Exact solutions to the sine-Gordon equation

Aktosun, Tuncay; Demontis, Francesco; Mee, Cornelis van der

doi:10.1063/1.3520596

Cited by 57 publications

(100 citation statements)

References 36 publications

(35 reference statements)

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“…It is also worth stressing that in the previous papers where the method of the triplets ( A, B, C) is used [32][33][34][35], the invertibility of the matrix P (solution of the Sylvester equation) is guaranteed by assuming the minimality of the triplet ( A, B, C), and that all the eigenvalues of the matrix A have positive real parts. In the present case, this is not true.…”

Section: Solving the Marchenko Equationsmentioning

confidence: 99%

“…[32][33][34][35], we now solve explicitly the Marchenko equation (74) in the reflectionless case, i.e., when ρ(z) ≡ 0 for all z ∈ R, and a finite number of discrete eigenvalues ζ n (see Appendix B for more details). In order to do so, let us represent the 2 × 2 matrix Marchenko kernel as follows:…”

Section: Solving the Marchenko Equationsmentioning

confidence: 99%

“…In Section 4 we will formulate and solve the inverse problem as a Riemann-Hilbert problem (RHP) in terms of a suitable uniform variable, and we will show that the asymptotic behavior of the scattering data ensures the algebraic-integral system of equations providing the solution of the inverse problem is well defined. Finally, in Section 5 we will formulate the inverse problem in terms of Marchenko integral equations, and we will develop the triplet method as a tool to obtain explicit multisoliton solutions by solving the Marchenko equations via separation of variables [32][33][34][35]. The crux of the method is to represent the Marchenko kernel as Ce −(y+z)A B, where (A, B, C) is a suitable matrix triplet.…”

Section: Introductionmentioning

confidence: 99%

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The Inverse Scattering Transform for the Defocusing Nonlinear Schrödinger Equations with Nonzero Boundary Conditions

Prinari²,

et al. 2012

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A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values q ± ≡ q 0 e iθ ± as x → ±∞ is presented. The direct problem is shown to be well posed for potentials q such that q − q ± ∈ L 1,2 (R ± ), for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann-Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.

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Section: Solving the Marchenko Equationsmentioning

confidence: 99%

Section: Solving the Marchenko Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Inverse Scattering Transform for the Defocusing Nonlinear Schrödinger Equations with Nonzero Boundary Conditions

Prinari²,

et al. 2012

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“…The classical SG equation is also used in a number of other physical applications. Among these are the motion of a rigid pendulum attached to a stretched wire Jin (2009); Wazwaz (2012), the dynamical properties of DNA molecules Wazwaz (2012), the stimulation of phonon modes Aktosun et al (2010), the signal dispersion in fiber optics for long-distance communications (Int sour. 2) and the phase difference across a long Josephson junction carrying a current infinitely under any applied voltage through two superconductors separated by an insulator layer Derks et al (2003).…”

Section: Afyon Kocatepe üNiversitesi Fen Ve Mühendislik Bilimleri Dermentioning

confidence: 99%

Approximate Soliton Solutions of Real Order Sine- Gordon Equations

Üzar¹

2017

AKU-J. Sci. Eng.

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In this study, the fractional Sine-Gordon (SG) equations (time-fractional, space-fractional and timespace-fractional) are solved using Homotopy Perturbation Method (HPM). The crucial point is the attained remarkable result from these solutions. While the solutions of classical and time-fractional SG equations are kink of type (Although of being the same type, they are different from each other), solution of the space-fractional SG equation is breather of type i.e., different types of soliton solutions are obtained using similar initial conditions for time and space fractional SG equation. Also these results show that some events such as vortex-antivortex couples in a Josephson junction or losses in signal dispersion of fiber optics communication can be modelled by fractional SG equations. In other words, this study may be very important for bringing to light the real behaviour of physical systems which have usually been described by classical SG equation. Because some physical events such as the memory effects of non-Markovian processes, the effects of non-Gaussian distribution, interactions between the systems and environment and some physical losses in the systems which are neglected in classical SG equation can be taken into account with fractional SG equations. Reel Mertebeli Sine-Gordon Denklemlerinin Yaklaşık Soliton Çözümleri Anahtar kelimelerCaputo kesirsel türev operatörü; HPM; SG denklemi; kesirli lineer olmayan denklemler ÖzetBu çalışmada kesirli Sine-Gordon denklemleri (zaman-kesirli, uzay-kesirli ve zaman-uzay-kesirli) homotopy pertürbasyon metodu (HPM) kullanılarak çözülmüştür. Bu çözümlerden dikkat çekici sonuçlar elde edilmiştir. Klasik ve zaman-kesirli SG denklemlerinin çözümü kink tipi iken (aynı tipte olmalarına rağmen, birbirlerinden farklıdır), uzay-kesirli SG denkleminin çözümü breather tiptir; başka bir ifadeyle zaman ve uzay kesirli SG denklemi için benzer başlangıç koşulları kullanıldığında farklı tip soliton çözümleri elde edilmiştir. Ayrıca bu sonuçlar josephson eklemlerindeki vorteks-antivorteks çifleri veya fiber optik iletişimde sinyal dağılımındaki kayıplar gibi bazı olayları kesirli SG denklemleri ile modelleyebileceğini göstermiştir. Diğer bir deyişle, bu çalışma genellikle klasik SG denklemleri ile tanımlanan fiziksel sistemlerin gerçek davranışlarına ışık tutabilir. Çünkü Markovian olmayan süreçlerin bellek etkileri, Gaussian olmayan dağılımların etkileri, sistem ile dış çevre arasındaki etkileşmeler ve klasik SG denkleminin ihmal ettiği sistemler içerisindeki bazı fiziksel kayıplar gibi bazı fiziksel olaylar kesirli SG denklemleri ile hesaba katılabilir.

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“…Use of symbolic manipulation software packages eases the task. The inverse tangent form for solitonic solutions, and several exact solutions of this form, which are expressible in terms of elementary functions are well known [38,39]. However, (14) is a special type of vortex solution to the elliptic sine-Gordon equation, describing two intersecting solitons with a +1 vortex (disclination) at their intersection.…”

mentioning

confidence: 99%