Common structure of several completely integrable non-linear equations

Abstract

The author consider several non-linear equations which are known to be completely integrable by the method of inverse scattering transform, ranging from sine-Gordon and the Dodd-Bullough equation (1976) to the axisymmetric, stationary, Einstein-Maxwell equations. Such equations are known to be invariant under a fundamental Lie group of symmetry (G), such as SL(2) or SL(3) in the simpler cases. The author's treatment starts with the reformulation of the given equations as a system involving the scalar invariants of the Lie group (G) only; this scalar formulation is not only (G) invariant, but also manifestly invariant under the continuous group associated with the emergence of a spectral parameter, and is thus expected to be more fundamental and simpler. In all the cases that the author has considered, including the Einstein-Maxwell system, vector pseudopotentials of dimension not higher than three have been derived; Backlund transformations have been found, which generate multisolitons starting from the vacuum, and have the general (reciprocal) form: X'=1/X where X is a component of one of the vector (or tensor) pseudopotentials. The author also mentions the result that Kinnersley's formulation (1977) of the Ernst (vacuum) equation (1968) is formally equivalent to the fluid dynamical problem of unidimensional ideal gas flow; the Dodd-Bullough equation is equivalent to another particular case of gas motion.