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

Physics and Mathematics of Nonlinear Phenomena 2013

Manakov, S.V.^1 ; Santini, P.M.^2

Landau Institute for Theoretical Physics, Chernogolovka, Russia^1

Department of Physics, University of Roma la Sapienza and INFN, Sezione di Roma, Rome, Italy^2

关键词: Cauchy problems;    Darboux transformations;    Dispersionless;    Implicit solutions;    Long time behavior;    One parameter family;    Recursion operators;    Spectral transform;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/482/1/012029/pdf
DOI  :  10.1088/1742-6596/482/1/012029

来源: IOP

【 摘 要 】

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 Backlund-Darboux transformations for integrable dispersionless PDEs.