The Cauchy problem for the Pavlov equation

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.

In the new version the analytical technique was essentially revised. The previous version contained a wrong statement about the solvability of the inverse problem for large data. This problem remains open

References in corpus (5)

• The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behavior, implicit solutions and wave breaking
• On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking
• A hierarchy of integrable PDEs in 2+1 dimensions associated with 2 - dimensional vector fields
• Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
• Regularization of Hele-Shaw flows, multiscaling expansions and the Painleve I equation

Cited by in corpus (1)

• Nonlocality and the inverse scattering transform for the Pavlov equation