Curve flows in Lagrange-Finsler geometry, bi-Hamiltonian structures and solitons

JOURNAL OF GEOMETRY AND PHYSICS

卷:59

Article

Anco, Stephen C.²  Vacaru, Sergiu I.¹ 

[1] Alexandru Ioan Cuza Univ, Fac Math, Iasi 700506, Romania

[2] Brock Univ, Dept Math, St Catharines, ON L2S 3A1, Canada

关键词: Curve flow;    (semi) Riemannian spaces;    Nonholonomic manifold;    Nonlinear connection;    Lagrange and Finsler geometry;    Bi-Hamiltonian;    Soliton equation;   

DOI  :  10.1016/j.geomphys.2008.10.006

来源: Elsevier

【 摘 要 】

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. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

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