k-symmetric AKS systems and flat immersions into spheres

• Autores: David Brander
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 77, Nº 2, 2008, págs. 299-319
• Idioma: inglés
• DOI: 10.1112/jlms/jdm109
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• Resumen

  • 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 2n � 1 can be addressed via this scheme, producing infinitely many real analytic solutions.