Blaszak, M.Gurses, M.Silindir, B.Szablikowski, B. M.2015-07-282015-07-2820080022-2488http://hdl.handle.net/11693/11622A general framework for integrable discrete systems on ℝ, in particular, containing lattice soliton systems and their q-deformed analogs, is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, in terms of which one can define algebra of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures and their continuous limits are constructed. The inverse problem based on the deformation quantization scheme is considered. © 2008 American Institute of Physics.EnglishQ-pseudodifferential SymbolsQ-analogHierarchiesAlgebraMatrixLimitArticle10.1063/1.2948962