Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields

S V Manakov,P M Santini

doi:10.1088/1742-6596/482/1/012029

Abstract

In this paper we review some results about the theory of integrable dispersionless PDEs arising as commutation condition of pairs of one-parameter families of vector fields, developed by the authors during the last years. We review, in particular, the basic formal aspects of a novel Inverse Spectral Transform including, as inverse problem, a nonlinear Riemann – Hilbert (NRH) problem, allowing one i) to solve the Cauchy problem for the target PDE; ii) to construct classes of RH spectral data for which the NRH problem is exactly solvable, corresponding to distinguished examples of exact implicit solutions of the target PDE; iii) to construct the longtime behavior of the solutions of such PDE; iv) to establish in a simple way if a localized initial datum breaks at finite time and, if so, to study analytically how the multidimensional wave breaks. We also comment on the existence of recursion operators and Backlünd – Darboux transformations for integrable dispersionless PDEs.

Highlights

Waves propagating in weakly nonlinear and dispersive media are well described by integrable soliton equations, like the Korteweg - de Vries [1], the Nonlinear Scrhodinger [2] equations and their integrable (2 + 1) dimensional generalizations, the Kadomtsev - Petviashvili [3] and Davey - Stewartson [4] equations respectively
The Inverse Spectral Transform (IST), introduced by Gardner, Green, Kruskal and Miura [5], is the spectral method allowing one to solve the Cauchy problem for such PDEs, predicting that a localized disturbance evolves into a number of soliton pulses + radiation, and solitons arise as an exact balance between nonlinearity and dispersion [6]-[9]
There is another important class of integrable PDEs, the so-called dispersionless PDEs, or PDEs of hydrodynamic type, arising in various problems of Mathematical Physics and intensively studied in the recent literature

Summary

Introduction

Waves propagating in weakly nonlinear and dispersive media are well described by integrable soliton equations, like the Korteweg - de Vries [1], the Nonlinear Scrhodinger [2] equations and their integrable (2 + 1) dimensional generalizations, the Kadomtsev - Petviashvili [3] and Davey - Stewartson [4] equations respectively. Due to the lack of dispersion, these multidimensional PDEs may or may not exhibit a gradient catastrophe at finite time, and their integrability gives a unique chance to study analytically such a mechanism With this motivation, a novel IST for vector fields, significantly different from that of soliton PDEs [6, 7, 9], has been recently constructed [35, 36, 37] i) to solve the Cauchy problem for dPDEs, ii) obtain the longtime behavior of solutions, iii) costruct distinguished classes of exact implicit solutions, iv) establish if, due to the lack of dispersion, the nonlinearity of the PDE is “strong enough” to cause the gradient catastrophe of localized multidimensional disturbances, and v) study analytically the breaking mechanism [35]-[43]. This paper is dedicated to the memory of Sergey, a great scientist and a loyal friend

Two basic examples

The IST for vector fields

Recursion operators and hierarchies of dispersionless PDEs