;DR

This paper applies methods from Lagrange-Finsler geometry to derive bi-Hamiltonian structures and soliton hierarchies from geometric flows on tangent bundles, linking geometric mechanics with integrable systems.

Contribution

It introduces a geometric framework for bi-Hamiltonian structures and soliton equations derived from Lagrangian and nonholonomic curve flows in tangent bundles.

Findings

Derived vector sine-Gordon and mKdV equations from geometric flows.

Classified tangent bundle geometries on symmetric spaces with N-connection structures.

Connected geometric curve flows to integrable soliton hierarchies.

Abstract

Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces $G/SO(n)$ provided with an N-connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/ vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrodinger maps on a tangent bundle.