Integrable systems and invariant curve flows in symplectic Grassmannian space

2017 ◽  
Vol 349 ◽  
pp. 1-11
Author(s):  
Junfeng Song ◽  
Changzheng Qu ◽  
Ruoxia Yao
Keyword(s):  
Integrable Systems ◽  
Invariant Curve ◽  
Curve Flows

scholarly journals On an integrable multi-component Camassa–Holm system arising from Möbius geometry

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210164
Author(s):  
Jing Kang ◽  
Xiaochuan Liu ◽  
Changzheng Qu
Keyword(s):  
Integrable Systems ◽  
Invariant Curve ◽  
Necessary Condition ◽  
Miura Transformation ◽  
Möbius Geometry ◽  
Two Component ◽  
De Vries ◽  
Curve Flows

In this paper, we mainly study the geometric background, integrability and peaked solutions of a ( 1 + n ) -component Camassa–Holm (CH) system and some related multi-component integrable systems. Firstly, we show this system arises from the invariant curve flows in the Möbius geometry and serves as the dual integrable counterpart of a geometrical ( 1 + n ) -component Korteweg–de Vries system in the sense of tri-Hamiltonian duality. Moreover, we obtain an integrable two-component modified CH system using a generalized Miura transformation. Finally, we provide a necessary condition, under which the dual integrable systems can inherit the Bäcklund correspondence from the original ones.

scholarly journals Integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n)

2015 ◽  
Vol 38 ◽  
pp. 1560071 ◽  
Author(s):  
Stephen C. Anco ◽  
Esmaeel Asadi ◽  
Asieh Dogonchi
Keyword(s):  
Symmetric Space ◽  
Integrable Systems ◽  
Hermitian Symmetric Space ◽  
Complex Matrix ◽  
Curve Flows ◽  
Frame Method ◽  
Parallel Frame ◽  
Natural Way ◽  
Sg Equation ◽  
Unitary Invariance

A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG) equation, involving a scalar variable coupled to a complex vector variable. The Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way from the beginning in formulating a complex matrix representation of the tangent space 𝔰𝔭(n)/𝔲(n) and its bracket relations within the symmetric Lie algebra (𝔲(n), 𝔰𝔭(n)).

Discrete Invariant Curve Flows, Orthogonal Polynomials, and Moving Frame

2020 ◽  
Author(s):  
Bao Wang ◽  
Xiang-Ke Chang ◽  
Xing-Biao Hu ◽  
Shi-Hao Li
Keyword(s):  
Orthogonal Polynomials ◽  
Affine Space ◽  
Affine Plane ◽  
Euclidean Plane ◽  
Invariant Curve ◽  
Moving Frame ◽  
High Dimensional ◽  
Moving Frames ◽  
Curve Flows ◽  
Explicit Expressions

Abstract In this paper, an orthogonal polynomials-based (OPs-based) approach to generate discrete moving frames and invariants is developed. It is shown that OPs can provide explicit expressions for the discrete moving frame as well as the associated difference invariants, and this approach enables one to obtain the corresponding discrete invariant curve flows simultaneously. Several examples in the cases of centro-affine plane, pseudo-Euclidean plane, and high-dimensional centro-affine space are presented.

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