The Cauchy problem for the Pavlov equation

TL;DR

This paper addresses the scattering and inverse scattering problems for the Pavlov equation, an integrable dispersionless PDE, and establishes the existence of global solutions for small initial data, advancing the mathematical understanding of such equations.

Contribution

It provides the first rigorous analysis of the scattering problem for the Pavlov equation and proves the existence of global solutions for small initial data.

Findings

Successfully formulated the scattering and inverse scattering framework for the Pavlov equation.

Proved the existence of global bounded solutions for small initial data.

Enhanced the mathematical theory of integrable dispersionless PDEs.

Abstract

Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature. This report is aiming to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.