Browsing by Subject "Integrable systems"
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Item Open Access Discrete symmetries and nonlocal reductions(Elsevier, 2020) Gürses, Metin; Pekcan, A.; Zheltukhin, K.We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.Item Open Access The Hirota direct method(2005) Pekcan, AslıThe search for integrability of nonlinear partial differential and difference equations includes the study on multi-soliton solutions. One of the most famous method to construct multi-soliton solutions is the Hirota direct method. In this thesis, we explain this method in detail and apply it to explicit examples.Item Open Access Integrable systems on regular time scales(2009) Yantır, Burcu SilindirWe present two approaches to unify the integrable systems. Both approaches are based on the classical R-matrix formalism. The first approach proceeds from the construction of (1 + 1)-dimensional integrable ∆-differential systems on regular time scales together with bi-Hamiltonian structures and conserved quantities. The second approach is established upon the general framework of integrable discrete systems on R and integrable dispersionless systems. We discuss the deformation quantization scheme for the dispersionless systems. We also apply the theories presented in this dissertation, to several well-known examples.Item Open Access On a transformation between hierarchies of integrable equations(Elsevier BV * North-Holland, 2006) Gürses, M.; Zheltukhin, K.A transformation between a hierarchy of integrable equations arising from the standard R-matrix construction on the algebra of differential operators and a hierarchy of integrable equations arising from a deformation of the standard R-matrix is given. © 2005 Elsevier B.V. All rights reserved.Item Open Access Zero curvature and Gel'fand-Dikii formalisms(2004) Silindir, BurcuIn soliton theory, integrable nonlinear partial differential equations play an important role. In that respect such differential equations create great interest in many research areas. There are several ways to obtain these differential equations; among them zero curvature and Gel’fand-Dikii formalisms are more effective. In this thesis, we studied these formalisms and applied them to explicit examples.