56 research outputs found
Construction and separability of nonlinear soliton integrable couplings
A very natural construction of integrable extensions of soliton systems is
presented. The extension is made on the level of evolution equations by a
modification of the algebra of dynamical fields. The paper is motivated by
recent works of Wen-Xiu Ma et al. (Comp. Math. Appl. 60 (2010) 2601, Appl.
Math. Comp. 217 (2011) 7238), where new class of soliton systems, being
nonlinear integrable couplings, was introduced. The general form of solutions
of the considered class of coupled systems is described. Moreover, the
decoupling procedure is derived, which is also applicable to several other
coupling systems from the literature.Comment: letter, 10 page
Zero curvature and Gel'fand-Dikii formalisms
Cataloged from PDF version of article.In 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.Silindir, BurcuM.S
Integrable systems on regular time scales
Ankara : The Department of Mathematics and the Institute of Engineering and Science of Bilkent University, 2009.Thesis (Ph.D.) -- Bilkent University, 2009.Includes bibliographical references leaves 98-102.We 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.Yantır, Burcu SilindirPh.D
Integrable discrete systems on R and related dispersionless systems
The general framework for integrable discrete systems on R in particular
containing lattice soliton systems and their q-deformed analogues is presented.
The concept of regular grain structures on R, generated by discrete
one-parameter groups of diffeomorphisms, through which one can define algebras
of shift operators is introduced. Two integrable hierarchies of discrete chains
together with bi-Hamiltonian structures are constructed. Their continuous limit
and the inverse problem based on the deformation quantization scheme are
considered.Comment: 19 page
Bi-Hamiltonian structures for integrable systems on regular time scales
A construction of the bi-Hamiltonian structures for integrable systems on
regular time scales is presented. The trace functional on an algebra of
-pseudo-differential operators, valid on an arbitrary regular time
scale, is introduced. The linear Poisson tensors and the related Hamiltonians
are derived. The quadratic Poisson tensors is given by the use of the recursion
operators of the Lax hierarchies. The theory is illustrated by
-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broer
hierarchies.Comment: 18 page
Integrable discrete systems on R and related dispersionless systems
A 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
R-matrix approach to integrable systems on time scales
A general unifying framework for integrable soliton-like systems on time
scales is introduced. The -matrix formalism is applied to the algebra of
-differential operators in terms of which one can construct infinite
hierarchy of commuting vector fields. The theory is illustrated by two
infinite-field integrable hierarchies on time scales which are difference
counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer
soliton systems are constructed as related finite-field restrictions.Comment: 21 page
Soliton solutions of q-Toda lattice by Hirota direct method
This paper presents the q-analogue of Toda lattice system of difference equations by discussing the q-discretization in three aspects: differential-q-difference, q-difference-q-difference and q-differential-q-difference Toda equation. The paper develops three-q-soliton solutions, which are expressed in the form of a polynomial in power functions, for the differential-q-difference and q-difference-q-difference Toda equations by Hirota direct method. Furthermore, it introduces q-Hirota D-operator and presents the q-differential-q-difference version of Toda equation. Finally, the paper presents its solitary wave like a solution in terms of q-exponential function and explains the nonexistence of further solutions in terms of q-exponentials by the virtue of Hirota perturbation
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