k ‐symmetric AKS systems and flat immersions into spheres | Zendy

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k ‐symmetric AKS systems and flat immersions into spheres

Author(s) -

Brander David

Publication year - 2008

Publication title -

journal of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.441

H-Index - 62

eISSN - 1469-7750

pISSN - 0024-6107

DOI - 10.1112/jlms/jdm109

Subject(s) - mathematics , quasiperiodicity , algebraic number , integrable system , euclidean space , pure mathematics , dimension (graph theory) , class (philosophy) , space (punctuation) , spheres , euclidean geometry , type (biology) , mathematical analysis , geometry , physics , computer science , quasiperiodic function , astronomy , ecology , artificial intelligence , biology , operating system

We define a large class of integrable nonlinear PDEs, k ‐symmetric AKS systems, with solutions that evolve on finite‐dimensional subalgebras of loop algebras and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the solution, and show that the problem of isometrically immersing n dimensional Euclidean space into a sphere of dimension 2 n – 1 can be addressed via this scheme, producing infinitely many real analytic solutions.

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