We assign a Riemannian metric to a system of nonlinear equations that describe the one-dimensional propagation of long magnetoacoustic waves (also called magnetosonic waves) in a cold plasma under the inference of a transverse magnetic field. The metric, which in general is expressed in terms of the density of the plasma and its speed across the magnetic field, when specialized to a particular solution of the nonlinear system (the Gurevich-Krylov (G-K) solution) is mapped explicitly to a Jackiw-Teitelboim (J-T) black hole metric, which is the main result. Dilaton fields, constructed from data involved in the G-K solution, are presented - which with the plasma metric provide for elliptic function solutions of the J-T equations of motion in 2d dilaton gravity. A correspondence between solutions of the nonlinear plasma system (whose Galilean invariance is also established) and certain solutions of a resonant nonlinear Schrödinger equation is set up, along with some other general background material to render an expository tone in the presentation.
Citation: Floyd L. Williams. From a magnetoacoustic system to a J-T black hole: A little trip down memory lane[J]. Communications in Analysis and Mechanics, 2023, 15(3): 342-361. doi: 10.3934/cam.2023017
We assign a Riemannian metric to a system of nonlinear equations that describe the one-dimensional propagation of long magnetoacoustic waves (also called magnetosonic waves) in a cold plasma under the inference of a transverse magnetic field. The metric, which in general is expressed in terms of the density of the plasma and its speed across the magnetic field, when specialized to a particular solution of the nonlinear system (the Gurevich-Krylov (G-K) solution) is mapped explicitly to a Jackiw-Teitelboim (J-T) black hole metric, which is the main result. Dilaton fields, constructed from data involved in the G-K solution, are presented - which with the plasma metric provide for elliptic function solutions of the J-T equations of motion in 2d dilaton gravity. A correspondence between solutions of the nonlinear plasma system (whose Galilean invariance is also established) and certain solutions of a resonant nonlinear Schrödinger equation is set up, along with some other general background material to render an expository tone in the presentation.
| [1] |
J. H. Lee, O. K. Pashaev, C. Rogers, W. K. Schief, The resonant nonlinear Schrödinger equation in cold plasma physics. Applications of Bäcklund-Darboux transformations and superposition principles, J. Plasma Physics, 73 (2007), 257–272. https://doi.org/10.1017/S0022377806004648 doi: 10.1017/S0022377806004648
|
| [2] | V. I. Karpman, Nonlinear waves in dispersive media, Pergamon, Oxford, 1975. https://doi.org/10.1016/B978-0-08-017720-5.50008-7 |
| [3] | A. I. Akhiezer, Plasma electrodynamics, Pergamon, Oxford, 1975. |
| [4] |
J. H. Lee, O. K. Pashaev, Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions and Hirota bilinear method, Theoretical and Mathematical Physics, 152 (2007), 991–1003. https://doi.org/10.1007/s11232-007-0083-3 doi: 10.1007/s11232-007-0083-3
|
| [5] |
O. K. Pashaev, J. H. Lee, Resonance solitons as black holes in Madelung fluid, Modern Physics Letters A, 17(2002), 1601–1619. https://doi.org/10.1142/S0217732302007995 doi: 10.1142/S0217732302007995
|
| [6] | F. Williams, Exploring a cold plasma - 2d black hole connection, Advances in Mathematical Physics, Hindawi. |
| [7] |
J. D'Ambroise, F. Williams, Relating some nonlinear systems to a cold plasma magnetoacoustic system, J. Modern physics, 11 (2020), 886–906. https://doi.org/10.4236/jmp.2020.116054 doi: 10.4236/jmp.2020.116054
|
| [8] | F. Williams, Some musings on theta, eta, and zeta: From $E_8$ to cold plasma to an inhomogeneous universe, accepted for publication in Springer Book series: Mathematical Physics Studies, Springer, 2023. |
| [9] | L. Martina, O. K. Pashaev, G. Soliani, Bright solitons as black holes, Physical Rev. D, 58 (1998). https://doi.org/10.1103/PhysRevD.58.084025 |
| [10] | K. Chandrasekharan, Elliptic functions, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. https://doi.org/10.1007/978-3-642-52244-4 |
| [11] | A. V. Gurevich, A. L. Krylov, A shock wave in dispersive hydrodynamics, Soviet Physics Doklady, 32 (1988), 73–74. |
| [12] | T. P. Horikis, P. G. Kevrekidis, F. Tsitoura, F. Williams, Solitary waves in the resonant nonlinear Schrödinger equation: Stability and dynamical properties, Physics Letters A, 2020. http://doi.org/10.1016/j-physleta.2020.126441 |
| [13] | J. D'Ambroise, F. Williams, Elliptic function solutions in Jackiw-Teitelboim dilaton gravity, Hindawi. https://doi.org/10.1155/2017/2154784 |
| [14] | F. Williams, Some selected thoughts old and new on soliton-black hole connections in 2d dilaton gravity, in: The sine-Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High Energy Physics, J.C.-Maraver, P.G.Kevrekidis, F. Williams, Editors, Springer (2014), 171–205. https://doi.org/10.1007/978-3-319-06722-3-8 |
| [15] | R. Jackiw, A two dimensional model for gravity, in: Quantum Theory of Gravity, S.Christensen, Editor, Adam Hilger Ltd. (1984), 403–420. |
| [16] | C. Teitelboim, The Hamiltonian structure of two-dimensional spacetime and its relation with the conformal anomaly, in: Quantum Theory of Gravity, S.Christensen, Editor, Adam Hilger Ltd. (1984), 327–344. |
| [17] | L. Martina, O. K. Pashaev, G. Soliani, Integrable dissipative structures in the gauge theory of gravity, Classical and Quantum Gravity, 14 (1977). https://doi.org/10.1088/0264-9381/14/12/005 |
Floyd L. Williams. From a magnetoacoustic system to a J-T black hole: A little trip down memory lane[J]. Communications in Analysis and Mechanics, 2023, 15(3): 342-361. doi: 10.3934/cam.2023017
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