The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions

TL;DR

This paper applies the inverse spectral transform to the Dunajski hierarchy of dispersionless PDEs, solving the Cauchy problem and analyzing the diverse long-term behaviors of solutions, including smooth evolution and finite-time singularities.

Contribution

It extends the inverse spectral transform method to the Dunajski hierarchy, providing formal solutions and insights into the long-time dynamics of these integrable dispersionless PDEs.

Findings

Solutions can exhibit either smooth evolution or gradient catastrophe.

Different equations in the hierarchy have markedly different long-term behaviors.

The method provides a formal approach to the Cauchy problem for these PDEs.

Abstract

In this paper we apply the formal Inverse Spectral Transform for integrable dispersionless PDEs arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S. V. Manakov and one of the authors, to one distinguished class of equations, the so-called Dunajski hierarchy. We concentrate, for concreteness, i) on the system of PDEs characterizing a general anti-self-dual conformal structure in neutral signature, ii) on its first commuting flow, and iii) on some of their basic and novel reductions. We formally solve their Cauchy problem and we use it to construct the longtime behavior of solutions, showing, in particular, that unlike the case of soliton PDEs, different dispersionless PDEs belonging to the same hierarchy of commuting flows evolve in time in very different ways, exhibiting either a smooth dynamics or a gradient catastrophe at finite time.