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\(n\)-dimension central affine curve flows. (English) Zbl 1441.53078

The paper under review deals with a particular family of integrable systems called \(n\)-dimensional central affine curvature flows. Originated from the affine geometry of curves in \(\mathbb R^n\), the flows in question are closely related to the \(A^{(1)}_{n-1}\)-KdV hierarchy, see [G. Marí Beffa, Asian J. Math. 12, No. 1, 1–33 (2008; Zbl 1173.37054)]. Applying this analytical correspondence and using geometric backgrounds, the authors reveal remarkable integrability features of central affine curvature flows. The main results are the following:
1) the Cauchy problems for central affine curvature flows with initial data having rapidly decaying central affine curvatures and with periodic initial data are solved;
2) Bäcklund transformations for central affine curvature flows are introduced, a corresponding permutability formula is proved and infinitely many explicit soliton solutions and rational solutions are constructed;
3) a bi-Hamiltonian structure and corresponding conservation laws for central affine curvature flows are described.

MSC:

53E10 Flows related to mean curvature
53A15 Affine differential geometry
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 1173.37054
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