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

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.

Curve Flows with a Global Forcing Term

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. We prove an analogue to Huisken’s distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below $$-\pi $$ - π and show that this condition is sharp. With that, we can exclude singularities in finite time for bounded forcing terms. For immortal flows of closed curves whose forcing terms provide non-vanishing enclosed area and bounded length, we show convexity in finite time and smooth and exponential convergence to a circle. In particular, all of the above holds for the area preserving curve shortening flow.

Lagrangian Curve Flows on Symplectic Spaces

A smooth map γ in the symplectic space R2n is Lagrangian if γ,γx,…, γx(2n−1) are linearly independent and the span of γ,γx,…,γx(n−1) is a Lagrangian subspace of R2n. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in R2n with respect to the symplectic group Sp(2n), (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s C^n(1)-KdV flows and A^2n−1(2)-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows.

Discrete Invariant Curve Flows, Orthogonal Polynomials, and Moving Frame

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.

Geometric curve flows in low dimensional Cayley–Klein geometries

Using the method of equivariant moving frames, we derive the evolution equations for the curvature invariants of arc-length parametrized curves under arc-length preserving geometric flows in two-, three- and four-dimensional Cayley–Klein geometries. In two and three dimensions, we obtain recursion operators, which show that the curvature evolution equations obtained are completely integrable.