Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine sl2 Gaudin model

In this work we will use a general procedure to construct higher local Hamiltonians for the affine sl(2 )Gaudin model. We focus on the first non-trivial example, the quartic Hamiltonians. We show by direct calculation that the quartic Hamiltonians commute amongst themselves and with the quadratic Hamiltonians which define the model. We go on to introduce a certain next-to-leading order semi-classical limit of the model. In this limit, we are able to write down the full hierarchy of higher local Hamiltonians and prove that they commute.