Deformations of non semisimple Poisson pencils of hydrodynamic type

TL;DR

This paper investigates the structure and parametrization of deformations in non semisimple Poisson pencils of hydrodynamic type, revealing invariant functions and conditions for unobstructed deformations.

Contribution

It characterizes second order deformations of non semisimple Poisson pencils, identifying invariant functions and exploring conditions for unobstructed deformations.

Findings

Second order deformations are mostly parametrized by two functions of one variable.

One function remains invariant under Miura transformations.

Deformations in exceptional cases involve four functions, with two invariants.

Abstract

We study deformations of two-component non semisimple Poisson pencils of hydrodynamic type associated with Balinski\v{\i}-Novikov algebras. We show that in most cases the second order deformations are parametrized by two functions of a single variable. It turns out that one function is invariant with respect to the subgroup of Miura transformations preserving the dispersionless limit and another function is related to a one-parameter family of truncated structures. In two expectional cases the second order deformations are parametrized by four functions. Among them two are invariants and two are related to a two-parameter family of truncated structures. We also study the lift of deformations of n-component semisimple structures. This example suggests that deformations of non semisimple pencils corresponding to the lifted invariant parameters are unobstructed.