Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations

Beals, Richard; Rabelo, Mauro Luiz; Tenenblat, Keti

doi:10.1002/sapm1989812125

Cited by 153 publications

(124 citation statements)

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“…Actually, the equation (1) appeared first a long time ago, as one of Rabelo's equations possessing a zero-curvature representation with a parameter [3] (we are grateful to Prof. E.G. Reyes for this essential reference; see also [4]). Recently, the Lax pair of the SPE, of the Wadati-Konno-Ichikawa (WKI) type, was rediscovered in [5], where also the second-order recursion operator of the SPE was presented and the chain of transformations…”

mentioning

confidence: 94%

Solitary wave solutions of the short pulse equation

Sakovich

2006

J. Phys. A: Math. Gen.

259

183

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Abstract. An exact nonsingular solitary wave solution of the Schäfer-Wayne short pulse equation is derived from the breather solution of the sine-Gordon equation by means of a transformation between these two integrable equations.The short pulse equation (SPE), which has the formup to scale transformations of its variables, was derived by Schäfer and Wayne [1] as a model equation describing the propagation of ultra-short light pulses in silica optical fibers. In contrast to the nonlinear Schrödinger equation (NLSE) which models slowly varying wave trains, the SPE approximates Maxwell's equations in the case when the pulse spectrum is not narrowly localized around the carrier frequency, that is when the pulse is as short as a few cycles of the central frequency. Numerical simulations made in [2] showed that the accuracy of the SPE approximation to Maxwell's equations steadily increases as the pulse shortens, whereas the NLSE approximation fails to be accurate for ultra-short pulses. Like the NLSE, the SPE is an integrable nonlinear partial differential equation. Actually, the equation (1) appeared first a long time ago, as one of Rabelo's equations possessing a zero-curvature representation with a parameter [3] (we are grateful to Prof. E.G. Reyes for this essential reference; see also [4]). Recently, the Lax pair of the SPE, of the Wadati-Konno-Ichikawa (WKI) type, was rediscovered in [5], where also the second-order recursion operator of the SPE was presented and the chain of transformationswas constructed which relates the SPE (1) with the sine-Gordon equation (SGE)Very recently, the bi-Hamiltonian structure of the SPE was discovered and the corresponding hierarchy and conservation laws were studied in [6,7]. The soliton solutions of the NLSE have played an important role in the recent development of fiber-optic communications. For this reason, one would definitely like to have exact mathematical expressions for the ultra-short light pulses governed by the SPE. In the present paper, we derive exact solitary wave solutions of the SPE (1) from the well-known ones of the SGE (3) by means of the transformation (2) between these two integrable equations.

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confidence: 94%

Solitary wave solutions of the short pulse equation

Sakovich

2006

J. Phys. A: Math. Gen.

259

183

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“…which may be used to solve the differential equation by inverse scattering [1], with η playing the role of a spectral parameter for the scattering problem. It is also shown in [5] that one can generate infinite sequences of conservation laws for the class of differential equations describing η pseudospherical surfaces by making use of the structure equations (3), although some of these conservation laws may end up being non-local.…”

Section: Introductionmentioning

confidence: 99%

Third-order differential equations and local isometric immersions of pseudospherical surfaces

Silva

Kamran

2016

Commun. Contemp. Math.

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The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa-Holm and Degasperis-Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in E 3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class of Chern and Tenenblat [3]. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the pde. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In a pair of earlier papers [13], [14] we have shown that this property fails to hold for all k-th order evolution equations ut = F (u, ux, ..., u x k ) and all other second order equations of the form uxt = F (u, ux), except for the sine-Gordon equation and a special class of equations for which the coefficients of the second fundamental form are universal, that is functions of x and t which are independent of the choice of solution u. In the present paper, we consider thirdorder equations of the form ut − uxxt = λuuxxx + G(u, ux, uxx), λ ∈ R, which describe pseudospherical surfaces. This class contains the Camassa-Holm and Degasperis-Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E 3 for which the coefficients of the second fundamental form depend on a jet of finite order of u, then these coefficients are universal in the sense of being independent on the choice of solution u. This result further underscores the special place that the sine-Gordon equations seems to occupy amongst integrable partial differential equations in one space variable.

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“…За рамками нашей работы осталось уравнение sin-Гордон, с которым связано, например, уравнение сверхкоротких импульсов [15]. Возможно, замечание 1 указы-вает способ построения преобразования Беклунда и в этом случае, но, скорее всего, удобнее перейти к спектральной задаче Захарова-Шабата.…”

Section: заключениеunclassified

Одевающая Цепочка Для Акустической Спектральной Задачи

Адлер¹,

Shabat²

2006

ТМФ

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Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations

Cited by 153 publications

References 8 publications

Solitary wave solutions of the short pulse equation

Solitary wave solutions of the short pulse equation

Third-order differential equations and local isometric immersions of pseudospherical surfaces

Одевающая Цепочка Для Акустической Спектральной Задачи

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