Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking

TL;DR

This paper develops a general method to solve certain integrable dispersionless PDEs using nonlinear Riemann Hilbert problems, providing implicit solutions and analyzing wave breaking phenomena for equations like dKP and heavenly.

Contribution

It introduces a systematic approach to construct solvable nonlinear Riemann Hilbert problems for integrable PDEs related to Hamiltonian vector fields, enabling explicit solution generation.

Findings

Constructed solvable NRH problems for integrable PDEs.

Identified classes of solutions with wave breaking and symmetry reductions.

Applied method to dKP and heavenly equations with explicit examples.

Abstract

We have recently solved the inverse spectral problem for integrable PDEs in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter $\lambda$. The associated inverse problem, in particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on a given contour of the complex $\lambda$ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional dispersionless Toda equations, are real PDEs associated with Hamiltonian vector fields. The corresponding NRH data satisfy suitable reality and symplectic constraints. In this paper, generalizing the examples of solvable NRH problems illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct implicit solutions of such PDEs parametrized by an arbitrary number of real functions of a single variable. Then we illustrate this theory on few distinguished examples for the dKP and heavenly equations. For the dKP case, we characterize a class of similarity solutions, a class of solutions constant on their parabolic wave front and breaking simultaneously on it, and a class of localized solutions breaking in a point of the $(x,y)$ plane. For the heavenly equation, we characterize two classes of symmetry reductions.