arXiv 0803.38662008-03-27math.DGmath-phmath.MPnlin.SI

Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems

Gloria Mari Beffa

备注：: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
发表信息：: SIGMA 4 (2008), 034, 23 pages
DOI：: 10.3842/SIGMA.2008.034

英文摘要：

In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discusses the close connection between different types of geometries and the type of equations they realize. In particular, we describe the direct relation between symmetric spaces and equations of KdV-type, and the possible geometric origins of this connection.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

有根树Connes-Kreimer Hopf代数的完全可积Lax对方程

Completely Integrable Lax Pair equations for Connes-Kreimer Hopf algebra of rooted trees

本文证明了由Bădiţoiu和Rosenberg引入的无穷小特征$\ ildeβ_0$对应的低维李代数Lax对方程具有完全可积性。

2023-12-01math-phmath.DGmath.MP

Transverse Hilbert Schemes and Completely Integrable Systems

Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $ω$ and a projection onto $\mathbb{C}$. We provide a full characterization of the completely integrable systems that arise in this way.

2017-06-06math.DGmath.SG

New examples of manifolds with completely integrable geodesic flows

We construct Riemannian manifolds with completely integrable geodesic flows, in particular various nonhomogeneous examples. The methods employed are a modification of Thimm's method, Riemannian submersions and connected sums.

1990-12-04math.DGmath.DS

Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$

I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold $S^2\times S^3$. In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, $Y^{p,q}$, discovered by physicists by showing that $Y^{p,q}$ and $Y^{p',q'}$ are inequivalent as contact structures if and only if $p\neq p'$.

2011-06-15math-phmath.DGmath.MP

On the Existence of a Codimension 1 Completely Integrable Totally Geodesic Distribution on a Pseudo-Riemannian Heisenberg Group

In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved by T. Hangan.

2010-02-28math.DG

Completely integrable curve flows on Adjoint orbits

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint $U$-orbits from flows in the soliton hierarchy associated to a compact Lie group $U$. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations…

2001-08-22math.DGnlin.SI

更多相关论文收起更多论文

Completely integrable torus actions on complex manifolds with fixed points

We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.

2012-12-15math.CVmath.DGmath.SG

The geometry of nondegeneracy conditions in completely integrable systems

Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions, we show the implication relations that exist between them, and we show the uniqueness of the…

2004-03-24math.DGmath.SG

A completely integrable flow of star-shaped curves on the light cone in Lorentzian $\mathbb{R}^4$

In this paper we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian $\mathbb{R}^4$, invariants under the centro-affine action of the Lorentzian group, is Poisson equivalent to the space of conformal differential invariants for curves in the Möbius sphere. We use this relation to find realizations of solutions of a complexly coupled system of KdV equations as flows of curves in the cone.

2017-11-01math.DG

双变量beta统计流形上的完全可积梯度系统

Completely Integrable Gradient System on the bivariate beta statistical manifold

本文研究三参数双变量beta统计流形上的完全可积梯度系统，证明其哈密顿性与Lax对表示，并利用Stirling近似揭示其伪黎曼几何与梯度流的线性化，凸显信息几何与动力系统的关联。

2025-08-06math.DG

Multi-Hamiltonian structure of Plebanski's second heavenly equation

We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.

2005-07-14gr-qchep-thmath-ph

Souriau-Fisher度量与在李群SO(2)和SO(3)上的完全可积系统

Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)

本文研究了Souriau-Fisher度量在SO(2)和SO(3)上的应用，探讨了2-余积对梯度系统可积性的影响，并展示了如何通过余积修改Fisher度量。

2026-01-22math.DG

Toric integrable geodesic flows

By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.

2001-11-03math.AGmath.DGmath.DS

Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.

2001-05-07math.DGmath.DSmath.SG

On Integrable Systems & Rigidity for PDEs with Symmetry

This thesis is divided into two parts. In the first part we study completely integrable systems, and their underlying structures, in detail. We study their deformation theory and the different equivalence relations surrounding it. We motivate the definition of weak equivalence (found in the literature) by studying different interpretations of the concept `singular Lagrangian foliation'. Finally, we prove a normal form theorem inspired by the Moser path method. In the second part we study the deformation theory of a PDE with a pseudogroup of symmetries in general. We prove that a solution is ri…

2017-12-03math.DGmath.SG