English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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

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