Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields
Abstract
In this paper we review some results about the theory of integrable dispersionless PDEs arising as commutation condition of pairs of oneparameter 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.
 Publication:

Journal of Physics Conference Series
 Pub Date:
 March 2014
 DOI:
 10.1088/17426596/482/1/012029
 arXiv:
 arXiv:1312.2740
 Bibcode:
 2014JPhCS.482a2029M
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 17 pages, 1 figure. Written rendition of the talk presented by one of the authors (PMS) at the PMNP 2013 Conference, in a special session dedicated to the memory of S. V. Manakov. To appear in the Proceedings of the Conference PMNP 2013, IOP Conference Series