Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case

TL;DR

This paper investigates the properties of bihamiltonian cohomologies related to hydrodynamic type structures and demonstrates the existence of deformations for the dispersionless KdV hierarchy through cohomology computations.

Contribution

It provides new theoretical results on bihamiltonian cohomologies and explicitly computes the third cohomology for the dispersionless KdV hierarchy, establishing deformation existence.

Findings

Computed the third cohomology for the bihamiltonian structure of dispersionless KdV.

Proved the existence of bihamiltonian deformations from any infinitesimal deformation.

Established theoretical properties of bihamiltonian cohomologies in hydrodynamic type structures.

Abstract

We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.