Browsing by Author "Pekcan, A."
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Item Open Access 2 + 1 KdV(N) equations(American Institute of Physics, 2011) Gürses, M.; Pekcan, A.We present some nonlinear partial differential equations in 2 + 1-dimensions derived from the KdV equation and its symmetries. We show that all these equations have the same 3-soliton solution structures. The only difference in these solutions are the dispersion relations. We also show that they possess the Painlevé property. © 2011 American Institute of Physics.Item Open Access (2+1)-dimensional local and nonlocal reductions of the negative AKNS system: soliton solutions(Elsevier, 2018) Gürses, Metin; Pekcan, A.Wefirstconstructa(2+1)dimensionalnegativeAKNShierarchyandthenwegiveallpossiblelocaland(discrete)nonlocalreductionsoftheseequations.WefindHirotabilinearformsofthenegativeAKNShierarchyandgiveone-andtwo-solitonsolutions.ByusingthesolitonsolutionsofthenegativeAKNShierarchywefindone-solitonsolutionsofthelocalandnonlocalreducedequations.Item Open Access Characteristic Lie algebra and classification of semidiscrete models(Springer New York LLC, 2007) Habibullin, I. T.; Pekcan, A.We study characteristic Lie algebras of semi-discrete chains and attempt to use this notion to classify Darboux-integrable chains. © Springer Science+Business Media, Inc. 2007.Item Open Access Complete list of Darboux integrable chains of the form t 1 x = t x + d ( t, t 1 )(2009) Habibullin, I.; Zheltukhina, N.; Pekcan, A.We study differential-difference equation (d/dx) t (n+1,x) =f (t (n,x),t (n+1,x), (d/dx) t (n,x)) with unknown t (n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, { t (n+k,x) } k=-∞ ∞, {(dk /d xk) t (n,x) } k=1 ∞, such that Dx F=0 and DI=I, where D x is the operator of total differentiation with respect to x and D is the shift operator: Dp (n) =p (n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f (u,v,w) =w+g (u,v). © 2009 American Institute of Physics.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 Integrable nonlocal reductions(Springer New York LLC, 2018) Gürses, Metin; Pekcan, A.We present some nonlocal integrable systems by using the Ablowitz-Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schrödinger (NLS) and modified Korteweg-de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to nonlocal Fordy-Kulish equations by utilizing the nonlocal reduction to the Fordy-Kulish system on symmetric spaces. We also consider the super AKNS system and then show that Ablowitz-Musslimani nonlocal reduction can be extended to super integrable equations. We obtain new nonlocal equations namely nonlocal super NLS and nonlocal super mKdV equations.Item Open Access Multi-component AKNS systems(Elsevier, 2022-12-31) Gürses, Metin; Pekcan, A.We study two members of the multi-component AKNS hierarchy. These are multi-NLS and multi-MKdV systems. We derive the Hirota bilinear forms of these equations and obtain soliton solutions. We find all possible local and nonlocal reductions of these systems of equations and give a prescription to obtain their soliton solutions. We derive also -dimensional extensions of the multi-component AKNS systems.Item Open Access Nonlocal hydrodynamic type of equations(Elsevier, 2020-03-01) Gürses, Metin; Pekcan, A.; Zheltukhin, K.We show that the integrable equations of hydrodynamic type admit nonlocal reductions. We first construct such reductions for a general Lax equation and then give several examples. The reduced nonlocal equations are of hydrodynamic type and integrable. They admit Lax representations and hence possess infinitely many conserved quantities.Item Open Access Nonlocal KdV equations(Elsevier, 2020) Gürses, Metin; Pekcan, A.Writing the Hirota-Satsuma (HS) system of equations in a symmetrical form we find its local and new nonlocal reductions. It turns out that all reductions of the HS system are Korteweg-de Vries (KdV), complex KdV, and new nonlocal KdV equations. We obtain one-soliton solutions of these KdV equations by using the method of Hirota bilinearization.Item Open Access Nonlocal modified KdV equations and their soliton solutions by Hirota Method(Elsevier, 2019) Gürses, Metin; Pekcan, A.We study the nonlocal modified Korteweg–de Vries (mKdV) equations obtained from AKNS scheme by Ablowitz–Musslimani type nonlocal reductions. We first find soliton solutions of the coupled mKdV system by using the Hirota direct method. Then by using the Ablowitz–Musslimani reduction formulas, we find one-, two-, and three-soliton solutions of nonlocal mKdV and nonlocal complex mKdV equations. The soliton solutions of these equations are of two types. We give one-soliton solutions of both types and present only first type of two- and three-soliton solutions. We illustrate our solutions by plotting their graphs for particular values of the parameters.Item Open Access Nonlocal nonlinear Schrödinger equations and their soliton solutions(American Institute of Physics, 2018-05-04) Gürses, Metin; Pekcan, A.We study standard and nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions, respectively. By using the Hirota bilinear method, we first find soliton solutions of the coupled NLS system of equations; then using the reduction formulas, we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)|2 for the standard and nonlocal NLS equations.Item Open Access On the classification of Darboux integrable chains(American Institute of Physics, 2008) Habibullin, I.; Zheltukhina, N.; Pekcan, A.We study differential-difference equation (d/dx) t (n+1,x) =f (t (n,x),t (n+1,x), (d/dx) t (n,x)) with unknown t (n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, { t (n+k,x) } k=-∞ ∞, {(dk /d xk) t (n,x) } k=1 ∞, such that Dx F=0 and DI=I, where D x is the operator of total differentiation with respect to x and D is the shift operator: Dp (n) =p (n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f (u,v,w) =w+g (u,v). © 2009 American Institute of Physics.Item Open Access The relationship between destination performance, overall satisfaction, and behavioral intention for distinct segments(Routledge, 2004) Baloglu, S.; Pekcan, A.; Chen, S.; Santos, J.Destination performance, visitor satisfaction, and favorable future behavior of visitors are key determinants of destination competitiveness. Most empirical work, assuming that overall tourist population is homogenous, investigates the relationships among product performance, satisfaction, and/or behavioral intentions in an aggregated manner. This study investigates these linkages for different segments of Canadian visitors of Las Vegas. The findings confirmed the mediating role of overall satisfaction for both segments and aggregated sample, and revealed variations in linkages and explanatory power of the models. The study concludes that the segment-based approach is more pragmatic because it provides segment-specific implications for destination management and marketing.Item Open Access Solutions of the extended Kadomtsev-Petviashvili-Boussinesq equation by the Hirota direct method(Taylor & Francis Asia Pacific (Singapore), 2009) Pekcan, A.We show that we can apply the Hirota direct method to some non-integrable equations. For this purpose, we consider the extended Kadomtsev-Petviashvili- Boussinesq (eKPBo) equation with M variable which is (uxxx-6uu x)x + a11uxx + 2ΣMk=2 a1kuxk + ΣMi,j=2aijuxixj=0, where aij = a ji are constants and xi = (x, t, y, z,⋯,x M). We will give the results for M = 3 and a detailed work on this equation for M = 4. Then we will generalize the results for any integer M > 4. © 2009 The Author(s).Item Open Access Superposition of the coupled NLS and MKdV systems(Elsevier, 2019) Gürses, Metin; Pekcan, A.Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method to these equations one obtains the same N-soliton solutions of the generating equation which differ only by the dispersion relations. Similar discussions can be made for the locally and nonlocally reduced equations as well. We give, as an example, AKNS system of equations in (1 + 1)-dimensions.Item Open Access Traveling wave solutions of degenerate coupled multi-KdV equations(American Institute of Physics, 2016) Gürses M.; Pekcan, A.Traveling wave solutions of degenerate coupled ℓ-KdV equations are studied. Due to symmetry reduction these equations reduce to one ordinary differential equation (ODE), i.e., (f′)2 = Pn(f) where Pn(f) is a polynomial function of f of degree n = ℓ + 2, where ℓ ≥ 3 in this work. Here ℓ is the number of coupled fields. There is no known method to solve such ordinary differential equations when ℓ ≥ 3. For this purpose, we introduce two different types of methods to solve the reduced equation and apply these methods to degenerate three-coupled KdV equation. One of the methods uses the Chebyshev’s theorem. In this case, we find several solutions, some of which may correspond to solitary waves. The second method is a kind of factorizing the polynomial Pn(f) as a product of lower degree polynomials. Each part of this product is assumed to satisfy different ODEs.Item Open Access Travelling wave solution of degenerate coupled KdV equations(A I P Publishing, 2014) Gürses M.; Pekcan, A.We give a detailed study of the traveling wave solutions of (l = 2) Kaup-Boussinesq type of coupled KdV equations. Depending upon the zeros of a fourth degree polynomial, we have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solutions. All such possible solutions are given explicitly in the form of Jacobi elliptic functions. Graphs of some exact solutions in solitary wave and periodic shapes are exhibited. Extension of our study to the cases l = 3 and l = 4 are also mentioned. (C) 2014 AIP Publishing LLC.Item Open Access Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations(2011) Ma W.-X.; Pekcan, A.The Kadomtsev-Petviashvili and Boussinesq equations (u xxx - 6uu x)x - ut x ± uyy = 0, (u xxx - 6uu x)x + u xx ± u tt = 0, are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. It is shown that the Kadomtsev-Petviashvili and Boussinesq equations and their dimensional reductions are the only integrable equations among a class of generalized Kadomtsev-Petviashvili and Boussinesq equations (u x1x1x1 - 6uu x1) x1 + σ M i, j=1 a iju xixj = 0, where the a i j's are arbitrary constants and M is an arbitrary natural number, if the existence of the three-soliton solution is required. © 2011 Verlag der Zeitschrift für Naturforschung, Tübingen.