Abstract
For certain 1 + 1-dimensional classical field theories, whose equations of motion can naturally be written in Lax form by introducing a quasi-dynamical spectral parameter (using sdiff2, the Lie algebra of symplectic diffeomorphisms of a two-dimensional manifold), the previously derived Poisson-commutativity of an infinite set of conserved charges also follows from the existence of an ‘γ-function’, whose functional form is given, and checked explicitly (as well as its Yang-Baxter equation) for an interaction that is exponential (respectively, inverse square) in the spatial derivative of the field.
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References
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By a different method, the integrability of these models was already shown in Bogoyavlensky, O. I.,Math. USSR Izv. 32, 245 (1989) where a hodograph transformation is used to prove the linearizability of the Lagrangian equations of motion.
See, e.g., Perelomov, A. M.,Integrable Systems of Classical Mechanics and Lie Algebras, Birkhäuser, Basel, 1990.