Abstract
It is pointed out that the Bäcklund transformations for a physically interesting class of nonlinear partial differential equations can be interpreted as generalisations of the Cauchy Riemann equations or as nonlinear Dirac equations. The generalisations are inhomogenisations of the Cauchy Riemann equations (or their hyperbolic analogue), whose condensed form makes the transformations easy to remember, which suggests ways to generalise to more than 2 dimensions, and which suggest that complex analysis techniques may be helpful in understanding the transformations.
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References
Crampin, M., Hogkin, L., McCarthy., P.J., and Robinson, D.W., ‘2-Manifolds of Constant Curvature, 3-Parameter Isometry Groups and Bäcklund Transformations’, preprint (1977).
McCarthy, P.J. and Newell, A. ‘Exact Solutions to φ xx + φ yy = sinh φ’, to be published.
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McCarthy, P.J. Bäcklund transformations as nonlinear Dirac equations. Lett Math Phys 2, 167–170 (1977). https://doi.org/10.1007/BF00398583
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DOI: https://doi.org/10.1007/BF00398583