RUS  ENG
Ïîëíàÿ âåðñèÿ
ÆÓÐÍÀËÛ // Symmetry, Integrability and Geometry: Methods and Applications

SIGMA, 2013, òîì 9, 009, 31 ñòð. (Mi sigma792)

Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
Aristophanes Dimakis, Folkert Müller-Hoissen

Ñïèñîê ëèòåðàòóðû

1. Arsie A., Lorenzoni P., “$F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard–Magri chains”, Int. Math. Res. Not. (to appear), arXiv: 1110.2461  crossref
2. Belinski V. A., Verdaguer E., Gravitational solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2001  mathscinet  zmath
3. Belinskiĭ V. A., Sakharov V. E., “Stationary gravitational solitons with axial symmetry”, Sov. Phys. JETP, 50 (1979), 1–9  mathscinet  adsnasa
4. Belinskiĭ V. A., Zakharov V. E., “Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions”, Sov. Phys. JETP, 48 (1978), 985–994  mathscinet  adsnasa
5. Breitenlohner P., Maison D., Gibbons G., “$4$-dimensional black holes from Kaluza–Klein theories”, Comm. Math. Phys., 120 (1988), 295–333  crossref  mathscinet  zmath  adsnasa  isi
6. Camacaro J., Cariñena J., “Alternative Lie algebroid structures and bi-differential calculi”, Applied Differential Geometry and Mechanics, eds. W. Sarlet, F. Cantrijn, University of Gent, 2003, 1–20
7. Chavchanidze G., “Non-Noether symmetries in Hamiltonian dynamical systems”, Mem. Differential Equations Math. Phys., 36 (2005), 81–134, arXiv: math-ph/0405003  mathscinet  zmath
8. Chen Y., Hong K., Teo E., “Unbalanced Pomeransky–Sen'kov black ring”, Phys. Rev. D, 84 (2011), 084030, 11 pp., arXiv: 1108.1849  crossref  adsnasa  isi
9. Chen Y., Teo E., “Rod-structure classification of gravitational instantons with $U(1)\times U(1)$ isometry”, Nuclear Phys. B, 838 (2010), 207–237, arXiv: 1004.2750  crossref  mathscinet  zmath  adsnasa  isi
10. Chruściel P., Cortier J., “Maximal analytic extensions of the Emparan–Reall black ring”, J. Phys. Conf. Ser., 229 (2010), 012030, 4 pp., arXiv: 0807.2309  crossref  mathscinet  adsnasa
11. Chruściel P. T., Eckstein M., Szybka S. J., “On smoothness of black saturns”, J. High Energy Phys., 2010:11 (2010), 048, 39 pp., arXiv: 1007.3668  crossref  mathscinet  isi
12. Cieśliński J. L., “Algebraic construction of the Darboux matrix revisited”, J. Phys. A: Math. Theor., 42 (2009), 404003, 40 pp., arXiv: 0904.3987  crossref  mathscinet  zmath
13. Crampin M., Sarlet W., Thompson G., “Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors”, J. Phys. A: Math. Gen., 33 (2000), 8755–8770  crossref  mathscinet  zmath  adsnasa
14. de Souza E., Bhattacharyya S. P., “Controllability, observability and the solution of $AX-XB=C$”, Linear Algebra Appl., 39 (1981), 167–188  crossref  mathscinet  zmath  isi
15. Dimakis A., Kanning N., Müller-Hoissen F., “The non-autonomous chiral model and the Ernst equation of general relativity in the bidifferential calculus framework”, SIGMA, 7 (2011), 118, 27 pp., arXiv: 1106.4122  mathnet  crossref  mathscinet  zmath  adsnasa  isi
16. Dimakis A., Müller-Hoissen F., “Bi-differential calculi and integrable models”, J. Phys. A: Math. Gen., 33 (2000), 957–974, arXiv: math-ph/9908015  crossref  mathscinet  adsnasa
17. Dimakis A., Müller-Hoissen F., “Bicomplexes and integrable models”, J. Phys. A: Math. Gen., 33 (2000), 6579–6591, arXiv: nlin.SI/0006029  crossref  mathscinet  zmath  adsnasa
18. Dimakis A., Müller-Hoissen F., “Bidifferential calculus approach to AKNS hierarchies and their solutions”, SIGMA, 6 (2010), 055, 27 pp., arXiv: 1004.1627  mathnet  crossref  mathscinet  zmath  isi
19. Dimakis A., Müller-Hoissen F., “Bidifferential graded algebras and integrable systems”, Discrete Contin. Dyn. Syst., 2009, suppl., 208–219, arXiv: 0805.4553  mathscinet  zmath
20. Dimakis A., Müller-Hoissen F., “Solutions of matrix NLS systems and their discretizations: a unified treatment”, Inverse Problems, 26 (2010), 095007, 55 pp., arXiv: 1001.0133  crossref  mathscinet  zmath  adsnasa  isi
21. Economou A., Tsoubelis D., “Multiple-soliton solutions of Einstein's equations”, J. Math. Phys., 30 (1989), 1562–1569  crossref  mathscinet  zmath  adsnasa  isi
22. Elvang H., Figueras P., “Black saturn”, J. High Energy Phys., 2007:5 (2007), 050, 48 pp., arXiv: hep-th/0701035  crossref  mathscinet  adsnasa
23. Elvang H., Rodriguez M. J., “Bicycling black rings”, J. High Energy Phys., 2008:4 (2008), 045, 30 pp., arXiv: 0712.2425  crossref  mathscinet  zmath
24. Emparan R., Reall H. S., “A rotating black ring solution in five dimensions”, Phys. Rev. Lett., 88 (2002), 101101, 4 pp., arXiv: hep-th/0110260  crossref  mathscinet  adsnasa  isi
25. Emparan R., Reall H. S., Black holes in higher dimensions, Living Rev. Relativ., 11, no. 6, 2008, 87 pp., arXiv: 0801.3471  zmath
26. Emparan R., Reall H. S., “Black rings”, Classical Quantum Gravity, 23 (2006), R169–R197, arXiv: hep-th/0608012  crossref  mathscinet  zmath  isi
27. Emparan R., Reall H. S., “Generalized Weyl solutions”, Phys. Rev. D, 65 (2002), 084025, 26 pp., arXiv: hep-th/0110258  crossref  mathscinet  adsnasa  isi
28. Evslin J., Krishnan C., “The black di-ring: an inverse scattering construction”, Classical Quantum Gravity, 26 (2009), 125018, 13 pp., arXiv: 0706.1231  crossref  mathscinet  zmath  adsnasa  isi
29. Figueras P., “A black ring with a rotating 2-sphere”, J. High Energy Phys., 2005:7 (2005), 039, 9 pp., arXiv: hep-th/0505244  crossref  mathscinet
30. Figueras P., Jamsin E., Rocha J. V., Virmani A., “Integrability of five-dimensional minimal supergravity and charged rotating black holes”, Classical Quantum Gravity, 27 (2010), 135011, 37 pp., arXiv: 0912.3199  crossref  mathscinet  zmath  adsnasa  isi
31. Frölicher A., Nijenhuis A., “Theory of vector-valued differential forms. I: Derivations in the graded ring of differential forms”, Proc. Koninkl. Ned. Acad. Wetensch. Ser. A, 59 (1956), 338–359  mathscinet
32. Frolov V. P., Goswami R., “Surface geometry of 5D black holes and black rings”, Phys. Rev. D, 75 (2007), 124001, 11 pp., arXiv: gr-qc/0612033  crossref  mathscinet  adsnasa  isi
33. Gauntlett J. P., Gutowski J. B., “Concentric black rings”, Phys. Rev. D, 71 (2005), 025013, 7 pp., arXiv: hep-th/0408010  crossref  mathscinet  adsnasa  isi
34. Griffiths J. B., Colliding plane waves in general relativity, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York  mathscinet
35. Gu C., Hu H., Zhou Z., Darboux transformations in integrable systems. Theory and their applications to geometry, Mathematical Physics Studies, 26, Springer, Dordrecht, 2005  mathscinet
36. Harmark T., “Stationary and axisymmetric solutions of higher-dimensional general relativity”, Phys. Rev. D, 70 (2004), 124002, 25 pp., arXiv: hep-th/0408141  crossref  mathscinet  adsnasa  isi
37. Hartwig R. E., “Resultants and the solution of $AX-XB=-C$”, SIAM J. Appl. Math., 23 (1972), 104–117  crossref  mathscinet  zmath
38. Hearon J. Z., “Nonsingular solutions of $TA-BT=C$”, Linear Algebra Appl., 16 (1977), 57–63  crossref  mathscinet  zmath
39. Herdeiro C., Rebelo C., Zilhão M., Costa M., “A double Myers–Perry black hole in five dimensions”, J. High Energy Phys., 2008:7 (2008), 009, 24 pp., arXiv: 0805.1206  crossref  mathscinet
40. Hollands S., Yazadjiev S., “Uniqueness theorem for 5-dimensional black holes with two axial Killing fields”, Comm. Math. Phys., 283 (2008), 749–768, arXiv: 0707.2775  crossref  mathscinet  zmath  adsnasa  isi
41. Hong K., Teo E., “A new form of the $C$-metric”, Classical Quantum Gravity, 20 (2003), 3269–3277, arXiv: gr-qc/0305089  crossref  mathscinet  zmath  adsnasa  isi
42. Horowitz G. (ed.), Black holes in higher dimensions, Cambridge University Press, Cambridge, 2012  adsnasa
43. Hoskisson J., Explorations of four and five dimensional black hole spacetimes, Ph. D. thesis, Durham University, 2009 http://etheses.dur.ac.uk/2115/
44. Hu Q., Cheng D., “The polynomial solution to the Sylvester matrix equation”, Appl. Math. Lett., 19 (2006), 859–864  crossref  mathscinet  zmath  isi
45. Iguchi H., Izumi K., Mishima T., “Systematic solution-generation of five-dimensional black holes”, Progr. Theoret. Phys. Suppl., 189 (2011), 93–125, arXiv: 1106.0387  crossref  adsnasa
46. Iguchi H., Mishima T., “Solitonic generation of vacuum solutions in five-dimensional general relativity”, Phys. Rev. D, 74 (2006), 024029, 17 pp., arXiv: hep-th/0605090  crossref  mathscinet  adsnasa  isi
47. Izumi K., “Orthogonal black di-ring solution”, Progr. Theoret. Phys., 119 (2008), 757–774, arXiv: 0712.0902  crossref  adsnasa  isi
48. Klein C., Richter O., Ernst equation and Riemann surfaces. Analytical and numerical methods, Lecture Notes in Physics, 685, Springer-Verlag, Berlin, 2005  mathscinet  zmath
49. Kodama H., Hikida W., “Global structure of the Zipoy–Voorhees–Weyl spacetime and the $\delta=2$ Tomimatsu–Sato spacetime”, Classical Quantum Gravity, 20 (2003), 5121–5140, arXiv: gr-qc/0304064  crossref  mathscinet  zmath  adsnasa  isi
50. Kramer D., Neugebauer G., “The superposition of two Kerr solutions”, Phys. Lett. A, 75 (1980), 259–261  crossref  mathscinet  adsnasa  isi
51. Lorenzoni P., “Flat bidifferential ideals and semi-Hamiltonian PDEs”, J. Phys. A: Math. Gen., 39 (2006), 13701–13715, arXiv: nlin.SI/0604053  crossref  mathscinet  zmath  adsnasa
52. Lorenzoni P., Magri F., “A cohomological construction of integrable hierarchies of hydrodynamic type”, Int. Math. Res. Not., 2005:34 (2005), 2087–2100, arXiv: nlin.SI/0504064  crossref  mathscinet  zmath
53. Marchenko V. A., Nonlinear equations and operator algebras, Mathematics and its Applications (Soviet Series), 17, D. Reidel Publishing Co., Dordrecht, 1988  crossref  mathscinet  zmath
54. Matveev V. B., Salle M. A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991  crossref  mathscinet
55. Mishima T., Iguchi H., “New axisymmetric stationary solutions of five-dimensional vacuum Einstein equations with asymptotic flatness”, Phys. Rev. D, 73 (2006), 044030, 6 pp., arXiv: hep-th/0504018  crossref  mathscinet  adsnasa  isi
56. Myers R. C., “Myers–Perry black holes”, Black Holes in Higher Dimensions, ed. G. Horowitz, Cambridge University Press, Cambridge, 2012, 101–133, arXiv: 1111.1903  crossref  zmath
57. Myers R. C., Perry M. J., “Black holes in higher-dimensional space-times”, Ann. Physics, 172 (1986), 304–347  crossref  mathscinet  zmath  adsnasa  isi
58. Nakamura Y., “Symmetries of stationary axially symmetric vacuum Einstein equations and the new family of exact solutions”, J. Math. Phys., 24 (1983), 606–609  crossref  mathscinet  zmath  adsnasa  isi
59. Nimmo J. J. C., Gilson C., Ohta Y., “Applications of Darboux transformations to the self-dual Yang–Mills equations”, Theoret. and Math. Phys., 122 (2000), 239–246  mathnet  crossref  mathscinet  adsnasa  isi
60. Pomeransky A. A., “Complete integrability of higher-dimensional Einstein equations with additional symmetry and rotating black holes”, Phys. Rev. D, 73 (2006), 044004, 5 pp., arXiv: hep-th/0507250  crossref  mathscinet  adsnasa  isi
61. Pomeransky A. A., Sen'kov R. A., Black ring with two angular momenta, arXiv: hep-th/0612005  zmath
62. Rogers C., Schief W. K., Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002  mathscinet
63. Sakhnovich A. L., “Dressing procedure for solutions of nonlinear equations and the method of operator identities”, Inverse Problems, 10 (1994), 699–710  crossref  mathscinet  zmath  adsnasa  isi
64. Sakhnovich A. L., “Generalized Bäcklund–Darboux transformation: spectral properties and nonlinear equations”, J. Math. Anal. Appl., 262 (2001), 274–306  crossref  mathscinet  zmath  isi
65. Sakhnovich A. L., “On the GBDT version of the Bäcklund–Darboux transformation and its applications to linear and nonlinear equations and Weyl theory”, Math. Model. Nat. Phenom., 5 (2010), 340–389, arXiv: 0909.1537  crossref  mathscinet  zmath  isi
66. Sakhnovich L. A., “Problems of factorization and operator identities”, Russ. Math. Surv., 41:1 (1986), 1–64  mathnet  crossref  mathscinet  zmath  isi
67. Sparano G., Vilasi G., Vinogradov A. M., “Vacuum Einstein metrics with bidimensional Killing leaves. I: Local aspects”, Differential Geom. Appl., 16 (2002), 95–120, arXiv: gr-qc/0301020  crossref  mathscinet  zmath  isi
68. Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E., Exact solutions of Einstein's field equations, Cambridge Monographs on Mathematical Physics, 2nd ed., Cambridge University Press, Cambridge, 2003  mathscinet
69. Szybka S. J., “Stable causality of Black Saturns”, J. High Energy Phys., 2011:5 (2011), 052, 9 pp., arXiv: 1102.3942  crossref  mathscinet  isi
70. Tan H. S., Teo E., “Multi-black-hole solutions in five dimensions”, Phys. Rev. D, 68 (2003), 044021, 11 pp., arXiv: hep-th/0306044  crossref  mathscinet  zmath  adsnasa  isi
71. Tangherlini F. R., “Schwarzschild field in $n$ dimensions and the dimensionality of space problem”, Nuovo Cimento, 27 (1963), 636–651  crossref  mathscinet  zmath
72. Tomimatsu A., Sato H., “New exact solution for the gravitational field of a spinning mass”, Phys. Rev. Lett., 29 (1972), 1344–1345  crossref  adsnasa
73. Tomizawa S., Ishihara H., Exact solutions of higher dimensional black holes, arXiv: 1104.1468
74. Tomizawa S., Morisawa Y., Yasui Y., “Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method”, Phys. Rev. D, 73 (2006), 064009, 8 pp., arXiv: hep-th/0512252  crossref  mathscinet  adsnasa  isi
75. Tomizawa S., Nozawa M., “Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method. II: Production of the black ring solution”, Phys. Rev. D, 73 (2006), 124034, 10 pp., arXiv: hep-th/0604067  crossref  mathscinet  adsnasa  isi
76. Tomizawa S., Uchida Y., Shiromizu M., “Twist of stationary black hole or ring in five dimensions”, Phys. Rev. D, 70 (2004), 064020, 5 pp., arXiv: gr-qc/0405134  crossref  mathscinet  adsnasa  isi
77. Verdaguer E., “Soliton solutions in spacetime with two spacelike Killing fields”, Phys. Rep., 229 (1993), 1–80  crossref  mathscinet  zmath  adsnasa  isi
78. Yazadjiev S. S., “5D Einstein–Maxwell solitons and concentric rotating dipole black rings”, Phys. Rev. D, 78 (2008), 064032, 11 pp., arXiv: 0805.1600  crossref  mathscinet  adsnasa  isi
79. Yazadjiev S. S., “Black saturn with a dipole ring”, Phys. Rev. D, 76 (2007), 064011, 8 pp., arXiv: 0705.1840  crossref  adsnasa  isi
80. Yazadjiev S. S., “Completely integrable sector in 5D Einstein–Maxwell gravity and derivation of the dipole black ring solutions”, Phys. Rev. D, 73 (2006), 104007, 7 pp., arXiv: hep-th/0602116  crossref  mathscinet  adsnasa  isi


© ÌÈÀÍ, 2026