scholarly journals Integrability and vesture for harmonic maps into symmetric spaces

2016 ◽  
Vol 28 (03) ◽  
pp. 1650006 ◽  
Author(s):  
Shabnam Beheshti ◽  
Shadi Tahvildar-Zadeh
Keyword(s):  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Dressing Method ◽  
Special Cases ◽  
Einstein Vacuum ◽  
Completely Reducible

After formulating the notion of integrability for axially symmetric harmonic maps from [Formula: see text] into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding [Formula: see text]-solitonic harmonic maps into a non-compact Grassmann manifold [Formula: see text] is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases [Formula: see text] and [Formula: see text] and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

EXACT SOLUTIONS OF NONLINEAR SU(N) PRINCIPAL σ-MODELS

1988 ◽  
Vol 03 (05) ◽  
pp. 1147-1154
Author(s):  
TIBOR KISS-TOTH
Keyword(s):  
Exact Solutions ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Static Solution ◽  
Finite Energy ◽  
Single Step ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Symmetric Static Solution

The superpotential for n-step soliton solution is derived in the case of an arbitrary dimensional projector for axially symmetric, static solution of nonlinear principal SU (N) σ-models. This was done by using an inverse scattering method developed by Belinski and Zakharov. Finite energy solutions are constructed for all SU (N) one soliton solutions generated by a single step.

scholarly journals SEMI-ANALYTIC EQUATIONS TO THE COX–THOMPSON INVERSE SCATTERING METHOD AT FIXED ENERGY FOR SPECIAL CASES

2008 ◽  
Vol 22 (23) ◽  
pp. 2191-2199 ◽  
Author(s):  
TAMÁS PÁLMAI ◽  
MIKLÓS HORVÁTH ◽  
BARNABÁS APAGYI
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Identical Particle ◽  
Inverse Scattering Problem ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Fixed Energy ◽  
Angular Momenta ◽  
Partial Waves ◽  
Special Cases

Solution of the Cox–Thompson inverse scattering problem at fixed energy1–3 is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.

scholarly journals ADDING A UNITON VIA THE DPW METHOD

2009 ◽  
Vol 20 (08) ◽  
pp. 997-1010 ◽  
Author(s):  
N. CORREIA ◽  
R. PACHECO
Keyword(s):  
Finite Type ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Point Of View ◽  
Type Property ◽  
Riemannian Symmetric Spaces

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

Travelling-wave-type and stationary soliton solutions of the Brans–Dicke equation

2002 ◽  
Vol 80 (9) ◽  
pp. 951-958 ◽  
Author(s):  
M H Dehghani ◽  
M Shojania
Keyword(s):  
Soliton Solutions ◽  
Lax Pair ◽  
Travelling Wave ◽  
Inverse Scattering Method ◽  
Time Dependent ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Wave Type ◽  
Killing Vectors ◽  
The One

Introducing the Lax pair, it is shown that the Brans–Dicke equation is integrable for space-times with two commuting Killing vectors. Using the inverse-scattering method given by Belinskii and Zakharov, the n soliton solutions for the case of the time-dependent metric are introduced. Specially, the one and two travelling-wave-type solitonic solutions are obtained. Also it is shown that the method could be applied to the case of stationary axially symmetric space-times with two commuting Killing vectors. PACS Nos.: 04.20jb, 04.50+h

scholarly journals Compatible flat metrics

2002 ◽  
Vol 2 (7) ◽  
pp. 337-370 ◽  
Author(s):  
Oleg I. Mokhov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Inverse Scattering ◽  
Nonlinear Partial Differential Equations ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Dressing Method ◽  
Partial Differential ◽  
Differential Reduction ◽  
Component Case

We solve the problem of description of nonsingular pairs of compatible flat metrics for the generalN-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is based on reducing to a special nonlinear differential reduction of the Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).

The Kerr solution

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Kerr Black Hole ◽  
Geodesic Equation ◽  
Einstein Equations ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Axially Symmetric ◽  
Observable Universe ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
The Many

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

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