Deformations of the Monge/Riemann hierarchy and approximately integrable systems

Abstract

Dispersive deformations of the Monge equation u_u=uu_x are studied using ideas originating from topological quantum field theory and the deformation quantization programme. It is shown that, to a high-order, the symmetries of the Monge equation may also be appropriately deformed, and that, if they exist at all orders, they are uniquely determined by the original deformation. This leads to either a new class of integrable systems or to a rigorous notion of an approximate integrable system. Quasi-Miura transformations are also constructed for such deformed equations.

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References in corpus (3)

• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Deformations of bihamiltonian structures of hydrodynamic type
• Deformations of dispersionless KdV hierarchies

Cited by in corpus (8)

• On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
• On Hamiltonian perturbations of hyperbolic systems of conservation laws
• Dispersive deformations of hydrodynamic reductions of 2D dispersionless integrable systems
• On bi-Hamiltonian deformations of exact pencils of hydrodynamic type
• Poisson bracket on 1-forms and evolutionary partial differential equations
• On integrable conservation laws
• On universality of critical behaviour in Hamiltonian PDEs
• On the Deformations of Dorfman's and Sokolov's Operators