Journal of Physics A Mathematical and Theoretical · 2011 · 11 citations · 20 references
Natural SciencesMathematicsPhysicsDifferential EquationsMathematical PhysicsTheoretical PhysicsHamiltonian MechanicsIntegral Equation FormulationsIntegral EquationsLie AlgebrasMultiple IntegralsSymplectic GeometryExplicit FormulaeKorteweg - De Vries EquationPartial Difference EquationsPoisson BracketsSine-gordon EquationSymplectic FormsSymplectic IntegrationSymplectic Maps
Closed form expressions in terms of multi-sums of products have been given in \cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called $(p,-1)$-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products.
20