Towards Lax formulation of integrable hierarchies of topological type (Q2447582)

The purpose of this paper is to establish an understanding of the connection between the following two approaches: {\parindent=6mm \begin{itemize} \item[-] The action of the Givental group on the Hirota quadratic equations (HQEs) of different hierarchies \item [-] The Dubrovin-Zhang hierarchies in Hamiltonian form \end{itemize}} This is done by looking at infinitesimal deformations of several copies of the KdV hierarchy. The deformed equations can be expressed as \(N\) copies of Lax (or Sato-Wilson) form in terms of pseudodifferential operators, where the non-trivial integral part is completely determined by the differential part. To derive the Sato-Wilson equations from the HQE, a fundamental lemma is proved. Then, the Givental group action on vertex operators is computed followed by a deformation of the Sato-Wilson equations. Finally, a comparison with deformations in Hamiltonian form is made.