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Algebraic properties of Gardner's deformations for integrable systems

Le : 06/12/2006 14h15
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Résumé : We propose a constructive definition of Gardner's deformations for completely integrable bi-Hamiltonians systems;the deformations determine recurrence relations for the Hamiltonians of the integrable hierarchies and also yield new evolutionary systems and exactly solvable hyperbolic equations.The suggested approach to Gardner's deformations through diagrams and the method of representing the Miura substitutions using the ambient Euler-Lagrange Liouville-type hyperbolic systems extend the class of deformable equations and allow to study the deformation cohomology.Secondly, we introduce two types of Gardner's adjoint integrable systems by inspecting how the deformations act of the Poisson structures of the hierarchies.New deformations of the (Kaup-)Boussinesq equations are thus found and an exactly solvable extension of the Liouville equation is discovered.