Multidimensional Hamiltonian Systems and their Deformations

Pioneered by our group in Loughborough, there has been recently a breakthrough in the classification of multi-dimensional integrable systems based on the method of hydrodynamic reductions and their dispersive deformations. In the current PhD project we plan to apply this techniques to the classification of Hamiltonian (dispersionless and dispersive) integrable systems. More concretely, the project will involve the following key steps:
1. Classification of integrable nonlocal Hamiltonian densities corresponding to the local Hamiltonian operator d/dx.
2. Investigating the structure of the corresponding integrable hierarchies involving higher nonlocalities using the integrability criterion based on the vanishing of the Haantjes tensor.
3. Proving that integrability based on the method of hydrodynamic reductions is equivalent to the requirement that the conformal structure defined by the principal symbol of the equation is Einstein-Weyl on every solution.
4. Construction of multi-dimensional dispersive deformations based on the method of deformations of hydrodynamic reductions and the method of quantisation of dispersionless Lax pairs.