B�cklund autotransformation for the equationu xt=eu?e?2u

Safin, S. S.; Шарипов, Руслан

doi:10.1007/bf01015902

Cited by 14 publications

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References 4 publications

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“…14 and 15. The Bäcklund transformation obtained by Sharipov and Yamilov 14 is a set of second order differential equations, which is very cumbersome for solving single soliton solutions 15 and is difficult to turn to our Bäcklund transformation ͑3.9͒-͑3.10͒.…”

Section: ͑31͒mentioning

confidence: 99%

Prolongation approach to Bäcklund transformation of Zhiber–Mikhailov–Shabat equation

Yang

1996

Journal of Mathematical Physics

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The prolongation structure of Zhiber-Mikhailov-Shabat ͑ZMS͒ equation is studied by using Wahlquist-Estabrook's method. The Lax pair for ZMS and Riccati equations for pseudopotentials are formulated respectively from linear and nonlinear realizations of the prolongation structure. Based on nonlinear realization of the prolongation structure, an auto-Bäcklund transformation of ZMS equation is obtained.

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Section: ͑31͒mentioning

confidence: 99%

Prolongation approach to Bäcklund transformation of Zhiber–Mikhailov–Shabat equation

Yang

1996

Journal of Mathematical Physics

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“…there exists a complicated auto-BT [118,14], and another, much simpler one will be published soon [36]. We only report here the classical, well established results.…”

Section: Tzitzéicamentioning

confidence: 99%

“…Since v 1 − v 2 = 2i sin u, the difference of (3.111) and (3.114) reads sin u = 2i(Log(ψ 1 /ψ 2 )) xt + sin U, sG(U ) = 0, (3.115) i.e., from the definition of the equation 118) and one must identify…”

Section: Sine-gordonmentioning

confidence: 99%

Painlevé Analysis for Nonlinear Partial Differential Equations

Musette¹

1999

The Painlevé Property

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La propriété de Painlevé, un siècle après The Painlevé property, one century laterAbstract. The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated Bäcklund transformation. A lot of remarkable properties are shared by these so-called "integrable" equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called "partially integrable" sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and systems of Riccati equations which are linearisable, as well as the importance of the Weierstrass elliptic function, for building solitary waves or more elaborate solutions.The Painlevé property, one century later, ed. R. Conte, CRM series in mathematical physics (Springer-Verlag, Berlin, 1998) 31 March 1998 solv-int/9804003

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“…17 The latter are either connected with solutions of Painlevé III 18,19 or solvable in elliptic functions. 20 Let us consider first the traveling wave solution of ͑3͒. We seek a solution of ͑3͒ in the form QϭQ͑x ͒, ϭ͑x͒, xϭϪc, which yields the system…”

Section: Similarity Reductions Of the Shg Equations And Painlevé IIImentioning

confidence: 99%

Second harmonic generation: Hamiltonian structures and particular solutions

Хуснутдинова

Steudel

1998

Journal of Mathematical Physics

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Articles you may be interested inSecond harmonic generation in random composites of particles with core-shell structure A numerical study of optical second-harmonic generation in a one-dimensional photonic structure For the equations describing second harmonic generation ͑SHG͒, we find a linear system different from one found by Kaup ͑but connected with it by a gauge trans-formation͒, which enables us to consider the SHG equations as related to the first negative flow in the coupled KdV hierarchy and to find therefore the Hamiltonian structures and the recursion operator. With the help of the recursion operator we write down the simplest commuting flows and consider their stationary points in order to find particular solutions of the SHG equations. We also investigate some particular invariant solutions of the SHG equations that can be of physical interest, and find their connections with invariant solutions of the Tzitzéica equation. The latter are either connected with Painlevé III or solvable in elliptic functions.

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B�cklund autotransformation for the equationu xt=eu?e?2u

Cited by 14 publications

References 4 publications

Prolongation approach to Bäcklund transformation of Zhiber–Mikhailov–Shabat equation

Prolongation approach to Bäcklund transformation of Zhiber–Mikhailov–Shabat equation

Painlevé Analysis for Nonlinear Partial Differential Equations

Second harmonic generation: Hamiltonian structures and particular solutions

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