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1.
Based upon the covariant prolongation
structures theory, we construct the sl(2,R)×R(ρ) prolongation structure for Konno-Asai-Kakuhata equation. By taking
two and one-dimensional prolongation spaces, we obtain the
inverse scattering equations given by Konno et al. and the
corresponding Riccati equation. The Bäcklund transformations are also presented. 相似文献
2.
The group-theoreti cal technique for generating stationary axisymmetric gravitational fields is approached by means of the prolongation structure theory for soliton systems. An sp(2)xc(t) structure is obtained via solving the fundamental equation for prolongation structures and the F-equation for Kinnersley-Chitre's generating function is naturally introduced as an inverse scattering equation. A homogeneous Hilbert problem(HRP) associated with the Geroch group K and a corresponding linear singular integral equation are derived based upon a general condition satisfied by the auto-Bäcklund transformations in the sense of prolongation structure theory. 相似文献
3.
An SL(2R) ×R1(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition. 相似文献
4.
In this papsr all the known Backlund trans- formations of the Ernst equation are de}tved by ua}ing prolongation: method. As tile nonlinear realizationsp and infinite-dimensional linear realization for the prolongation structure are used, the inverse scattering equation (Lax-pair) is also derived. And the connection between BT and Lax-pair becomes evident. A systematic procedure for deriving the non-Linear realizations of the algebras is sugested based on the prolongation structure. The two knowniBackl and transformations as well as the Lax-pair of the Ernst-Maxwell equations are obtained bu the same method. 相似文献
5.
XIE Tao LI Min-Li 《理论物理通讯》2008,50(9):565-567
Based upon the covariant prolongation structures theory, we construct the sl(2, R)×R(p) prolongation structure for Konno-Asai-Kakuhata equation. By taking two and one-dimensional prolongation spaces, we obtain the inverse scattering equations given by Konno et al. and the corresponding Riccati equation. The Baecklund transformations are also presented. 相似文献
6.
The prolongation structure methodologies of Wahlquist--Estabrook [Wahlquist H D and Estabrook F B 1975 J. Math. Phys. 16 1] for nonlinear differential equations are applied to a variable-coefficient KdV equation. Based on the obtained prolongation structure, a Lie algebra with five parameters is constructed. Under certain conditions, a Lie algebra representation and three kinds of Lax pairs for the variable- coefficient KdV equation are derived. 相似文献
7.
In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. From the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation. 相似文献
8.
《Journal of Nonlinear Mathematical Physics》2013,20(3-4):201-215
Abstract Bäcklund transformations, which are relations among solutions of partial differential equations–usually nonlinear–have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure. The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables. This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example. The Zakharov-Shabat method is also discussed. Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented. 相似文献
9.
M. Leo R.A. Leo L. Martina F.A.E. Pirani G. Soliani 《Physica D: Nonlinear Phenomena》1981,4(1):105-112
The Estabrook-Wahlquist prolongation of a one-parameter family of evolution equations is carried out explicitly. It is shown that a non-Abelian prolongation algebra is obtained only for that parameter value for which the prolonged equation belongs to a known class of integrable equations. A Bäcklund transformation is constructed for the integrable case, and periodic, kink and bell-type solutions are displayed. 相似文献
10.
We investigate the coupled inhomogeneous nonlinear Schrodinger equation by the covariant prolongation structure theory, and obtain its Lax's representation. Moreover, we present the corresponding Riccati equations, Backlund transformation, and one-soliton solution. 相似文献
11.
S. StalinM. Senthilvelan 《Physics letters. A》2011,375(43):3786-3788
In this Letter, we formulate an exterior differential system for the newly discovered cubically nonlinear integrable Camassa-Holm type equation. From the exterior differential system we establish the integrability of this equation. We then study Cartan prolongation structure of this equation. We also discuss the method of identifying conservation laws and Bäcklund transformation for this equation from the identified exterior differential system. 相似文献
12.
The hidden symmetry and integrability of the long-short wave equation in (2 1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable. 相似文献
13.
It is shown that the proper geometrical framework for the nonlinear evolution equations (NEEs) and the soliton equations Should be the fibre bundle theory, the principal bundle and its associated bundle and their connection theory. Based upon the requirement of covariance of the geometrical quantities, a covariant generic geometry theory for the prolongation strutures of the NEEs is proposed and the fundamental equations for the prolongation structures are presented. From the fundamental equations it immediately follows that the comections corresponding to these NEEs always flat but with torsion and the covariant formulae satisfied by the conservation quantities associated with these NEEs are obtained. The prolongation structure of the MKdV equation, as an example, is concretely worked out by means of the covariant theory of the prolongation structure presented in this paper. 相似文献
14.
We discuss a new kind of nonlinear Schrödinger equation from the viewpoint of prolongation theory. It is shown that the equation possess a Lax pair with a 3 × 3 matrix structure. It is further demonstrated that by a multiple scale perturbation of Zakharovet al. it can be reduced to the usual KdV equation. 相似文献
15.
16.
We consider a system of equations, a two component generalisation of the KdV equation, which Hirota and Satsuma recently conjectured to be integrable. Using the Wahlquist-Estabrook prolongation technique, we derive a scattering problem for this system. 相似文献
17.
《Physics letters. A》1997,234(5):358-360
We investigate in detail the integrability of fermionic extensions of the Korteweg-de Vries equation by using the prolongation structure method. The integrable one- and two-parameter systems are obtained. The previously known integrable systems are their special cases. 相似文献
18.
基于与Kogelnik同样的物理模型,但是用薄光栅的振幅传输理论,得到了与Kogelnik相同的耦合波方程.所以说这一开拓方法将薄光栅与厚光栅的两种理论联系起来了. 相似文献
19.
Yoshishige Haraoka 《Journal of Nonlinear Mathematical Physics》2013,20(4):70-84
We extend the notion of deformation to inverse operations of restrictions of completely integrable systems to regular or singular locus, and call the extended notion prolongation. We show that a prolongability determines uniquely a Fuchsian ordinary differential equation of rank three with three regular singular points. This seems similar to that the deformation equation determines the accessory parameters as a function of the geometric moduli. Relations between prolongations and middle convolutions is also studied. 相似文献
20.
BAI Yong-Qiang WU Ke GUO Han-Ying ZHAO Wei-Zhong 《理论物理通讯》2007,48(4):591-600
Based on noncommutative differential calculus, we present a theory of prolongation structure for semidiscrete non/inear evolution equations. As an illustrative example, a semi-discrete model of the non/inear SchrSdinger equation is discussed in terms of this theory and the corresponding Lax pairs are also given. 相似文献