The Maxwell equations as a Bäcklund transformation
Main Article Content
Abstract
Bäcklund transformations (BTs) are a useful tool for integrating nonlinear partial differential equations (PDEs). However, the significance of BTs in linear problems should not be ignored. In fact, an important linear system of PDEs in Physics, namely, the Maxwell equations of Electromagnetism, may be viewed as a BT relating the wave equations for the electric and the magnetic field, these equations representing integrability conditions for solution of the Maxwell system. We examine the BT property of this system in detail, both for the vacuum case and for the case of a linear conducting medium.
Downloads
Article Details
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
References
C. J. Papachristou, Symmetry and integrability of classical field equations, http://arxiv.org/abs/0803.3688.
C. J. Papachristou, Potential symmetries for self-dual gauge fields, Phys. Lett. A 145 (1990) 250.
C. J. Papachristou, Recursion operator and current algebras for the potential SL(N,C) self-dual Yang-Mills equation, Phys. Lett. A 154 (1991) 29.
C. J. Papachristou, Lax pair, hidden symmetries, and infinite sequences of conserved currents for self-dual Yang-Mills fields, J. Phys. A 24 (1991) L 1051.
C. J. Papachristou, Symmetry, conserved charges, and Lax representations of nonlinear field equations: A unified approach, Electron. J. Theor. Phys. 7, No. 23 (2010) 1.
C. J. Papachristou, B. K. Harrison, Backlund-transformation-related recursion operators: Application to the self-dual Yang-Mills equation, J. Nonlin. Math. Phys., Vol. 17, No. 1 (2010) 35.
C. J. Papachristou, Symmetry and integrability of a reduced, 3-dimensional self-dual gauge field model, Electron. J. Theor. Phys. 9, No. 26 (2012) 119.
D. J. Griffiths, Introduction to Electrodynamics, 3rd Edition (Prentice-Hall, 1999).
E. C. Zachmanoglou, D. W. Thoe, Introduction to Partial Differential Equations with Applications (Dover, 1986).