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Nonlinear electron solutions and their characteristics at infinity

Published online by Cambridge University Press:  17 February 2009

Hilary Booth
Hilary Booth
Affiliation:
Centre for Bioinformation Science, Australian National University, Canberra ACT 0200, Australia; e-mail: hilary.booth@wintermute.anu.edu.au.
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Abstract

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The Maxwell-Dirac equations model an electron in an electromagnetic field. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equation, resulting in a nonlinear system of partial differential equations (PDE's). Well-behaved solutions, within reasonable Sobolev spaces, have been shown to exist globally as recently as 1997 [12]. Exact solutions have not been found—except in some simple cases.

We have shown analytically in [6, 18] that any spherical solution surrounds a Coulomb field and any cylindrical solution surrounds a central charged wire; and in [3] and [19] that in any stationary case, the surrounding electron field must be equal and opposite to the central (external) field. Here we extend the numerical solutions in [6] to a family of orbits all of which are well-behaved numerical solutions satisfying the analytic results in [6] and [11]. These solutions die off exponentially with increasing distance from the central axis of symmetry. The results in [18] can be extended in the same way. A third case is included, with dependence on z only yielding a related fourth-order ordinary differential equation (ODE) [3].

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 51 - 59
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Ablowitz, M. J., Ramani, A. and Segur, H., “A connection between nonlinear evolution equations and ordinary differential equations of P-type. I”, J. Math. Phys. 21 (1980) 715–721.CrossRefGoogle Scholar
[2]Ablowitz, M. J., Ramani, A. and Segur, H., “A connection between nonlinear evolution equations and ordinary differential equations of P-type. II”, J. Math. Phys. 21 (1980) 1006–1018.CrossRefGoogle Scholar
[3]Booth, H. S., “The static Maxwell-Dirac equations”, Ph. D. Thesis, University of New England, 1998.Google Scholar
[4]Booth, H. S., “Various ODE solutions to the static and 1 + 1 Maxwell-Dirac equations”, (in preparation).Google Scholar
[5]Booth, H. S., Jarvis, P. and Legg, G., “Algebraic solution for the vector potential in the Dirac equation”, J. Phys. A: Math. Gen. 34 (2001) 5667–5677.CrossRefGoogle Scholar
[6]Booth, H. S. and Radford, C. J., “The Dirac-Maxwell equations with cylindrical symmetry”, J. Math. Phys. 38 (1997) 1257–1268.CrossRefGoogle Scholar
[7]Chadam, J., “Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac system in one space dimension”, J. Funct. Anal. 13 (1973) 173–184.CrossRefGoogle Scholar
[8]Cieliebak, K. and Séré, E., “Pseudoholomorphic curves and multiplicity of homoclinic orbits”, Duke Math. J. 77 (1995) 483–518.CrossRefGoogle Scholar
[9]Das, A., “General solutions of Maxwell-Dirac equations in 1 + 1 dimensional space-time and a spatially confined solution”, J. Maths. Phys. 34 (1993) 3986–3999.CrossRefGoogle Scholar
[10]Das, A. and Kay, D., “A class of exact plane wave solutions of the Maxwell-Dirac equations”, J. Maths. Phys. 30 (1989) 2280–2284.CrossRefGoogle Scholar
[11]Esteban, M., Georgiev, V. and Séré, E., “Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations”, Calc. Var. 4 (1996) 265–281.CrossRefGoogle Scholar
[12]Flato, M., Simon, J. C. H. and Taflin, E., Asymptotic Completeness, Global Existence and the Infrared Problem for the Maxwell-Dirac Equations, Memoirs of the American Mathematical Society (Amer. Math. Soc., 1997).CrossRefGoogle Scholar
[13]Georgiev, V., “Small amplitude solutions of the Maxwell-Dirac equations”, Indiana Univ. Math. J. 40 (1991) 845–883.CrossRefGoogle Scholar
[14]Glassey, R. T. and Strauss, W. A., “Conservation laws for the Maxwell-Dirac and Klein-Gordon-Dirac equations”, J. Math. Phys. 20 (1979).CrossRefGoogle Scholar
[15]Gross, L., “The Cauchy problem for the coupled Maxwell and Dirac equations”, Comm. Pure Appl. Math. 19 (1966) 1–5.CrossRefGoogle Scholar
[16]Lisi, A. G., “A solitary wave solution of the Maxwell-Dirac equations”, J. Phys. A 28 (1995) 5385–5392.CrossRefGoogle Scholar
[17]Moroz, I. M., Penrose, R. and Tod, P., “Spherically-symmetric solutions of the Schrödinger-Newton equations”, Class. Quantum Gray. 15 (1998) 2733–2742.CrossRefGoogle Scholar
[18]Radford, C. J., “Localised solutions of the Dirac-Maxwell equations”, J. Math. Phys. 37 (1996) 4418–4433.CrossRefGoogle Scholar
[19]Radford, C. J. and Booth, H. S., “Magnetic monopoles, electric neutrality and the static Maxwell-Dirac equations”, J. Phys. A: Math. Gen. 32 (1999) 5807–5822.CrossRefGoogle Scholar
[20]Schwinger, J., “Gauge invariance and mass. II”, Phys. Rev. 128 (1962) 2425–2429.CrossRefGoogle Scholar
[21]Wakano, M., “Intensely localized solutions of the Dirac-Maxwell field equations”, Prog. T Phys. 35 (1966) 1117–1141.CrossRefGoogle Scholar
[22]MATLAB is a trademark of The Mathworks, Inc.Google Scholar
[23]NAG is a registered trademark of The Numerical Algorithms Group, Inc.Google Scholar