2. Topological obstructions to complete integrability

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

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An analytical scheme on complete integrability of 2D biophysical excitable systems

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2021.125924 ◽

2021 ◽

pp. 125924

Author(s):

Argha Mondal ◽

Kshitish Ch. Mistri ◽

M.A. Aziz-Alaoui ◽

Ranjit Kumar Upadhyay

Keyword(s):

Complete Integrability ◽

Analytical Scheme ◽

Excitable Systems

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Complete integrability of the Kadomtsev-Petviashvili equations in 2 + 1 dimensions

Physica D Nonlinear Phenomena ◽

10.1016/0167-2789(86)90191-0 ◽

1986 ◽

Vol 18 (1-3) ◽

pp. 305-307 ◽

Cited By ~ 15

Author(s):

Zhuhan Jiang ◽

R.K. Bullough ◽

S.V. Manakov

Keyword(s):

Complete Integrability

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Complete integrability of vector fields in RN

Journal of Geometry and Physics ◽

10.1016/j.geomphys.2021.104308 ◽

2021 ◽

pp. 104308

Author(s):

Jaume Llibre ◽

Rafael Ramírez ◽

Valentín Ramírez

Keyword(s):

Vector Fields ◽

Complete Integrability

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Conformal geodesics on gravitational instantons

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000463 ◽

2021 ◽

pp. 1-32

Author(s):

MACIEJ DUNAJSKI ◽

PAUL TOD

Keyword(s):

Magnetic Field ◽

Geodesic Flow ◽

Isometry Group ◽

Three Dimensional ◽

First Integrals ◽

Complete Integrability ◽

Conformal Killing ◽

Gravitational Instantons ◽

Geodesic Equations ◽

Jacobi Equations

Abstract We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

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