Multidimensional Hamiltonian Systems and their Deformations

由我们在拉夫堡的小组率先，最近在基于流体动力约简及其弥散变形的方法的多维可积系统分类方面取得了突破。在目前的博士项目中，我们计划将该技术应用于哈密顿（无色散和色散）可积系统的分类。更具体地说，该项目将涉及以下关键步骤：对应于局部哈密顿算子d/d的可积非局部哈密顿密度的分类。利用基于Haantjes张量消失的可积性准则研究了涉及高非定域的相应可积层次结构。证明了基于水动力约简方法的可积性等价于方程主符号所定义的共形结构在每一个解上都是爱因斯坦-魏尔的要求。基于水动力约简变形法和无色散Lax对量化方法的多维色散变形构造。

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.