On a Laplace Sequence of Nonlinear Integrable Ernst-Type Equations

Schief, W. K.; Rogers, C.

doi:10.1007/978-1-4612-2434-1_16

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…The Laplace invariants of systems of equations have previously been considered in a few different contexts [12,1,13,14]. It has been established [15] that the chains of the Laplace invariants for the most well-known hyperbolic systems having complete sets of integrals -the open Toda chains, are finite.…”

Section: Integrals and Generalized Laplace Invariants Of Reduced Systemsmentioning

confidence: 99%

On non-Abelian Toda A2(1) model and related hierarchies

Demskoi

Lee

2009

Journal of Mathematical Physics

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We study limiting cases of the two known integrable chiral-type models with tree-dimensional configuration space. One of the initial models is the non-Abelian Toda A(1) 2 model and the other was found by means of the symmetry approach by A.G. Meshkov and one of the authors. The Cintegrability of the reduced models is established by constructing their complete sets of integrals and general solutions. A description of the generalized symmetry algebras of these models is given in terms of operators mapping integrals into symmetries. The integrals of the Liouvilletype systems are known to define Miura-type transformations for their generalized symmetries. This fact allowed us to find a few new systems of the Yajima-Oikawa type. We present a recursion operator for one them.

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Section: Integrals and Generalized Laplace Invariants Of Reduced Systemsmentioning

confidence: 99%

On non-Abelian Toda A2(1) model and related hierarchies

Demskoi

Lee

2009

Journal of Mathematical Physics

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“…In [6], it has been shown that the Ernst-type equation (9) admits special solutions in terms of particular Painleve III transcendents which are associated with generalized Weingarten surfaces of revolution. It may be verified that the underlying particular Painleve III equation constitutes a degenerate case of the Painleve V equation.…”

Section: Helicoids and The Painleve V Equationmentioning

confidence: 99%

“…The second sheet E_ turns out to be another generalized where the harmonic function v is defined as in (3). The transformation (19) which, by construction, leaves invariant the Ernst-type equation (9) represents the analogue of the Laplace-Darboux-type transformation j£?_ for the Ernst-Weyl equation (cf. (2)).…”

Section: Sphere Congruences and A Backlund Transformationmentioning

confidence: 99%

“…The above 'Ernst-Weyl' equation has also been identified [7] as a canonical 2+0-dimensional reduction of the 2+1dimensional Loewner-Konopelchenko-Rogers (LKR) integrable system [5]. This connection has been exploited in [9] to construct a Laplace-Darboux-type invariance of the nonlinear Ernst-Weyl equation. Indeed, if (E, p) is a solution of the Ernst-Weyl equation (1) then the Laplace-Darboux-type transforms…”

Section: Introductionmentioning

confidence: 99%

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On the geometry of the Painlevé V equation and a Bäcklund transformation

Schief

2002

ANZIAM J.

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“…In general, the studies of Painlevé equations and Ermakov-type systems have proceeded independently. Thus, the only known hybrid solitonic-Ermakov system seems to be that obtained in [54,55] where a 2+1-dimensional Ernst-type system of general relativity as derived in [56], suitably constrained, leads to a novel composition of the integrable 2+1-dimensional sinh-Gordon equation of [30,31] and of a generalised Ermakov-Ray-Reid system. The work of [2,[36][37][38] on Ermakov-Painlevé II systems has recently been augmented by the introduction in [47] of prototype Ermakov-Painlevé IV systems via a symmetry reduction of a coupled derivative resonant NLS triad.…”

Section: Introductionmentioning

confidence: 99%