On the solutions of the second heavenly and Pavlov equations

TL;DR

This paper applies inverse scattering and Riemann-Hilbert techniques to analyze solutions of the heavenly and Pavlov equations, demonstrating their global existence, long-term behavior, and implicit solution classes.

Contribution

It extends inverse scattering methods to the heavenly and Pavlov equations, showing solutions do not break and characterizing their long-term dynamics and implicit solutions.

Findings

Localized solutions of heavenly and Pavlov equations do not break over time.

The long-term behavior of these solutions is explicitly constructed.

A class of implicit solutions for the heavenly equation is characterized.

Abstract

We have recently solved the inverse scattering problem for one parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, like the heavenly equation of Plebanski, the dispersionless Kadomtsev - Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one dimensional vector fields, like the Pavlov equation. We also showed that the associated Riemann-Hilbert inverse problems are powerfull tools to establish if the solutions of the Cauchy problem break at finite time,to construct their longtime behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; ii) constructing the longtime behaviour of the solutions of their Cauchy problems; iii) characterizing a distinguished class of implicit solutions of the heavenly equation.